Discontinuity, Nonlinearity, and Complexity

Volume 7 Issue 4 December 2018

ISSN 2164-6376 (print) ISSN 2164-6414 (online)

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky University of Colorado, Boulder, and University of North Carolina, Chapel Hill, USA Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitri Volchenkov Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA & Sichuan University of Science and Engineering, Sichuan, Zigong 643000, China Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Department of Mathematics and Physics University of Wisconsin-Parkside 900 Wood Rd, Kenosha, WI 53144 Tel: 1(262) 595-2517 Email: [email protected]

J. A. Tenreiro Machado Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr. Antonio Bernardino de Almeida, 431, 4249-015 Porto, Portugal Fax: 351-22-8321159 Email: [email protected]

Dumitru Baleanu Department of Mathematics Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

Alexander N. Pisarchik Center for Biomedical Technology Technical University of Madrid Campus Montegancedo 28223 Pozuelo de Alarcon, Madrid, Spain E-mail: [email protected]

Josep J. Masdemont Department of Mathematics. Universitat Politecnica de Catalunya. Diagonal 647 (ETSEIB,UPC) Email: [email protected]

Marian Gidea Department of Mathematical Sciences Yeshiva University New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Edgardo Ugalde Instituto de Fisica Universidad Autonoma de San Luis Potosi Av. Manuel Nava 6, Zona Universitaria San Luis Potosi SLP, CP 78290, Mexico Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 Email: [email protected]

Elbert E.N. Macau Laboratory for Applied Mathematics and Computing, National Institute for Space Research, Av. dos Astronautas, 1758 C. Postal 515 12227-010 - Sao Jose dos Campos - SP, Brazil Email: [email protected], [email protected]

Michael A. Zaks Institut für Physik Humboldt Universität Berlin Newtonstr. 15, 12489 Berlin Email: [email protected]

Mokhtar Adda-Bedia Laboratoire de Physique Ecole Normale Supérieure de Lyon 46 Allée d’Italie, 69007 Lyon, France Email: [email protected]

Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202, USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on the inside back cover

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 7, Issue 4, December 2018

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

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Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 355-364

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

On I -lacunary Summability Methods of Order α in Intuitionistic Fuzzy n-normed Spaces E. Savas¸† Department of Mathematics, Istanbul Ticaret University, Sütlüce-Istanbul, Turkey Submission Info Communicated by D. Volchenkov Received 4 August 2017 Accepted 18 May 2018 Available online 1 January 2019

Abstract In this paper we introduce and study the notion I -statistical convergence of order α , and I -lacunary statistical convergence of order α with respect to the intuitionistic fuzzy n-normed space and also we investigate their relationship and some inclusion theorems are proved.

Keywords Ideal Filter I -statistical convergence I -lacunary statistical convergence Statistical convergence of order α

©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Zadeh [1] has introduced the concept of fuzzy sets and fuzzy set operations. Subsequently several authors have discussed various aspects of its theory and applications such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming etc. In recent years, there has been an increasing interest in various mathematical aspects of operations defined on fuzzy sets. Based on these, sequences of fuzzy numbers have been introduced by several authors and they have obtained many important properties. The class of all p-summable convergent sequences of fuzzy numbers is introduced by Nanda [2]. Later on statistical convergence of sequences of fuzzy numbers are introduced by Nuray and Savas(see, [3]). Recently bounded variation for fuzzy numbers is studied by Tripathy et al in ([4, 5]). As the set of all real numbers can be embedded in the set of all fuzzy numbers, many results in reals can be considered as a special case of those fuzzy numbers. However, since the set of all fuzzy numbers is partially ordered and does not carry a group structure, most of the facts known for the sequences of real numbers may not valid in fuzzy setting. Therefore, this theory is not a trivial extension of what has been known in real case. The theory of intuitionistic fuzzy sets was introduced by Atanassov [6]; it has been extensively used in decision making problems [7]. The concept of an intuitionistic fuzzy metric space was introduced by Park [8]. † Corresponding

author. Email address: [email protected] ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.12.001

356

E. Savas¸ / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 355–364

Furthermore, Saadati and Park [9] gave the notion of an intuitionistic fuzzy normed space. Some works related to the convergence of sequences in several normed linear spaces in a fuzzy setting can be found in [10–14]. The idea of convergence of a real sequence has been extended to statistical convergence by Fast [15] (see also [16]) as follows. Let K be a subset of N. Then the asymptotic density of K is denoted by δ (K) := limn→∞ 1n |{k ≤ n : k ∈ K}| , where the vertical bars denote the cardinality of the enclosed set. A number sequence x = (xk )k∈N is said to be statistically convergent to L if for every ε > 0, δ ({k ∈ N : |xk − L| ≥ ε }) = 0. If (xk )k∈N is statistically convergent to L we write st-lim xk = L. It is doubtless that the study of statistical convergence and its various generalizations has become an active research area since late 90’s of the last century. Statistical convergence turned out to be one of the most active areas of research in summability theory after the ˇ at [18]. work of Fridy [17] and Sal´ In [19], P. Kostyrko et al introduced the concept of I -convergence of sequences in a metric space and studied some properties of such convergence. Note that I -convergence is an interesting generalization of statistical convergence. The reader is referred to [20–29] for more details. In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [30]. A lacunary sequence is an increasing integer sequence θ = {kr }r∈N∪{0} such that k0 = 0 and hr = kr . kr − kr−1 → ∞, as r → ∞. Let Ir = (kr−1 , kr ] and qr = kr−1 A sequence x = (xk ) of real numbers is said to be lacunary statistically convergent to L (or Sθ -convergent to L) if, for any ε > 0, 1 lim |{k ∈ Ir : |xk − L| ≥ ε }| = 0, r→∞ hr where |A| denotes the cardinality of A ⊂ N. In [30], the relation between lacunary statistical convergence and statistical convergence was established, among other things. Recently, Mohiuddine and Aiyub [31] introduced the concept lacunary statistical convergence in random 2-normed space. In [32], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. Also lacunary statistically convergent double sequences in probabilistic normed space was studied by Mohiuddine and Savas¸ in [33]. In [12], generalized statistical convergence in intuitionistic fuzzy 2-normed space was studied by Savas. Mursaleen, et al. studied the ideal convergence of double sequences in intuitionistic fuzzy normed spaces (see, [34]). Moreover, some different results of lacunary statistical convergence in intuitionistic fuzzy n-normed linear is given in [14]. More results on this convergence can be found in [28, 35]. Recently in [22] we used ideals to introduce the concepts of I -statistical convergence and I -lacunary statistical convergence with respect to the intuitionistic fuzzy norm (µ , v) which naturally extend the notions of the above mentioned convergence. On the other hand in [36, 37] a different direction was given to the study of statistical convergence where the notion of statistical convergence of order α , 0 < α < 1 was introduced by replacing n by nα in the denominator in the definition of statistical convergence. In this paper, the notions of ideal statistical convergence of order α and ideal lacunary statistical convergence of order α , where 0 < α < 1 are introduced in an intuitionistic fuzzy n- normed linear space and some important results are obtained. Throughout the paper, N will denote the set of all natural numbers. First we need some basic definitions used in the paper. The following definitions and notions will be needed in the sequel. Definition 1. ([38]) A triangular norm (t-norm) is a continuous mapping ∗ : [0, 1] × [0, 1] → [0, 1] such that (S, ∗) is an abelian monoid with unit one and c ∗ d ≤ a ∗ b if c ≤ a and d ≤ b for all a, b, c, d ∈ [0, 1]. Definition 2. ([38]) A binary operation 3 : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-conorm if it satisfies the following conditions: (i) 3 is associate and commutative, (ii) 3 is continuous,


357

(iii) a30 = a for all a ∈ [0, 1], (iv) a3b ≤ c3d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. In this case (µ , v) is called an intuitionistic fuzzy norm. Then observe that (X , µ , v, ∗, 3) is an intuitionistic fuzzy normed space. Before proceeding further, we should recall some notations. In [39], Gähler introduced the following concept of n-normed space. Let n ∈ N and X be a real linear space of dimension d ≥ n ≥ 2. A real valued function ||., ..., .|| : X n → R satisfying the following four properties: (N1 ) ||x1 , x2, ..., xn || = 0 if and only if x1 , x2, ..., xn are linearly dependent vectors, (N2 ) ||x1 , x2, ..., xn || = ||x j1 , x j2 , ..., x jn || for every permutation ( j1 , j2 , ..., jn ) of (1, 2, ..., n), (N3 ) ||α x1 , x2, ..., xn || = |α |||x1 , x2, ..., xn || for all α ∈ R, (N4 ) ||x, x′ , x2 , ..., xn || ≤ ||x, x2 , ..., xn || + ||x′ , x2 , ..., xn || for x, x′ , x2 , ..., xn ∈ X is called an n-norm on X and the pair (X , ||., ..., .||) is called a linear n−normed space. The concept of a 2-normed space was developed by Gähler ([39, 40]) in the mid of 1960’s while that of an n−normed space can be found in Misiak [41]. Since then, many others have studied this concept and obtained various results; see for instance Gunawan [42]. We now have Definition 3. (see, [14]). An IFnNLS is the five-tuple (X , µ , v, ∗, ◦), where X is a linear space over a field F, ∗ is a continuous t-norm, ◦ is a continuous t-conorm, µ , v are fuzzy sets on X n × (0, ∞), µ denotes the degree of membership and v denotes the degree of non-membership of (x1 , x2 , ..., xn ,t) ∈ X n × (0, 1) satisfying the following conditions for every (x1 , x2 , ..., xn ) ∈ X n and s,t > 0 : (i) µ (x1 , x2 , ..., xn ,t) +v(x1 , x2 , ..., xn ,t) ≤ 1, (ii) µ (x1 , x2 , ..., xn ,t) > 0, (iii) µ (x1 , x2 , ..., xn ,t) = 1 if and only if x1 , x2 , ..., xn are linearly dependent, (iv) µ (x1 , x2 , ..., xn ,t) is invariant under any permutation of x1 , x2 , ..., xn , (v) µ (x1 , x2 , ..., cxn ,t) = µ (x1 , x2 , ..., xn , |c|t ) if c 6= 0, c ∈ F, (vi) µ (x1 , x2 , ..., xn , s) ∗µ (x1 , x2 , ..., x′n ,t) ≤ µ (x1 , x2 , ..., xn + x′n , s + t), (vii) µ (x1 , x2 , ..., xn ,t) : (0, ∞) → [0, 1] is continuous in t, (viii) lim µ (x1 , x2 , ..., xn ,t) = 1 and limµ (x1 , x2 , ..., xn ,t) = 0, t→∞

t→0

(ix) v(x1 , x2 , ..., xn ,t) < 1, (x) v(x1 , x2 , ..., xn ,t) = 0 if and only if x1 , x2 , ..., xn are linearly dependent, (xi) v(x1 , x2 , ..., xn ,t) is invariant under any permutation of x1 , x2 , ..., xn , (xii) v(x1 , x2 , ..., cxn ,t) = v(x1 , x2 , ..., xn , |c|t ) if c 6= 0, c ∈ F, (xiii) v(x1 , x2 , ..., xn , s) ◦ v(x1 , x2 , ..., x′n ,t) ≥ v(x1 , x2 , ..., xn + x′n , s + t), (xiv) v(x1 , x2 , ..., xn ,t) : (0, ∞) → [0, 1] is continuous in t, (xv) lim v(x1 , x2 , ..., xn ,t) = 0 and limv(x1 , x2 , ..., xn ,t) = 1. t→∞

t→0

358


Example 1. Let (X , ||.||) be an n-normed space. Also let a ∗ b = ab and a ◦ b = min{a + b, 1} for all a, b ∈ [0, 1], ||x1 ,x2 ,...,xn || µ (x1 , x2 , ..., xn ,t) = t+||x1 ,xt2 ,...,xn || and v(x1 , x2 , ..., xn ,t) = t+||x . Then (X , µ , v, ∗, ◦) is an IFnNS. 1 ,x2 ,...,xn || The following definition has given in [14]. Definition 4. Let (X , µ , v, ∗, ◦) be an IFnNS. We say that a sequence x = {xk } in X is convergent to L ∈ X with respect to the intuitionistic fuzzy n-norm (µ , v) if, for every ε > 0 , t > 0 and y1 , y2 , ..., yn−1 ∈ X , there exists k0 ∈ N such that µ (y1 , y2 , ..., yn−1 , xk − L,t) > 1− ε and v(y1 , y2 , ..., yn−1 , xk − L,t) < ε for all k ≥ k0 . It is denoted (µ ,v)

by (µ , v)n − lim x = L or xk → n L as k → ∞.

We also need the following definitions and notions in this paper. Definition 5. A family I ⊂ 2N is said to be an ideal of N if the following conditions hold: (a) A, B ∈ I implies A ∪ B ∈ I , (b) A ∈ I , B ⊂ A implies B ∈ I , Definition 6. A proper ideal I is said to be admissible if {n} ∈ I for each n ∈ N. Throughout I will stand for a proper admissible ideal in N × N. 2 I -Statistical and I -Lacunary statistical convergence of order α on IFnNS In this section we deal with the ideal statistical convergence of order α and ideal lacunary statistical convergence of order α on the intuitionistic fuzzy n-norm spaces. We are ready to begin. Definition 7. ([19]) Let I ⊂ 2N be a proper admissible ideal in N. The sequence {xn }n∈N of elements of R is said to be I -convergent to L ∈ R if, for each ε > 0, the set A (ε ) = {n ∈ N : |xn − L| ≥ ε } ∈ I . Definition 8. ([20]) A sequence x = (xk ) is said to be I -statistically convergent to L or S (I )-convergent to L if, for each ε > 0 and δ > 0, 1 {n ∈ N : |{k ≤ n : |xk − L| ≥ ε }| ≥ δ } ∈ I n or equivalently if for each ε > 0

δI (A (ε )) = I - lim δn (A (ε )) = 0,

where A (ε ) = {k ≤ n : |xk − L| ≥ ε } and δn (A (ε )) =

|A(ε )| n .

In this case we write xk → L (S (I )). The class of all I -statistically convergent sequences will be denoted simply by S (I ) . Let I = I f = {A ⊆ N : A is a finite subset}. Then I f is an admissible ideal in N and I -statistically convergent is the statistical convergence. Using the notation as given above, we proceed to our main definitions and results. Definition 9. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n− normed space. Then, a sequence x = (xk ) is said to be I -statistically convergent of order α to L ∈ X or S(I )α -convergent to L, where 0 < α ≤ 1, with respect to (µ , v)n if, for each ε > 0, t > 0 and δ > 0, and for y1 , y2 , ..., yn−1 ∈ X , such that 1 k ≤ n : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or n∈N: α ≥δ ∈I. v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε n (µ ,v) In this case we write xk → n L S(µ ,v)n (I )α . The class of all I -statistically convergent of order α sequences on an intuitionistic fuzzy n− normed space will be denoted by simply S(µ ,v)n (I )α .


359

Remark 1. For I = I f = {A ⊆ N : A is a finite subset}. S(µ ,v)n (I )α -convergence coincides with statistically convergence of order α with respect to (µ , v)n . For an arbitrary ideal I and for α = 1 it coincides with I -statistically convergence with respect to (µ , v)n . When I = I f in and α = 1 it becomes only statistically convergence with respect to (µ , v)n , (see, [43]). Definition 10. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n− normed space and θ be a lacunary sequence. A sequence x = (xk ) is said to be I -lacunary statistically convergent of order α to L ∈ X or Sθ (I )α -convergent to L, where 0 < α ≤ 1, with respect to (µ , v)n if, for any ε > 0, t > 0 and δ > 0, for y1 , y2 , ..., yn−1 ∈ X , such that 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or r∈N: α ≥δ ∈I. v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε hr ( µ ,v)n

( µ ,v)

In this case, we write xk → L(Sθ n (I )α ). The class of all I -lacunary statistically convergent sequences of ( µ ,v) order α on an intuitionistic fuzzy n− normed space will be denoted by Sθ n (I )α .

Remark 2. For α = 1 the definition coincides with I -lacunary statistical convergence with respect to (µ , v)n , ( see, [43]). For I = I f = {A ⊆ N : A is a finite subset}, I -lacunary statistically convergent sequences of order α on an intuitionistic fuzzy n− normed space coincides with lacunary statistically convergent sequences of order α on an intuitionistic fuzzy n− normed space, Further it must be noted in this context that lacunary statistical convergence of order α with respect to (µ , v)n has not been studied till now. We now turn our attention to the following theorem. Theorem 1. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n− normed space and 0 < α ≤ β ≤ 1. Then S(µ ,v)n (I )α ⊂ S(µ ,v)n (I )β . Proof. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n- normed space and 0 < α ≤ β ≤ 1. Then |{k ≤ n : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε | nβ |{k ≤ n : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε | ≤ nα and so for any δ > 0, and for t > 0 and y1 , y2 , ..., yn−1 ∈ X , |{k ≤ n : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε }| ≥ δ} nβ |{k ≤ n : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε }| ≥ δ }. ⊂{n ∈ N : nα {n ∈ N :

Hence if the set on the right hand side belongs to the ideal I then obviously the set on the left hand side also belongs to I . This shows that S(µ ,v)n (I )α ⊂ S(µ ,v)n (I )β . As with theorem 1, we can state the following Corollary. Corollary 2. If a sequence is I -statistically convergent of order α to L for some 0 < α ≤ 1 with respect to (µ , v)n then it is I -statistically convergent to L with respect to (µ , v)n i.e. S(µ ,v)n (I )α ⊂ S(µ ,v)n (I ). Similarly we can show that Theorem 3. Let 0 < α ≤ β ≤ 1. Then, ( µ ,v) ( µ ,v) (i) Sθ n (I )α ⊂ Sθ n (I )β . ( µ ,v)n

(ii) In particular Sθ

(µ ,v)n

(I )α ⊂ Sθ

(I ).

360


We define (µ ,v)

Definition 11. Let θ be a lacunary sequence. Then x = (xk ) is said to be Nθ n (I )α -convergent to L ∈ X with respect to (µ , v)n if, for any ε > 0, δ > 0, t > 0 and y1 , y2 , ..., yn−1 ∈ X , such that ) ( 1 µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ∈I. r∈N: α ∑ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε hr k∈I r

( µ ,v)

This convergence is denoted by xk → L(Nθ n (I )α ), and the class of such sequences of order α on an intu( µ ,v) itionistic fuzzy n− normed space will be denoted simply by Nθ n (I )α . Theorem 4. Let θ be a lacunary sequence. Then ( µ ,v) ( µ ,v) (a) xk arrowL(Nθ n (I )α )arrowxk arrowL(Sθ (I )α ), and ( µ ,v) ( µ ,v) (b) Nθ 2 (I )α is a proper subset of Sθ n (I )α . ( µ ,v)

Proof. (a) If ε > 0 and xk → L(Nθ n (I )α ), we can write for t > 0 and y1 , y2 , ..., yn−1 ∈ X , µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ∑ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε k∈Ir µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ≥ ∑ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε k∈I ,µ (y ,y ,...,y ,x −L,t)≤1−ε r

1

2

n−1

k

or v(y1 ,y2 ,...,yn−1 ,xk −L,t)≥ε

and so

k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ≥ε v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε

1 µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ∑ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε ε hαr k∈I r 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ≥ α . v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε hr

Then, for any δ > 0 and t > 0 and y1 , y2 , ..., yn−1 ∈ X , 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or r∈N: α ≥δ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε hr   1  hr ∑ µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − εδ or    ∈I. ⊆ r ∈ N : k∈I1r     hr ∑ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ εδ k∈Ir

This proves the result. ( µ ,v) (µ ,v) (b) In order √ to establish that the√ inclusion Nθ n (I )α ⊆ Sθ n (I )α is proper, let θ be given, and define xk to be 1, 2, ..., hr for the first hr integers in Ir and xk = 0 otherwise, for all r = 1, 2, .... Then, for any ε > 0 and t > 0 and y1 , y2 , ..., yn−1 ∈ X , √ hr 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − 0,t) ≤ 1 − ε or , ≤ α v (y1 , y2 , ..., yn−1 , xk − 0,t) ≥ ε hr hr and for any δ > 0, for t > 0 and y1 , y2 , ..., yn−1 ∈ X , we get p [ hαr ] 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − 0,t) ≤ 1 − ε or r∈N: α ≥ δ ⊆ {r ∈ N : hα ≥ δ }. v (y1 , y2 , ..., yn−1 , xk − 0,t) ≥ ε hr r


361 ( µ ,v)n

Since the set on the right-hand side is a finite set and so belongs to I , it follows that xk → 0(Sθ On the other hand, p p hαr hαr + 1 1 1 µ (y1 , y2 , ..., yn−1 , xk − 0,t) ≤ 1 − ε or = . . ∑ v (y1 , y2 , ..., yn−1 , xk − 0,t) ≥ ε hαr k∈I hr 2 r

(I )α ).

Then    r∈N:

 1   ∑ µ (y1, y2 , ..., yn−1 , xk − 0,t) ≤ 1 − 4 or  k∈Ir 1 1     hαr ∑ v (y1 , y2 , ..., yn−1 , xk − 0,t) ≥ 4   k∈Ir ) ( p √ hαr hr + 1 1 ≥ = {m, m + 1, m + 2, ...} = r∈N: α hr 2 1 hαr

( µ ,v)n

for some m ∈ N which belongs to F(I ), since I is admissible. So xk 9 0(Nθ ( µ ,v)n

Remark 3. It is known that (ii) x ∈ l∞ ( µ ,v)n

(iii) Sθ

(µ ,v)n

(I ) ∩ l∞

(µ ,v)n

= Nθ

( µ ,v)n

and xk → L(Sθ (µ ,v)n

(I ) ∩ l∞

(I )α ).

(µ ,v)n

(I )) ⇒ xk → L(Nθ

(I ),

.

An Open Problem. We do not yet know whether these results remain true for 0 < α < 1 is not clear. We, therefore, choose to leave it is an open problem. We will now investigate the relationship between I -statistical and I -lacunary statistical convergence of order α . We now prove the inclusion theorem Theorem 5. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n− normed space. For any lacunary sequence θ , I statistical convergence of order α with respect to (µ , v)n implies I -lacunary statistical convergence of order α with respect to (µ , v)n if lim infqαr > 1. r

Proof. Suppose first that lim infqαr > 1. Then there exists σ > 0 such that qαr ≥ 1 + σ for sufficiently large r r which implies that hαr σ ≥ . α kr 1+σ Since xk → L S(µ ,v)n (I )α , for every ε > 0, t > 0, and for sufficiently large r, we have 1 k ≤ kr : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε krα 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or ≥ α v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε kr 1 kr : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or σ . ≥ . v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε 1 + σ hαr Then for any δ > 0, and for y1 , y2 , ..., yn−1 ∈ X , we get 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or r∈N: α ≥δ v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε hr 1 k ≤ kr : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or δα ⊆ r∈N: α ≥ (1 + α ) ∈ I . v (y1 , y2 , ..., yn−1 , xk − L,t) ≥ ε kr

362


Remark 4. The converse of this result is true for α = 1 (see Theorem 3 [14]). However for α < 1 it is not clear and we leave it as an open problem. We now present the following theorem which specify the sufficient conditions for the converse relation of Theorem 7 to be true. For the next the result we assume that the lacunary sequence θ satisfies the condition that S for any set C ∈ F(I), {n : kr−1 < n < kr , r ∈ C} ∈ F(I).

Theorem 6. Let (X , µ , v, ∗, ♦) be an intuitionistic fuzzy n− normed space. For a lacunary sequence θ satisfying the above condition, I -lacunary statistical convergence of order α with respect to (µ , v)n implies I -statistical hαi+1 convergence of order α , 0 < α < 1, with respect to (µ , v)n if sup∑r−1 i=0 (kr−1 )α = B(say) < ∞. r

Proof. xk → L S(µ ,v)n (I )α and for ε , δ , δ1 > 0, and for t > 0 and y1 , y2 , ..., yn−1 ∈ X , define the sets 1 k ∈ Ir : µ (y1 , y2 , ..., yn−1 , xk − L,t) ≤ 1 − ε or C= r∈N: α m . In the second case, our notion of modularity is associated with the idea of genetic redundancy, whereby the fitness of a genotype is similar in the presence of different copy numbers of a given gene. The extreme limit of this is when the landscape is associated with an “OR” function, so that the fitness of a type is the same whether there is one or multiple copies of a gene. The intuition of a module in this context is that, in the presence of redundancy with multiple copy number, one, or maybe more, genes can be removed or mutated without affecting the fitness of the type. Thus, a gene acts as a module as it can be changed independently without affecting the fitness of the type. As we will see, this corresponds to a system with a maximal degree of negative epistasis.

5 Recombination in an exact two-locus model A full, exact analysis for loci with arbitrary landscape and population is prohibitively difficult so we will consider the case of two loci with = 2. Note that by two loci here we do not necessarily imply that they represent “genes”. They may represent any two structural units, such as exons, introns or other motifs, or nucleotides themselves, that can be separated or recombined by crossover and which can be characterized, as an approximation, by a fitness landscape that is independent of the rest of the genome. In this case fx1 x2 = F (0) +

2

(1)

∑ Fi

i1 =1

1

(2)

xi1 + F12 x1 x2 .

(2)

(7) (2)

(1)

(1)

For an additive (modular) landscape F12 = 0. For a multiplicative landscape F(0) F12 = F1 F2 . For a redun(2) (1) (1) dant (modular) landscape, Fi j = −Fi = −Fj which, as mentioned, can be understood in terms of a Boolean (1)

(1)

“OR”, fitness, being the same if either one or both alleles are optimal. For a NIAH landscape F1 = F2 = 0 which, in contrast to the redundant landscape, corresponds to a Boolean “AND”, as fitness is only different if both alleles are optimal. For two loci all genotypes can be characterized by a multi-index I = i j, with i, j ∈ {0, 1, . . . , C }, where C + 1 is the cardinality of the alphabet that labels the loci, or alleles in the case of genes. For = 2, there is only one non-trivial crossover maske m = 01, and its conjugate, that lead to the BBs i∗ and ∗ j. The sum over masks in the general expression for the SWLD coefficient is thus reduced to only one term: Δi j = Pij − Pi∗ P∗ j = Pij − (Pii + Pii¯)(Pj j + Pj¯ j ),

(8)

with Δi j = Δi¯ j¯ = −Δii¯ = −Δ j¯ j where i¯ is the bit complement of i, and thus the evolution equations in the twoallele, two-locus problem are: (9) Pi j (t + 1) = Pij (t) − pc Δi j The general parametrized two-locus, two allele landscape is f = a + b1 x1 + b2 x2 + cx1 x2 , where c is the measure of the additive epistasis between the two loci. Taking the genotype I = 00 as the wild type, the genotypes I = 01 and 10 as single mutants and I = 11 as a double mutant, which is the optimal genotype, there are just three main landscape categories for the two-bit, two-locus model: 1. The wild type and the double mutant are the anti-optimum and optimum respectively. 2. One of the single mutants (10 or 01) is the antioptimum. 3. The two lowest fitness phenotypes are the single mutants. e

The masks m = 00 and 11 correspond to cloning, where both offspring loci come from a single parent.

Christopher R. Stephens / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 365–381

373

In the first two landscape types, a generic population will always eventually evolve towards the global optimum. In the third type, the population may converge to the optimum or the suboptimal wild type 00 depending on the initial population and the recombination probability.f From Equations (4) and (5) we have fi j DPi j (t) = ( ¯ − 1) − pcΔi j (t), f (t)

(10)

f 2 (t) f 2 (t) − pc ∑ fi j Δi j (t). D f¯r+s (t) = ¯ s+r − ¯s fs+r (t) fs (t) ij

(11)

1 Δ11 (t) = ¯2 (a(a + b1 + b2 + c)P11(t)P00 (t) − (a + b1 )(a + b2 )P01 (t)P10 (t)). f (t)

(12)

For the optimal genotype

To develop some intuition for how the characteristics of the landscape affect our metrics we set Pi j (0) = 1/4, i.e., a homogeneous population with no initial bias for one genotype versus another. As the parameter a just sets the scale for the landscape we can without loss of generality for fitness proportional selection set a = 1. We will also set b1 = b2 = b so that both single mutants have the same fitness. In this case, Δ11 (0) =

(c − b2 ) . (1 + b + c/4)2

(13)

For a multiplicative landscape c = b2 and Δ11 = 0, as is well known. For an additive landscape c = 0 and therefore Δ11 (t) = −b2 /(1 + b) < 0. In this case recombination leads to a higher frequency of the optimal genotype in the next generation than selection alone. For a deceptive landscape, b < 0, but c > −2b and so Δ11 (t) > 0 and recombination in this region of the parameter space leads to a lower frequency of the optimal genotype in the next generation. In terms of BBs, for deceptive landscapes, the marginal fitnesses are such that f1∗ < f0∗ and f∗1 < f∗0 , and so the reason why recombination is unfavorable is that the necessary mutant alleles for constructing the optimal genotype are deleterious relative to the corresponding alleles of the genotype 00. For additive epistasis, such that c > b2 , we have Δ11 (t) > 0 and recombination once again leads to a lower frequency of the optimal genotype in the next generation than selection alone. Generally, if we take c − b2 < 0 as signifying negative multiplicative epistasis then we see that in such landscapes recombination has a positive effect in terms of our Δ metric and on the contrary for positive multiplicative epistasis. Note that the additive limit c = 0 corresponds to negative multiplicative epistasis. Interestingly, equation (13) shows that the greatest benefit from recombination, i.e., the minimum value of Δ11 , is associated with landscapes with negative additive epistasis, i.e., c < 0. Maximum negative epistasis is given by the minimum value of c, c = −b. In this case Δ11 (t) = −b(1 + b)/(1 + 3b/4)2 . Why would this maximum negative epistasis be associated with the utility of recombination, at least in terms of metric (4)? Examining equation (12) we see that the first term, proportional to P11 (t)P00 (t), corresponds to elimination of the optimal genotype 11 by recombining it with the suboptimal genotype 00, whereas the term proportional to P01 (t)P10 (t) corresponds to construction of 11 via recombination of the single mutants 10 and 01. It is the competition between these two effects that measures the benefits of recombination in terms of (4). Additive landscapes with c = 0 reduce the impact of destruction without compromising the positive effect of reconstruction. Negative epistasis, on the other hand, does not affect the construction of the optimal genotype by recombining the single mutants, but it does minimize the effect of destruction of the optimal genotype. The f

The latter two landscape categories are known as deceptive landscapes of Type I and Type II respectively in the Genetic Algorithm literature [33]. It has been proved [34] that Type I systems always converge to the global optimum whereas Type II systems converge to the optimum or double mutant depending on the population and recombination probability.

374


maximal effect is when c = −b and corresponds to a Boolean “OR” landscape, where f01 = f10 = f11 > f00 . This is the situation where there is genetic redundancy, as the fitness of the optimal phenotype requires the presence of only one optimal allele not both. At this naive level we also see that the benefit of recombination is not restricted to small negative multiplicative epistasis but, rather, the larger the additive negative epistasis the larger the benefit conferred by it.

6 Exact numerical results in a two locus model For the two-locus model there are 7 parameters to be considered. In such a high dimensional space, visualization of the resulting graphs requires separation into several distinct cases. We set pc = 0.5 in the following as pc just affects the magnitude of the effects of recombination but not whether it is beneficial or not, as this is principally controlled by the sign of Δ. We fix b1 = b2 = b and set a = 1 and display the results as functions of b and c. The valid region, all fitnesses positive, with the genotype 11 as optimum, is given by b > −1, c > −2b and c > −b. The deceptive region is given by b < 0. For ease of interpretation we also show lines associated with the multiplicative limit b2 = c (yellow) and the additive limit c = 0 (green). Note that both the additive and multiplicative limits require b > 0. The NIAH landscape is given by b = 0, c > 0 and lies on the border that separates non-deceptive and deceptive landscapes. The point b = 0, c = 0 corresponds to a flat fitness landscape where there is no selection pressure. Two kinds of graphs are provided, one that displays the value of the SWLD coefficient in different generations, and another that displays Df¯r+s (Equation (5)), defined as the change in average fitness between generation t and generation t + 1 in a population evolving with both selection and recombination minus the change in average fitness of the same population but evolving with selection only. In the graphs we show four representative time slices - t = 1, 2, 6, and 10 generations after the initial one. The plane Δ11 = 0 that separates the recombination advantageous/disadvantageous regimes is displayed (turquoise in the online version). For a given generation, those values of b and c where Δ11 < 1 are shaded in red (below the Δ11 = 0 plane), while those where Δ11 > 1 correspond to a darker shading (above the Δ11 = 0 plane). We will show explicitly the case of a uniform initial population, where all genotypes have the same initial frequency, 0.25, as the resultant behavior is qualitatively similar to that found for other populations [35]. Figure 3 shows the result for Δ. An important point here is that, given the ample presence of the optimal genotype in the initial population, there is no search regime and so the dynamics begins and remains in the modular regime. With no population bias we can see the role played by the multiplicative limit with, at t = 1, Δ being positive for landscapes with positive multiplicative epistasis and, particularly, deceptive landscapes. It is negative for weakly positively epistatic, additive and negatively epistatic landscapes. As evolution progresses we can see that the relative advantage diminishes such that at t = 10 the advantage of recombination is only noticeable for larger negative epistasis. In terms of average population fitness, in Figure 4 we see an analogous story: at t = 1 average population fitness is increased only for landscapes with negative multiplicative epistasis, up to the additive limit, but is, in fact, negative for negative additive epistasis. However, as evolution progresses, once again, we see the dominant role played by modular landscapes - i.e., weakly positively epistatic, additive and negatively epistatic landscapes.

7 Comparing recombination distributions as watchmakers In the previous section we restricted to a two-locus model so that all possible landscapes could be examined to see exactly for which landscapes there was a benefit from recombination. However, beyond = 2, to consider all landscapes is not feasible. We saw that there was a benefit for quasi-modular landscapes, where epistasis between the loci was low, and for negatively epistatic (redundant) landscapes. The fact that only two loci were considered, however, made the modularity of the landscape in terms of the structure of the genomes somewhat trivial, as


Δ at t=1

4

Δ at t=2

4

c

2

c

2

375

0

−2 −1

0

0

b

1

−2 −1

2

Δ at t=6

4

4

b

1

2

1

2

Δ at t=10

c

2

c

2

0

0

−2 −1

0

0

b

1

−2 −1

2

0

b

Fig. 1 Value of Δ at different generations for two-locus two-allele system as a function of fitness landscape, characterized by b and c. The initial population is Pi j (0) = 0.25. The Δ = 0 plane has been marked to distinguish between conditions in which recombination is favorable (Δ < 0) or not. The curve on the plane is c = b 2 , the condition for a multiplicative landscape.

there are only two indivisible modules and only one crossover point associated with them. To understand further the relationship between modularity and recombination we will now consider a set of modular landscapes with a non-trivial landscape within each module. Explicitly, we will consider landscapes of the form f (I) =

∑ cξ δξ (I)

ξ ∈S

where S is the set of landscape blocks that enter in the fitness function, cξ is the order of the schema and δξ (I) = 1 if the schema ξ belongs to the string I and is zero otherwise. By landscape block here we mean a set of, contiguous in the present case, loci that contribute to the fitness function independently of the other loci. In other words, there is epistasis within the block but not between blocks. Within a given block, the fitness function will be of NIAH type, and so the full landscape is a linear sum of NIAH landscapes. Such modular landscapes are much like the watch problem, with each module corresponding to a watch part. By varying the number of modules we can vary the number of watch parts. If there are loci and n landscape blocks, where the size of a block is /n, then when n = the landscape is completely additive with each locus being its own landscape block, while for n = 1 the full landscape is NIAH. In the latter the watch can only be built by putting all the parts together. We consider binary alleles where the optimal sequence is all ones. We have mentioned that building block hierarchies are associated with a preferred set of search algorithms. With that in mind we will use two metrics to measure performance in these landscapes. The first one is Run Length Distributions (RLDs) which are curves that show the characteristic behaviour of a given search algorithm on a given fitness landscape. RLDs are obtained by performing many runs (here, we use 100) of an algorithm, evaluating the “time/effort” (we use here the number of fitness evaluations), associated with a given run, needed

376


Df¯r+s at t=1

Df¯r+s at t=2

2

2 c

4

c

4

0

−2 −1

0

0

b

1

−2 −1

2

Df¯r+s at t=6

0

b

1

2

Df¯r+s at t=10

2

2 c

4

c

4

0

−2 −1

0

0

b

1

2

−2 −1

0

b

1

2

Fig. 2 Value of D f¯r+s at different generations for the two-locus two-allele system as a function of fitness landscape, characterized by b and c. The initial population is Pi j (0) = 0.25. The D f¯r+s = 0 plane has been marked to distinguish between conditions in which recombination is favorable (D f¯r+s > 0) or not.

to find the optimum, sorting them from minimum to maximum time/effort and graphing the resultant curve. The curves are noisy when the number of runs is small but, as this number grows, noise is reduced, and the curves are smoother. RDLs provide more information than simple summary statistics, such as min, max and average effort, since one can see at a glance the proportion of runs that are expected to find the optimum below any fixed level of effort, as well as phenomena that affect only a fraction of the runs. Our second metric is a statistical measure comparing average run times between two different algorithms. Explicitly, τ B − τ A ε (A, B) = 10( 2 ) (14) (σB + σA2 )1/2 where in the numerator we have the difference in average run times over the 100 runs. If we assume a normal distribution then, |ε | > 1.96 corresponds to a 95% confidence interval, which we will take to mean that it is statistically significant that algorithm A is leading to lower/higher average run times than algorithm B. As a statistical test ε is just the t-test for two distributions with unequal variances. In terms of the different algorithms we consider, first we will take a “meta”-evolution, where instead of considering a dynamics with recombination and mutation for a single, “typical” fixed value of the mutation rate and/or a particular recombination distribution, rather, different mutation rates and recombination distributions will be used. We will also consider another algorithm, the Rank-GA [36, 37], where, instead of a uniform mutation rate being applied to all the population, different rates are applied to different members of the population according to their fitness rank. Here, the fittest population member at a given generation will be given zero rate, while the least fit will be assigned the highest mutation rate considered. Considering both a meta-evolution and a Rank GA will allow us to better understand the interaction between landscape blocks and Building Blocks


377

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and the advantages and disadvantages of recombination in search. The population size is chosen to be small = ( + 1), given that we know what happens in the infinite population limit. In Figures 3 and 4 we see the results for a 12-locus problem, where the landscape is chosen to be modular, consisting of three landscape blocks, where each block has a NIAH landscape, with the needle having twice the fitness of the hay. We consider populations evolving with recombination and, now, mutation aswell, with mutation at different rates, and with a crossover rate of 0.9. However, distinct to the two locus case, we will consider different numbers of crossover points uniformly spread through the genome, where the end of the gene sequence was also considered as a possible crossover point. Thus, for two crossover points the first crossover is at the beginning of the string and therefore does not cut a landscape block, while the second one is at the 5th cutting point and does cut a landscape block. Similarly, for 3 cutting points, they are distributed at points 0, 4 and 8, so all of them respect the landscape blocks. For 4 cutting points they are at 0, 3, 6 and 9 so one respects the landscape blocks and three do not. For 6 cutting points, the corresponding points are 0, 2, 4, 6, 8 and 10, of which three respect the blocks and three do not. Finally, for 12 cutting points 3 preserve the landscape blocks and 9 cut them. In Figure 3 we see the RLDs for different values of the parameters. So, what do these results tell us? Who’s the clever watchmaker? It’s the Rank GA. Why? Well, first, let’s think about what strategy corresponds to the foolish watchmaker. This would be random search, with mutation rate 0.5. We can see that there are many other strategies corresponding to a dynamics, associated with different but fixed mutation rates and crossover points, that are better than the foolish watchmaker; but each has it’s defects. For instance, we see that search is improved with a larger number of crossover points but, in this case, the optimal mutation rate is somewhat lower, as the exploratory capacity of crossover can compensate some of the search of mutation, as can be seen in the improving performance of the zero mutation rate algorithm. However, with the simple algorithm “you can’t have your cake and eat it”, which is, after all, the fundamental problem of exploration versus exploitation. In other words, mutation and recombination can be balanced to give a good exploration in order to find landscape blocks in the first place but then, just like the foolish watchmaker, don’t exploit them and allow them to be disrupted. A recombination distribution that cuts only at the boundaries of landscape blocks would be optimal if we had a good supply of blocks in the population, but would be very suboptimal in getting

378


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blocks in the first place. The Rank GA, in contrast, with its elitism for the fittest genome and low mutation rate for those near to it in fitness rank, preserves landscape blocks once they are produced, while at the same time allowing for a more ample search for the unfit strings. In this sense, it acts like the clever watchmaker by trying to put to one side useful subparts. However, what we can see is that the performance of the Rank GA deteriorates as a function of the number of crossover points. This is associated with the second characteristic of the clever watchmaker: It is no good recognizing and preserving blocks and then failing to recombine them, i.e., by recombining Building Blocks that are not also landscape blocks. In this sense the Building Blocks associated with recombination are not respecting the landscape blocks. When recombination is restricted to the landscape block boundaries, however, we now see the full strategy of the clever watchmaker: blocks are first identified and preserved and then only these blocks are recombined together. In Figure 4 we use ε to confirm this. The Rank GA clearly outperforms the Simple GA in a statistically significant way (ε > 2) for any mutation rate of the Simple GA and for any number of crossover points. However, we see that the biggest benefit is when crossover is restricted to block boundaries. Note that distinct to the two locus case we can here distinguish two types of building block - Building Blocks that are schemata defined by the recombination distribution and landscape blocks which are the modules of the fitness landscape. The relationship between the two is such that a requirement for the efficient use of Building Blocks is the existence of modules in the fitness landscape - landscape blocks - and a recombination distribution whose associated Building Blocks respect these blocks. In other words, recombination should disrupt these blocks as little as possible. However, blocks still have to be found in the first place. This antagonism between needing ample exploration in order to find blocks in the first place and then restricted exploration afterwards in order to exploit them was shown to be at the root of why the performance of a standard evolutionary algorithm in modular landscapes was not as efficient as it was thought it should be. It is very difficult, if not impossible, for the simple algorithm with fixed mutation and recombination rates to adjust its exploration/exploitation balance


379

in this way. However, for a Rank GA that uses a heterogeneous mutation rate we saw that it was possible to preserve blocks while simultaneously searching for others. In this case the optimal recombination distribution is one that does not disrupt any already constructed landscape block, for instance, by cutting only at the boundaries of landscape blocks. These insights lead to several interesting hypotheses: i) that modularity and recombination have coevolved as recombination works best in such landscapes; ii) that the recombination distribution has evolved so as to respect module boundaries, i.e., recombination hotspots there; iii) that recombination and modularity lead to the formation of complex hierarchies of effective degrees of freedom.

8 Conclusions As discussed in the introduction, genetic recombination and sexual reproduction remain a puzzle as far as having a full, intuitive understanding of why it is so prevalent, with no generally accepted explanation of its benefits. Previous work [29, 30, 38, 39], both analytical and numerical, has hinted at the fact that recombination seems to be especially useful in the context of quasi-additive landscapes, while other work has shown a role for weak, negative multiplicative epistasis. However, these analyses did not cover the full parameter space of the considered models, and so there is always doubt that the landscapes or initial populations considered were not representative and therefore any identified benefits of recombination were not “universal” but, rather, tied to the specific scenario considered. To counter these arguments, in [35] we considered a simple two-locus model where all possible fitness landscapes and populations could be considered in a space of three landscape parameters. The result is consistent with the previous results of [29], where it was shown that there are two important, but distinct, regimes in which recombination is beneficial in terms of both the metrics that we have used to characterize its benefits. The first of these is the search regime, which is associated with conditions where the fittest genotype is either not present or only at low frequency. This regime is relatively well known, especially in the Genetic Algorithms literature, where search is a fundamental goal. In this regime the benefit from recombination is relatively independent of the fitness landscape. However, exactly how beneficial it is does depend on both the landscape and the actual population. The second regime we termed the modular regime and is associated with weakly additively epistatic landscapes, i.e., quasi-additive landscapes. However, the fact that we have here analyzed the set of possible landscapes and populations, allows us to go beyond this restricted analysis and observe and characterize several important universal properties of recombination. More importantly, within the confines of the assumptions of the initial model itself, we have performed an exact numerical integration of the dynamical equations, without any approximations or further assumptions, that allows us to see how any benefits of recombination change as a function of time. The results shown in section 6 illustrated the utility of recombination in the modular regime, showing that there is a clear association between the sign of the epistasis in the landscape and the sign of Δ. Production of the optimal genotype (11) is more favorable in the presence of negative additive epistasis (c < 0) than for positive additive epistasis (c > 0) for beneficial mutations. It is also disfavored when single mutants (01 and 10) are less fit (b < 0) than the suboptimal genotype 00, i.e., in the case of deceptive BBs. What is more, by following the dynamics across multiple generations for different initial populations, we saw that recombinative evolution itself is directed towards favoring landscapes that are more and more modular, more and more negatively epistatic. This is a universal feature that is independent of the initial population. In terms of the increase in average population fitness relative to selection only dynamics, we see a profoundly interesting dynamic, as can be seen in Figure 2, where the initial population is homogeneous. There, we see that recombination is disfavored initially (t = 1) for any positively multiplicatively epistatic landscape - including deceptive landscapes - and for any additively negatively epistatic landscape. However, very quickly the universal tendency towards favoring quasi-additive and negatively additively epistatic landscapes emerges. Thus, it is only in the modular regime that both our metrics show a benefit for recombination. We believe that these results link the existence and utility of recombination to two other very important

380


concepts in modern evolutionary thought - the ubiquity of modularity and, relatedly, the ubiquity of genetic redundancy, and thereby offer a quite universal explanation of why recombination is so widespread. In turn we have argued that modularity is fundamental as with the paradigm of the blind watchmakers it allows for the construction of complex objects through a hierarchy of building blocks at different scales. In other words, our conclusion is that recombination is so widespread because it leads to important evolutionary benefits only for systems that are modular and/or redundant and it is precisely such landscapes that seem to be the norm.

Acknowledgements This work was partially supported by DGAPA PAPIIT grant IN113414 and by a special CONACyT grant to the Centro de Ciencias de la Complejidad.

References [1] Maynard Smith, J. (1971), What use is sex? Journal of Theoretical Biology, 30(2), 319-335. [2] Eshel, I. and Feldman, M. (1970), On the evolutionary effect of recombination, Theoretical Population Biology, 1(1), 88-100. [3] Fisher, R. (1930), The genetical theory of natural selection, Clarendon Press. [4] Kondrashov, A. (1988), Deleterious mutations and the evolution of sexual reproduction, Nature, 336(6198), 435-440. [5] Muller, H. (1932), Some genetic aspects of sex, The American Naturalist, 66(703), 118-138. [6] Felsenstein, J. (1974), The evolutionary advantage of recombination, Genetics, 78(2), 737. [7] Barton, N. and Charlesworth, B. (1998), Why sex and recombination? Science, 281(5385), 1986. [8] Watson, R., Weinreich, D., and Wakeley, J. (2011), Genome structure and the benefit of sex, Wiley Online Library. [9] Otto, S. and Lenormand, T. (2002), Resolving the paradox of sex and recombination, Nature Reviews Genetics, 3(4), 252-261. [10] Feldman, M. (1972), Selection for linkage modification: I. Random mating populations. Theoretical Population Biology, 3(3), 324-346. [11] Otto, S. and Feldman, M. (1997), Deleterious mutations, variable epistatic interactions, and the evolution of recombination, Theoretical Population Biology, 51(2), 134-147. [12] Barton, N. (1995), A general model for the evolution of recombination, Genetical Research, 65(2), 123-144. [13] Liberman, U. and Feldman, M. (2008) On the evolution of epistasis iii: the haploid case with mutation, Theoretical Population Biology, 73(2), 307-316. [14] Zhivotovsky, L., Feldman, M., and Christiansen, F. (1994), Evolution of recombination among multiple selected loci: A generalized reduction principle. Proceedings of the National Academy of Sciences, 91(3), 1079. [15] Charlesworth, B. (1990), Mutation-selection balance and the evolutionary advantage of sex and recombination, Genet. Res., 55(3), 199-221. [16] Keightley, P. and Otto, S. (2006), Interference among deleterious mutations favours sex and recombination in finite populations. Nature, 443(7107), 89-92. [17] Pepper, J. (2000), The evolution of modularity in genome architecture, Proceedings of the Artificial Life, 7, 9-12. [18] Christiansen, F., Otto, S., Bergman, A., and Feldman, M. (1998), Waiting with and without recombination: the time to production of a double mutant, Theoretical Population Biology, 53(3), 199-215. [19] Rice, W., et al. (2002), Experimental tests of the adaptive significance of sexual recombination, Nature Reviews Genetics, 3(4), 241-251. [20] Stephens, C.R. and Mora-Vargas, J. (2000), Effective fitness as an alternative paradigm for evolutionary computation i: General formalism, Genetic Programming and Evolvable Machines, 1(4), 363-378. [21] Stadler, P. and Stephens, C. (2003), Landscapes and effective fitness, Comments on Theoretical Biology, 8(4-5), 389431. [22] Simon, H. (1996), The Sciences of the Artificial, MIT Press. [23] Holland, J. (1992), Adaptation in Natural and Artificial Systems. MIT Press. [24] Schlosser, G. and Wagner, G. (2004), Modularity in Development and Evolution, University Of Chicago Press. [25] Eigen, M. (1971), Selforganization of matter and the evolution of biological macromolecules, Die Naturwissenschaften, 10, 465-523. [26] Stephens, C. and Poli, R. (2007), Coarse-grained dynamics for generalized recombination, IEEE Transactions on Evolutionary Computation, 11(4), 541-557. [27] Stephens, C.R. and Waelbroeck, H. (1999), Schemata evolution and building blocks, Evolutionary Computation, 7,


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109-124. [28] Stephens, C.R. (2002), The renormalization group and the dynamics of genetic systems, Acta Phys. Slov., 52, 515-524. [29] Stephens, C., Arenas, E., Cervantes, J., Peralta, B., Ricalde, E., and Segura, C. (2006), When are building blocks useful? In Artificial Intelligence, 2006. MICAI’06. Fifth Mexican International Conference on, IEEE, 217-228. [30] Stephens, C. and Cervantes, J. (2007), Just what are building blocks? Foundations of Genetic Algorithms, 15-34. [31] Chryssomalakos, C. and Stephens, C.R. (2004), Covariant genetic dynamics, Evolutionary Computation, 15(3), 291320. [32] Weinberger, E.D. (1991), Fourier and Taylor series on fitness landscapes, Biological Cybernetics, 65, 321-330. [33] Goldberg, D.E. (1989), Genetic algorithms and Walsh functions: Part I. A gentle introduction. Complex Systems, 3, 123-152. [34] Takahashi, Y. (1998), Convergence of simple genetic algorithms for the two-bit problem, Bio Systems, 46(3), 235. [35] Del R´ıo, M.B., Stephens, C.R., and Rosenblueth, D.A. (2015), Fitness landscape epistasis and recombination, Advances in Complex Systems, 18(07n08):1550026. [36] Cervantes, J. and Stephens, C. (2009), Limitations of existing mutation rate heuristics and how a rank ga overcomes them, Evolutionary Computation, IEEE Transactions on, 13(2), 369-397. [37] Cervantes, J. and Stephens, C.R. (2008), Rank based variation operators for genetic algorithms, ACM, 905-912. [38] Livnat, A., Papadimitriou, C., Dushoff, J., and Feldman, M.W. (2008), A mixability theory for the role of sex in evolution, Proceedings of the National Academy of Sciences, 105(50), 19803-19808. [39] Livnat, A., Papadimitriou, C., Pippenger, N., and Feldman, M.W. (2010), Sex, mixability, and modularity, Proceedings of the National Academy of Sciences, 107(4), 1452-1457.



Complex Inference Networks: A New Tool for Spatial Modelling Christopher R. Stephens1,2†, Raúl Sierra Alcocer3 , Constantino González Salazar1,4† 1

Centro de Ciencias de la Complejidad and Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior, A. Postal 70-543, México D.F. 04510 3 CONABIO, Liga Perif´ erico - Insurgentes Sur 4903, Parques del Pedregal, Ciudad de México. C.P. 14010 4 Departamento de Ciencias Ambientales, CBS Universidad Aut´ onoma Metropolitana, Unidad Lerma, Estado de México 52006, México

2

Submission Info Communicated by D. Volchenkov Received 25 June 2017 Accepted 11 May 2018 Available online 1 January 2019 Keywords Networks Inference Complexity Building block

Abstract All systems - physical, biological, ecological and social - are composed of hierarchies of building blocks - atoms, molecules, cells, tissues, individuals, species etc. - with corresponding interactions, wherein the presence of interactions - attractive and repulsive - affects the relative spatio-temporal distribution of the building blocks. In physical systems, in particular, the structure of the building blocks and the nature of their interactions has been deduced via systematic observations of their positions in space and time. Unfortunately, Complex Adaptive Systems are highly multi-factorial, so that, unlike many physical systems, it is impossible to systematically observe and characterize each and every interaction that exists in such systems. In this paper we discuss a general framework - Complex Inference Networks - wherein interactions, particularly in Complex Adaptive Systems, may be studied and characterized using position data about their building blocks. We compare and contrast physical versus Complex Adaptive Systems and give as an explicit example the identification of disease hosts in the ecology of emerging and neglected diseases, where it has been possible to discover previously unknown ecological interactions from species co-occurrence data. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Possibly the most important characteristic of all systems - physical, chemical, biological, social etc. - is that they are divisible into parts - “building blocks” - and that these building blocks have interactions. Moreover, these building blocks are arranged hierarchically in scale, wherein building blocks at one scale are composed of constituent building blocks associated with a smaller/greater distance/energy scale, and which, in their turn, can aggregate to form building blocks at a greater/smaller distance/energy scale. Furthermore, the interactions † Corresponding

author. Email address: [email protected](Christopher R. Stephens), [email protected](Constantino González Salazar)

ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.12.003

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between building blocks associated with a given scale are stronger/weaker than those that are associated to the smaller/larger scale. The most well known and most understood such hierarchy is that associated with fundamental matter. Molecules are composed of atoms, which are composed of electrons and nuclei, which are composed of neutrons and protons... etc. This hierarchy, passing to smaller/greater distance/energy scales, also extends in the opposite direction when we consider astronomical scales, with the scales of planet-satellite systems being smaller than that of solar systems which is less than that of galactic and then cosmological scales. In the case of fundamental matter our understanding of the associated interactions has been gleaned by observing the relative positions of the building blocks of fundamental matter as a function of space and time, passing from data to phenomenology and taxonomy and then to theory. A well known and canonical example of this process is that of celestial dynamics, wherein spatial data on the motion of the celestial bodies was collected by Tycho Brahe, was then closely examined by Johannes Kepler, who deduced the set of phenomenological laws that bear his name, and which in their turn were deduced by Isaac Newton as natural consequences of the law of universal gravitation. This logic has been at the heart of our understanding of all the fundamental interactions: gravitational, electromagnetic, weak and strong nuclear forces, all of which lead to deviations in the positions of matter in their presence relative to their absence. Our understanding of Complex Adaptive Systems (CAS) is very much less than that of fundamental matter. Two reasons for that are that, unlike physical matter, CAS are not classifiable in terms of a small number of fundamental properties, such as charge, mass etc. Secondly their interactions are not as universal. These characteristics are, in their turn, related to the extreme degree of multi-factoriality associated with CAS. An implication of this is that to characterize them phenomenologically requires a huge amount of spatial data, and that it is only recently that such data has begun to be available due to the Data Revolution of the last couple of decades. The volumes associated with such data and its associated multi-factoriality also imply that new methodologies, such as data mining, as opposed to traditional scientific computing, are required to model them. Over the last ten years or so in particular there has been a continuous and spectacular growth in the availability of spatial data. We are getting better and better at collecting and sharing data. Through the use of location aware devices, greater amounts of spatial data are being generated every day, ranging from scientific data, such as species collection data, or satellite weather services - to casual data from mobile applications. Also, with the creation of open databases like GBIF (www.gbif.org), or governmental data services like www.data.gov, there is a lot of data that is publicly available. All these data contain implicit information about the interactions that shape our world and society. The challenge is to find better tools with which to transform that data into knowledge, in ways that are useful for a wider range of users, not only the professional analyst [1]. Multivariate geospatial data are complex to explore, and it is usually necessary to use more than one visual perspective, or space, to analyze them. For example, in [2] the authors use three visual spaces to explore spatial interaction data: the multivariate, the geographic and the network graph spaces. Each of these spaces has its pros and cons, with each being more efficient at showing particular types of patterns in the data. Exploration in the network space however, is usually considered only when spatial data is embedded in a network structure. While the geographical space allows one to study the spatial distribution of the variables of interest, the multivariate space allows one to identify multivariate relations or interesting multivariate profiles between these variables using techniques such as parallel coordinate plots. A major problem with this type of analysis emerges when the number of variables is large, as then the visual exploration of potential relations is much more difficult. Although new techniques for analysing spatial data sets have been developed in the last decade, most of them seem to be focused on analysing small collections of large point sets and do not adjust well to analysing large collections of point sets, where each point set corresponds to one variable and visual exploration is limited to a few characteristic features of the data. As data mining is chiefly associated with seeking correlations or patterns in spaces with many variables, as opposed to a few, where a more classical statistical approach is appropriate, spatial data mining requires tools that are suitable for considering large numbers of spatial distributions considered as potential predictive variables. One technique for doing this is network graph analysis where we


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can use the network graph space to explore potential relationships. Network graphs offer a rich structure that has been recognized in many fields as a helpful visual representation, due to their ability to represent complex systems of relationships in a visually insightful and intuitive way. Network graphs also provide a strong mathematical framework for algorithmic and statistical analysis, mainly from graph theory and, more recently, from complex systems. In particular, geographic information science has a long tradition in network analysis for the study of spatial networks [3]. Spatial networks are usually network graphs defined by a set of nodes representing spatial locations and a set of links representing some type of connection between locations. In these cases, network graphs are used as a third analysis space to explore the topological and geometrical properties of infrastructure networks [4], or to understand the dynamics of flows between spatial locations [5]. Although, network graphs are consistently used in spatial networks, their use has not extended to spatial data mining in general when data are not already explicitly embedded in a network structure. In this paper we discuss a recently developed methodology to build inferential network graphs - Complex Inference Networks (CIN) – that can help as spatial data exploration tools to infer, identify and characterize the presence of interactions from spatial data. Such CINs have been shown to be very useful in identifying ecological interactions, especially in the case of the ecology of emerging and neglected diseases. A motivation of this work is to make this work available to a wider audience, as we believe that CINs can be extremely useful as exploratory analysis tools in spatial data mining in many other contexts, and especially in the context of CAS, where the interaction structure is so rich that a direct theoretical modelling approach is unlikely to be useful. CINs also bring information about potential interactions between variables that is difficult to see using other visual representations. Learning about them can support the construction of new hypothesis about the system under study and provide insight about the principal predictive drivers of a spatial distribution.

2 Building blocks and interactions 2.1

In physical systems

The relationship between building block hierarchies and the characterization of interactions between building blocks is clearest and most quantifiable in physics and, to a somewhat lesser extent, in chemistry. In the case of physics, or rather astronomy, our first encounter with building blocks was to note that celestial bodies were spatially discrete objects with a predictable dynamics. Much human activity depended on this predictability, but their taxonomy was founded on a terrestrial viewpoint, whereby planets, sun and moon were not differentiated in terms of understanding their role in the context of a higher order building blocks - planet/satellite and solar system. Based on the detailed and precise observations of Brahe it was for Kepler to deduce from that data the “laws” that capture the phenomenological regularities associated with the movement of the planets. However, there was no notion of that movement being associated with the concept of interaction between celestial bodies. It was for Newton to notice that if one object is in orbit around another then it is in a state of acceleration and therefore, according to his laws of motion, must be subject to a force, i.e., there must exist an interaction between the two bodies. Using Kepler’s laws and his own laws of motion it was in fact possible to deduce the precise nature of this interaction, being associated with a postulated law of gravitation wherein the force of interaction between two celestial bodies was proportional to the product of their masses and the inverse square of their distance. Moreover, given that mass is a positive number, it was possible to see that the interaction is such as to always give rise to an attractive force between bodies. The apocryphal tale of the falling apple is in the same vein, the falling to earth of the apple being a consequence of the interaction between apple and earth. Moreover, it was possible to postulate and verify that the nature of this interaction was truly universal in that it was associated with an interaction between any two objects with mass and depended on one, and only one, property of the associated objects - their mass. The crucial lesson from this is that it was possible to deduce the nature of the interaction between objects, both qualitatively and quantitatively by simply observing the relative

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positions in space and time of these objects. Without going into further detail, our understanding of the nature of the other types of interactions between physical objects has followed the same path. By observing the relative spatial positions of objects of distinct types we have been able to deduce the existence of four fundamental interaction types: gravity, electromagnetism and the strong and weak nuclear forces; each possessing a high degree of universality, wherein the nature of the interaction depends only on a very small number of parameters: mass and different types of charge. In the case of some of the fundamental interactions, as in the case of electromagnetism, it is possible to also have repulsion as well as attraction. By noting the relative spatial positions of electrically charged bodies it was possible to deduce analogs of Kepler’s and Newton’s laws, such as Coulomb’s law - the inverse square law for the force between charged bodies - which then were further subsumed into Maxwell’s equations, which fundamentally characterize the electromagnetic interaction. It is the relative strength of the fundamental interactions as a function of scale that then determines the nature of the building block hierarchies that we see. For example, at the atomic and molecular level it is the electromagnetic interaction that dominates while at the nuclear scale and below it is the strong and weak nuclear forces and, at astronomical scales, the gravitational interactions. It is the existence of distinct interaction scales and distinct charge types that is responsible for the fact that the different forces have very different regimes in which they dominate and therefore very different phenomenologies. The electromagnetic force, for instance, has little to no role to play in celestial dynamics because matter at such a macroscopic scale is effectively chargeless. On the other hand, gravitational interactions are negligible at the atomic level due to the very small masses of electrons and nuclei. Building blocks hierarchies are also characterized by the relative positions of their constituents. For instance, in a gas of molecules the average distance between molecules is much larger than the average distance between the atoms that constitute the molecule. Similarly, the average distance between the nucleons in one of the nuclei of the atoms is less than the average interatomic distance. Similarly, in terms of celestial dynamics the distance between the moon and the earth is much less than that between the earth and the sun. In either case, by observing the positions in time of building blocks at different scales we may understand the hierarchy of interactions between them. Thus, we may state that experimental observation at different scales of the relative positions of physical objects and their subsequent spatial modelling has allowed us to completely characterize, both qualitatively and quantitatively, the nature of the interactions between objects in physics and to see that they can be categorized into just a very few fundamental categories, that are associated with a very small number of parameters, and to note where and when each one predominates. It is, in fact, the case that our success in modelling physical systems owes a very great deal to the fact that these fundamental interactions have quite different regimes in which they predominate. Thus, in physics, building blocks are largely the same at one level of description until we pass to the next level, whereupon there is a clear differentiation between them as we can have more combinations. Thus, there are more species of molecule than there are of atom, there are more species of atom than there are of nuclear particle etc. 2.2

In complex adaptive systems

Just as in physics, the concepts of building blocks and their interactions are at the very heart of the study of CAS. Distinct to physics however, neither the building blocks nor their interactions are associated with a simple, universal taxonomy. In analogy with physics we may try to characterize interactions in CAS into categories. For instance, in ecology we would naturally consider as candidates: mutualism, commensalism, parasitism, competition etc. We may further drive the analogy and consider mutualism to be associated with an “attractive” interaction and competition with a “repulsive” interaction. The question remains, however, as to what degree may we classify and understand, both qualitatively and quantitatively, ecological interactions? We emphasized that in physics a key to understanding the nature of the physical interactions was to study the relative positions of physical bodies and their properties. A huge simplification came with the fact that each physical body could


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be characterized by one of four fundamental properties: mass, electrical, strong and weak charges. What are the relevant properties or labels in the ecological context? In physics, it is the nature of the objects of study in terms of matter and energy that is of relevance. However, in ecology it is the nature of the objects as organisms that is relevant. As has been noted [6], a huge distinction in this context is that physical objects “are” while organisms “do”. For example, electrical charge is a static property of an individual physical body, while the label “predator” is associated with an action associated with the relationship between one organism - the predator - and another the prey. If we think of an individual organism as an ecological “building block” then, unlike the case in physics, building block comes with a potentially unlimited number of labels. Some, phenotypic traits such as size or body mass, are more analogous to physical properties, while others, such as predator, prey, parasite, vector, reservoir etc. refer more to a relationship with other organisms. Although biotic and ecological interactions in general are very complex, it is reasonable, to state that the spatio-temporal distributions of taxa, or other ecological variables, should reflect all of the factors and their causal interactions that determine them. In [7, 8], our co-occurrence based metodology was applied to the case of an important neglected disease - Leishmaniasis. The degree of co-occurrence between taxa was taken as an observable measure with which potential interactions could be inferred. Although co-occurrence is not equal to direct biological interaction, a significantly non-random co-occurrence distribution is a necessary condition for a biotic interaction between taxa, and as such it can be used to formulate hypotheses that can be checked experimentally. However, it is clearly not a sufficient condition. In the spirit of niche modelling [9], a biotic variable that co-occurs with a target taxon can be understood as being a niche component in the same sense as any abiotic variable, such as temperature. In fact, one would generally expect a closer causal relation between biotic variables than with abiotic variables. For example, the distribution of prey species for a predator, such as a carnivore, should influence the latter’s distribution more significantly than temperature or precipitation. In the important explicit case of emerging diseases, for many, the predominant interaction between disease vector and disease host is due to the former feeding on the latter. This obviously requires a coincidence in space and time. Species that offer blood meals can maintain the presence of vector populations independently of the capacity to harbour a given pathogen. In other words, the interaction between host and vector is a necessary but not sufficient condition for the transmission of the pathogen.

3 Direct versus indirect interactions An important concept when discussing interactions is that of direct versus indirect interaction. This can be illustrated in the physical context by considering the difference between, say, a gas and a solid. For a gas of helium atoms there are strong electromagnetic interactions between the nucleus of a given atom and its constituent electrons. The interactions between one atom and another are negligible. However, the interactions between atoms may be non-negligible and that is what can give rise to molecules. Thus, the interaction between an electron and a proton in an hydrogen atom is direct, being describable directly in terms of the electrostatic attraction between the positively charged proton and the negatively charged electron. Two hydrogen atoms may form an hydrogen molecule. In this case, the interaction between the atoms, although it has its origin in the electrostatic interaction, is not directly describable as a simple interaction between two point-like hydrogen atoms but, rather, between two objects that are not point-like but possess structure. The resultant effective interaction between the hydrogen atoms results in the formation of a covalent bond, where electron sharing leads to a stable balance of attractive and repulsive forces between the atoms. Importantly, the direct electrostatic interaction between two protons would be negative, leading to a repulsion between the protons. However, the effect of the intermediating electrons in the covalent bond is such as to lead to an effective attraction between the protons. Unlike the direct electrostatic interaction between the two protons this interaction is indirect and is a consequence of the presence of other intermediating elements - the electrons. In a potential abusive use of ecological terminology we could say that the repulsive “competitive” interaction between individuals of the proton species were turned into an attractive interaction by the “facilitation” of individuals of the electron species. The resulting indirect interaction

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between the protons due to the presence of the electrons we can then think of, and mathematically characterize, as an emergent, direct “effective” interaction. In CAS the degree of indirection of the interactions between building blocks is much higher than in physical systems and is, in fact, a hallmark of their complexity. In such systems it is extremely difficult to disentangle the true causal relationships between variables and confounding variables abound. For instance, socio-economic status as an important risk factor for many diseases. In the case of emerging or neglected disease many factors affect the presence of a disease or its vectors. An important measure is the total number of encounters between vector and potential hosts, which depends on many factors, including the abundance of both species. However, a key factor is their co-occurrence, as proxied by the geographical overlap between them, as the greater the overlap the greater the probability of an encounter. Thus, for two host species, identical in all respects except their relative geographical overlap with the vectors, the host species with the larger overlap will be epidemiologically more important. Thus, vector-mammal geographical overlap is a necessary but not sufficient condition for both a feeding interaction and a pathogen transmission interaction. Of course, there may be geographical overlap between species, and/ or with other factors, due to other reasons than a direct biotic interaction. In fact, one would expect a hierarchy of direct and indirect effects. For instance, one may expect temperature and precipitation to potentially affect the distribution of a carnivore, such as a polar bear, directly, as well as indirectly through its effects on vegetation, which in turn affect the distributions of prey species etc.

4 Co-occurrence as an indicator of interactions We have laid out the thesis that hierarchical building blocks and their mutual interactions are a fundamental property of all systems, the chief difference being the nature of the building blocks at a given scale and the precise nature of their interactions. We have also shown that our understanding of the nature of the interactions between building blocks is largely a result of observing their relative positions. A fundamental characteristic of basically all known interactions is their locality. This locality can be observed of course. In physics, the “range” of an interaction can be precisely defined, once the interaction has been quantified. Thus, the very fact that the moon orbits, i.e. is constantly closer to, the earth rather than the sun shows that the moon’s motion depends more on its gravitational interaction with the earth than with the sun. Similarly, the fact that the earth orbits, i.e., is constantly closer to, the sun than to another star shows that the interaction between the earth and the sun is stronger than the sun with another star. More basically, to a good approximation, the moon interacts with the earth, but not other elements of the solar system, while the earth interacts with the sun but not with other stars. As another example, if we heat hydrogen gas, the hydrogen molecules will eventually dissociate into hydrogen atoms and, if we keep heating the gas to even higher temperatures, the atoms will ionize and we will have a gas of free electrons and protons. The state of the system and the nature of the interactions can be deduced by observing the relative positions of the relevant building blocks, quantum mechanical subtleties notwithstanding. Thus, the fact that in the case of hydrogen atoms versus hydrogen ions the electrons are bound to the protons means that their positions are highly non-random, being correlated with the positions of the protons. In the same way for hydrogen molecules, hydrogen atoms are bound together so that their relative positions are highly non-random with hydrogen atoms in the same molecule being correlated while those in distinct molecules are uncorrelated. As stated, the notion that physical interactions between objects can be inferred from their relative positions has been fundamental. Compared to biology and ecology however, the detailed nature of the interactions in physics are much simpler, much more universal and much more direct. In all cases however, the nature of interaction is so as to affect the spatio-temporal distributions of building blocks. Thus, a necessary condition for there to exist an interaction is that the spatio-temporal distributions of the corresponding building blocks is altered, with positive “attractive” interactions causing building blocks to be closer together than they would otherwise be, while with negative “repulsive” interactions they tend to be further apart. A generic notion to


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characterize this difference between these different behaviours is that of co-occurrence, wherein we expect building blocks to co-occur more or less than expected according to the degree of interaction between them.

5 Quantifying co-occurrence To quantify this notion, we must first address the question of how to define a co-occurrence. Here, for simplicity, we will restrict to co-occurrences in space rather than time, or space and time, though the formalism is readily extendable to both. If two objects are to be considered as co-occurring we need some notion of spatial resolution and, then, to determine what is the appropriate scale. To quantify spatial co-occurrence we require a notion of locating objects in space and an associated distance measure for those objects. We wish that any two or more objects can be classified as co-occurring or not independently of where they are located. One way to achieve this is, in the space of interest S , to define a partition of S into N elementary units Si such that S = ∪i Si . For instance, a grid of rectangular cells of a fixed size. Census block groups used by the U.S. census would be another example. An object of type A, Am (x), located at x is defined to be present or not in the spatial cell i according to if x ∈ i or not. Thus, Am (x) is a Boolean variable. We define a co-occurrence for two objects Am (x) and Bn (y), located at x and y, as an indicator function I(Am (x), Bn (y)) = 1 ⇐⇒ x ∈ Si and y ∈ Si , i.e., both objects are in the same spatial cell. If we are considering unique objects then the co-occurrence can only occur in one cell. However, if we consider object types, A and B, such as species of atom or biological species, then multiple co-occurrences are possible, depending on the spatial distribution of the objects. The number of co-occurrences NAB is given by 1 (1) NAB = ∑ ∑ I(Am (x), Bn (y)). 2 m n In this way we count multiple object pairs within the same spatial cell. We can also count by simply counting in each cell occurrence of a given type, independently of how many examples of the type there are in the cell. In this case the number of co-occurrences is given by NAB = ∑ I(A(i), B(i)),

(2)

i

where the indicator function is I(A(i), B(i)) = 1 ⇐⇒ A and B are present in the spatial cell i. Given NAB we may determine the probability of a co-occurrence P(AB) = NAB /N. We may also determine the conditional probabilities P(A|B) = NAB /NB and P(B|A). We may take P(AB) and P(A|B) as measures of the degree of co-occurrence of the types A and B. 5.1

Cell size

The first step in our methodology was to introduce a spatial grid for determining co-occurrences. However, a general problem for any methodology that uses such a notion of spatial discretization is how the level of aggregation of the data can affect the results of the analysis. This problem was identified in 1934 [10], and it is known as the “modifiable areal unit problem”(MAUP) [11]. Although MAUP has been an active research area since the 80s [12], it is still an open challenge to develop systematic methods to detect appropriate sampling scales [1]. In our methodology MAUP is the consequence of the choice of grid resolution, which is allied with the question of optimally sampling events on that grid. The general effect of cell size can be appreciated if we consider the limits of very large or very small cells. There are two general considerations: One is the effect of cell size on the effective size of the statistical sample to be analyzed, and the other is to do with its effect on the number of co-occurrences. For the former, as we are doing statistical analysis and hypothesis testing it is natural to take advantage of the samples at our disposal as much as possible. The first step of our process is to map events to spatial cells as statistical analysis is done at the level of cells not events. If we have N events then the maximum size of the sample of cells is also N.

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However, if the cell size is such that multiple, assuming they are independent, events occur in a given cell then the effective sample size is reduced. For instance, for a random distribution of events, if the cell size is such that on average the number of events per cell is 4 then a reduction in the cell size by a factor of 2 will probably lead to cells where the expected number of events per cell is closer to one. In other words, all else being equal, the number of cells should naturally scale as the number of events. For the case of co-occurrences, for a finite set of events if we go to the limit of very small cells it is clear that eventually we will end up with zero co-occurrences. On the other hand, in the limit of very large cells we will end up with only one co-occurrence as all the events will be in one cell. The choice of cell size has been investigated empirically [8], where it has been determined that an optimal resolution exists that maximises the number of co-occurrences and, fortunately, the sensibility of our methods to the spatial grid size in characterizing interactions is relatively small.

6 Inferring interactions If P(AB) or P(A|B) is larger/smaller than expected we will use that to infer that there must exist an interaction, either direct or indirect, between A and B. However, in order to determine if, say, P(A|B) is large or small we need to specify a baseline from which to measure it. There are several relatively equivalent baselines. For instance, we can quantify P(A|B) relative to P(A) = NA /N, the probability to find the type A independently of the presence of B. Another benchmark would be to measure P(AB) relative to P(A)P(B). Both benchmarks contain the same conceptual starting point, that co-occurrence (correlation) is measured relative to a distribution where there is no correlation given that P(A|B) = P(A) if P(AB) = P(A)P(B). We thus take as a measure of interaction between the object types A and B, ∆(AB) = (P(AB) − P(A)P(B)). If ∆(AB) > 0 then the types A and B co-occur more frequently than would be expected in the absence of interaction and less frequently than expected in the case ∆(AB) < 0. No interaction is defined as ∆(AB) = 0. The maximum and minimum values of ∆(AB) are 1 and −1. The maximum value corresponds to the situation that two object types are always found together on the spatial grid, while the minimum value is when the two object types are never found together on the grid. We will take ∆(AB) 6= 0 to infer that there is an “interaction” between the types A and B. In the case ∆(AB) > 0 we will say there is a positive, or attractive, interaction and in the case ∆(AB) < 0 a negative, or repulsive, interaction. To gain intuition for this we can consider the case of celestial dynamics, where A represents the moons of the planets and B the planets themselves. Taking a uniform spatial grid with cells of size say 106 km we would find that objects of type “moon” co-occur with objects of type “planet” but that there are no planet-planet co-occurrences. This would lead to us to infer, correctly, that there are attractive interactions between moons and planets and that planets have an effective repulsion that is a consequence of the fact that they all orbit the sun. If P(AB), P(B) and P(A) are determined from finite samples, then there is a possibility that the population estimates of P(AB), P(B) and P(A) have sampling errors, such that ∆(AB) 6= 0. In this case a statistical significance measure must be used to determine the degree of interaction. As we are considering objects as Boolean variables a binomial test is appropriate, where the diagnostic is NB (P(A|B) − P(A)) ε (A|B) = p (NB P(A)(1 − P(A))

(3)

which measures the statistical dependence of A on B relative to the null hypothesis that the distribution of A is independent of B. As the sampling distribution of the null hypothesis is a binomial distribution where, in this case, every cell is given a probability P(A) of having a point collection of A. The numerator of equation (3) then, is the difference between the actual number of co-occurrences of A and B relative to the expected number if the distribution of point collections were obtained from a binomial with sampling probability P(A). As we are talking about a stochastic sampling the numerator must be measured in appropriate units. As the underlying null hypothesis is that of a binomial distribution, it is natural to measure the numerator in standard


391

deviations of this distribution and that forms the denominator of equation (3). In general, the null hypothesis will always be associated with a binomial distribution as in each cell we are carrying out a Bernoulli trial (coin flip). However, the sampling probability can certainly change. The quantitative values of ε (A|B) can be interpreted in the standard sense of hypothesis testing by considering the associated p-value as the probability that |ε (A|B)| is at least as large as the observed one and then comparing this p-value with a required significance level. In the case where N > 5 − 10 then a normal approximation for the binomial distribution should be adequate, in which case ε (A|B) = 1.96 would represent the standard 95% confidence interval. When a normal approximation is not accurate then other approximations to the cumulative probability distribution of the binomial must be used. Although it is ∆(AB) that determines the degree of interaction, a diagnostic such as ε is necessary in order to quantify the degree of statistical significance of the observed interaction.

7 Networks and building blocks Networks have a long history, particularly in representing geographic information, such as infrastructure, communication and transportation networks. Network graphs are popular analysis tools because they allow one to visualise relationships between multiple objects in a way such that is easy for the human eye to detect patterns, and also because they provide a mathematical structure that allows them to be analyzed algorithmically. Over the last few decades they have become an important tool in many scientific areas, especially in biology [13] and ecology [14, 15], and are considered to be a key component of Complexity Science. However, in the vast majority of cases their local structure - in the sense of two nodes and a link as the base element - represents an already known relation, such as in a molecular interaction network [13], or a food web [16], or in a contact network representing ticks, vertebrates and pathogens, as in [17]. In this case the local structure of the network, i.e., the individual nodes and links, only represents what is already known and it is the emergent, topological, network-wide properties that lead to new insights and predictions. In general, the nodes of a network represent building blocks and the links represent interactions between them, with the link being weighted if we wish to indicate the strength of the interaction. However, as emphasized, typically, the set of potential interactions between the building blocks of a CAS is much, much greater than our capacity to observe them. In such systems, there are many important interactions still to be discovered and quantified. The question is how can such interactions be inferred without the need for detailed interaction specific observations?

8 Complex inference networks 8.1

Deducing the nature of interactions in complex adaptive systems

As we have stated, building blocks, especially in CAS, may be associated with multiple labels for their characterization, each of which may potentially be linked to an interaction. In ecology, as discussed, there are multiple potential labels for an organism and therefore potentially multiple corresponding interactions. For example, a small mammal may be simultaneously a food source for a blood-sucking insect, a host for a pathogen that the insect transmits and a predator for the same insect, to name but three. Crucially though, all of these interactions require the co-occurrence of both parties, and each one will leave its imprint on the relative spatio-temporal distributions of the relevant species. Conversely, the relative distributions of two species can be used to deduce whether or not there is an interaction between them. It does not necessarily determine the nature of that interaction. For instance, whether it is direct or indirect, or on what other factors it depends. 8.2 8.2.1

Applications Inference in natural language processing

In Natural Language Processing (NLP), network graphs are constructed for scenarios like topic discovery [18] or to visually compare content differences in versions of a text [19]. These type of networks are usually ref-

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ered as word co-occurrence networks, where nodes represent words and two words are linked if they co-occur significantly. Word co-occurrence is defined if two words are at less than a given distance from each other in a sentence or a document, and they co-occur significantly if they co-occur more than expected, for example, if P(W1 ,W2 ) > P(W1 )P(W2 ) [20]. Our work is in line with this kind of methodology, but with emphasis on the spatio-temporal context. In NLP network graphs are used in different subjects like word adjacency networks, semantic networks, word association networks and syntactic networks [21]. In [22] word co-occurrence networks are built for an Information Retrieval System, in this methodology two words co-occur if they are at most at a distance of 50 words in the same text, and a measure of the association strength between the words is developed based on their frequency of co-occurrence on a text database. In [23] the authors build a word co-occurrence network using t-scores to compute the significance of the word associations. The previous examples in this paragraph build word co-occurrence networks in order to analyze characteristics of documents in these cases, however, visualization is not the goal, network graph structures are used more as mathematical objects, for example to demonstrate that language has an underlying small world structure [20], or to select the right meaning of a term given its context [23]. In [19] word co-occurrence networks are used to characterise versions of a text in order to visually assess the changes that the document has suffered. 8.2.2

Emerging diseases

To illustrate the methodology, we will consider an application to the study of species interactions in the context of emerging diseases [7] and, in particular, Leishmaniasis, a neglected disease with 350 million people in 98 countries worldwide living at risk of developing one of the many forms of the disease [24]. It is caused by infection with one of several different species of protozoan parasites of the genus Leishmania, which maintain their life cycle through transmission between an insect vector (sandflies - genus Lutzomyia) and a mammalian host. The Lutzomyia carry the parasite after feeding from infected mammals (hosts). Subsequently, the Lutzomyia can infect other mammals, in particular, it may bite humans and infect them. Understanding the interactions between Lutzomyia and the parasite reservoirs is key to understanding the ecosystem that supports this disease and potentially prevent its spread. However, there are not enough data about which mammals are acting as reservoirs, exhaustive studies across all mammal species being prohibitively costly in both time and money. In Mexico, until recently with the advent of the predictive approach shown herein, it was thought that there were only a relatively small number of hosts of Leishmaniasis. In fact, only 8 species of 438 mammal species distributed in Mexico had been confirmed as hosts of this important pathogen. The approach using CINs was to infer potential reservoirs from geo-referenced point collection data for both mammals and Lutzomyia and study the potential species interactions from the resulting ε network graph. The data consisted of 427 point sets of species data in Mexico - collected over the last 100 years - which all together add up to 35,397 points. Since the study was focused on learning about the mammals that may be acting as feeding resources for the Lutzomyia, we are interested in the one sided relationships of Lutzomyia with a list of mammal species. In order to do that, we divided the point sets into two classes, 427 mammal species and 11 species of Lutzomyia. This means we have 11 spatial boolean features, for which we would like to understand how they are affected by the other 427 spatial boolean features. It is clear that directly, visually exploring the 427 × 11 co-distributions is not an option. Given the objectives of the research, the relationships between species are filtered in two ways. First, instead of exploring all the species’ pairwise combinations, the analysis only considers relationships between ordered pairs (Li , M j ), where Li is a Lutzomyia and M j a mammal, because the main goal is to discover by statistical inference mammals on which the Lutzomyia depend. Second, we set 2 as the minimum ε for an association to be considered significant - we could also consider those relationships with ε ≤ −2, for repulsive relations, but it is less interesting to know where there is little to no risk of presence of Lutzomyia. After creation of a uniform spatial grid, using a window size of 25km, we found that, from the 4697 potential statistical associations, only 521 are statistically significant, also that from the 427 mammal species, only 187 had a significant statistical association with at least one Lutzomyia. In order to determine first the most significant inferred interactions with Lutomyias as a genus we considered co-occurrences of each mammal species with


393

Fig. 1 Inferred relationships between potential hosts of Leishmaniasis and Lutzomyia vectors as a genus

any species of Lutzomyia and calculated ε for each one, subsequently ranking them from highest to lowset. The highest values of ε correspond to those mammals with the most statistically significant geographic attraction to Lutzomyias. The 25% of highest ranked mammals can be seen in the list in Figure 1. The full set of relationships between Lutzomyia and mammals can be seen in Figure 2. One should note that the resulting ε network graph is directed and weighted since ε is not a symmetric relation. For the visualization of the Complex Inference Network, the nodes were first distributed using a standard force-directed layout algorithm, usually the preferred method for automatic network graph layout. We begin our analysis from the resulting layout. We can identify distinct areas of the network centred on different subgroups of Lutzomyia. At the upper right we have one Lutzomyia node (L. Arthophora) surrounded by a set of mammal nodes, most of which are exclusively connected only to this species. Another section of the network, at the left, is composed of four Lutzomyia nodes and its corresponding neighbours, and finally the third rough area would be on the other six Lutzomyia, and a large set of surrounding neighbours. These areas are in close connection with the corresponding geographic distributions of the species.

394


Fig. 2 Complex Inference Network for potential vectors and hosts of the neglected disease Leishmaniasis

Although we consider 2 a reasonable threshold for ε one could take it just as a starting point and pick other limits. This helps to simplify the network in order to explore different levels of association among the species. This results in a simpler network, but still one that maintains the general shape of the structure. It is important to note that the topology of this network graph is in correspondence with the species geographic distributions. For example, the upper nodes correspond to L. panamensis, L. ovallesi, L. olmeca olmeca and L. gomezi, all species mostly found to the south east of Mexico in the Yucatan peninsula. A bit lower on the network graph, but in the same general area, we find L. cruciata and L. shannoni which are present in the peninsula but also spread towards the centre of the territory. They share some space with L. longipalpis, which in the network graph is reflected by a shared cluster of neighbours. On the other side we have L. Anthophora a species indigenous only to the north of Mexico and the United States. So, in this case, the network structure is closely related to a regionalization of the dataset. In this case, connections between subnetworks represent the overlap of distributions that may then explain ways in which the disease could jump from one region to another. There are other things that can be observed from the network structure. We can note, for example, that, other than the L. Anthophora subnetwork, the network graph is connected, meaning that there is a possibility that the parasite could propagate from one species to any other in the network. The degree of a node can be an useful cue. If the node is a Lutzomyia node, the degree represents the number of mammals with which the vector shares important statistical relationships and are potential hosts, a high degree vector node indicating that this vector may be exploiting many mammals. On the other side, if it is a mammal node then a high degree means there is a higher risk that this species is a host, since potentially many different vector species may exchange parasites


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with it [7]. To exemplify this, we consider 4 mammals that gave positive results to tests for the presence of the Leishmania parasite - Peromyscus yucatanicus, Ototylomys phyllotis, Reithrodontomys gracilis and Heteromys gaumeri - looking at their nodes in the network we can see that they are highly connected. In summary, CINs - inferred network graphs - can be a useful tool for the exploratory analysis of statistical relationships between a very large number of spatial variables. They provide a structure that summarises properties of the system which other exploratory methods cannot. The network graph structure allows one to compare all the nodes at once and identify properties, like high connectivity, that are important to understand the potential interaction dynamics, of the ecosystem in this case. The structure of this network graph led to hypotheses about which mammals could be reservoirs - mammals strongly connected to one or many Lutzomyia - or that may carry the parasite to other regions. Also, the network graph structure is related to the geographic distributions, and groups of nodes may represent geographic regions that are shown in a compact way since we are subtracting spatial information from the visualization and the analyst can focus on the topology of the relations. It is also important to note that the networks can serve not only in the exploratory analysis phase of data mining but also in the explicit construction of predictive models. In the present case this was done by using ε as a “score” function for each mammal. The higher the score the higher the probability that the corresponding mammal species is an important disease host. Thus, using ε one forms a ranked list of mammal species that serves as a predictive model. For example, if random collections are taken from highly ranked species versus those of low rank one would predict that those that are infected with the parasite should be more for the high rank mammals. A large experimental group has in fact carried out this work, confirming that the high ranked mammals are much more likely to be hosts. In this way, 23 new species of host for Leishmania have been discovered including, for the first time in Mexico, bat species [25].

9 Conclusions In this paper we discussed a new methodology - Complex Inference Networks - for the exploratory analysis of the interrelations between large collections of boolean spatial features, and how this could be used to discover and characterize interactions between the building blocks of CAS. The methodology allows one to find and explore statistical associations from the perspective of network graphs, which are a natural way to describe the relationships between a large set of entities that supports both visual and algorithmic analysis. The methodology integrates naturally three kinds of filters. Node type selection (variable selection), relationship type and edge magnitude, which allow the exploration of networks with different structures from the same dataset. These are important tools for the analysis, because with them one can approach the data from different points of view and from different levels of complexity. The methodology is also modular. For example, we could change the local co-occurrence model or the statistical measure and test how consistent it is with ε and the process would be the same in essence. We can also integrate multiple variable types from multiple resolutions to increase the dimensions of our local cooccurrence model. For example, if we have elevation in addition to longitude and latitude, we could consider rectangular 3D boxes as the local co-occurrence model, notice that we do not say cubes, this is because one may find more appropriate a finer resolution for the elevation axis. Analogously, we could do the same if data have a time variable. Extra dimensions, of course can make the analysis more complex, and the choice of the box dimensions more difficult. Using species data and a study on the ecosystem that supports the transmission of Leishmaniasis, our methodology shows that it is possible to construct CINs, where the local structure - two nodes and a link represents an interaction, where the interaction is inferred from the co-distributions of the building blocks, such as species, that form the nodes. Thus, in such networks both the local structure and the emergent global properties are associated with new information. Such networks can therefore help in the definition of new directions for further analysis, and the construction of hypothesis from the network’s properties. As we have seen, it also leads to explicit prediction models.

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Acknowledgements This work was supported by DGAPA-PAPIIT grants IN113414 and IG200217 and by the CONABIO.

References [1] Andrienko, G., et al., (2010), Space, time and visual analytics, International Journal of Geographical Information Science, 24(10), 1577-1600. [2] Guo, D. (2009), Flow mapping and multivariate visualization of large spatial interaction data, IEEE Transactions on Visualization and Computer Graphics, 15(6), 1041-1048. [3] Curtin, K.M. (2007), Network analysis in geographic information science: review, assessment, and projections, Cartography and Geographic Information Science, 34(2), 103-111. [4] Xie, F. and Levinson, D. (2007), Measuring the structure of road networks, Geographical Analysis, 39(3), 336-356. [5] Rae, A. (2009), From spatial interaction data to spatial interaction information? Geovisualisation and spatial structures of migration from the 2001 UK census, Computers, Environment and Urban Systems, 33(3), 161-178. [6] Stephens, C.R. (2015), What Isn’t Complexity, arXiv:1502.03199 [nlin.AO]. [7] Stephens, C.R., Heau, J.G., González, C., Ibarra-Cerdena, C.N., Sánchez-Cordero, V., and González-Salazar, C. (2009), Using biotic interaction networks for prediction in biodiversity and emerging diseases, PLoS One, 4, e5725, DOI:10.1371/journal.pone.0005725. [8] Sierra, R. and Stephens, C.R. (2012), Exploratory analysis of the interrelations between co-located boolean spatial features using network graphs, Int. J. Geogr. Inf. Sci., 26, 444-468, DOI:10.1080/13658816.2011.594799. [9] Townsend Peterson, A., Soberón, J., Pearson, R.G., and Anderson, R.P., Mart´ınez-Meyer, E., Nakamura, M., and Araújo, M.B. (2011), Ecological Niches and Geographic Distributions, Princeton University Press. [10] Gehlke, C.E. and Biehl, K. (1934), Certain effects of grouping upon the size of the correlation coefficient in census tract material, Journal of the American Statistical Association, 29(185), 169-170. [11] Openshaw, S., (1983), The modifiable areal unit problem, Concepts and techniques in modern geography, Norfolk, UK: Geo Books. [12] Dungan, J.L., et al. (2002), A balanced view of scale in spatial statistical analysis, Ecography, 25(5), 626-640. [13] Sanchez., C, Lachaize., C, Janody. F, et al. (1999), Grasping at molecular interactions and genetic networks in Drosophila melanogaster using FlyNets, an Internet database, Nucleic Acids Res., 27(1), 89-94. [14] Proulx, S.R., Promislow, D.E.L., and Phillips, P.C. (2005), Network thinking in ecology and evolution, Trends Ecol Evol., 20, 345-353. [15] Okuyama, T. and Holland, J.N. (2008), Network structural properties mediate the stability of mutualistic communities, Ecol Lett., 11, 208-216. [16] Pimm, S.L., Lawton, J.H., and Cohen, J.E. (1991), Food web patterns and their consequences, Nature, 350, 669-674. [17] Estrada-Peña, A., de la Fuente, J., Ostfeld, R.S., and Cabezas-Cruz, A. (2015), Interactions between tick and transmitted pathogens evolved to minimise competition through nested and coherent networks, Sci. Rep. Nature Publishing Group 5, 10361. [18] Madani, O. and Yu, J. (2010), Discovery of numerous specific topics via term co-occurrence analysis, In: Proceedings of the 19th ACM international conference on Information and knowledge management, CIKM 10, 26-30 October, Toronto, ON, Canada. New York, NY: ACM, 1841-1844. [19] Magnusson, C. and Vanharanta, H. (2003), Visualizing sequences of texts using collocational networks, In: P. Perner and A. Rosenfeld, eds. Machine learning and data mining in pattern recognition. Vol. 2734 of Lecture Notes in Computer Science. Berlin, Heidelberg: Springer Berlin/Heidelberg, chap. 24, 291-304. [20] Cancho, R.F. and Solé, R.V. (2001), The small world of human language, Proceedings of the Royal Society of London, Series B: Biological Sciences, 268(1482), 2261-2265. [21] Antiqueira, L., et al. (2007), Strong correlations between text quality and complex networks features, Physica A: Statistical and Theoretical Physics, 373, 811-820. [22] Veling, A. and Van Der Weerd, P. (1999), Conceptual grouping in word co-occurrence networks, In: Proceedings of the 16th international joint conference on Artificial intelligence, Volume 2, Stockholm, Sweden. San Francisco, CA: Morgan Kaufmann Publishers Inc., 694-699. [23] Edmonds, P. (1997), Choosing the word most typical in context using a lexical co-occurrence network, In: Proceedings of the 35th annual meeting of the association for computational linguistics and eighth conference of the European chapter of the association for computational linguistics, ACL 98, Madrid, Spain. Stroudsburg, PA: Association for Computational Linguistics, 507-509. [24] World Health Organization, (2010), Control of the leishmaniases, World Health Organ Tech Rep Ser., 949, 22-26. [25] Berzunza-Cruz, M., Rodr´ıguez-Moreno, A., Gutiérrez-Granados, G., González-Salazar, C., Stephens, C.R., HidalgoMihart, M., et al. (2015), Leishmania (L.) mexicana Infected Bats in Mexico: Novel Potential Reservoirs, Plos Negl Trop Dis., 9, 1-15, DOI:10.1371/journal.pntd.0003438.



MS-Stability Analysis of Predictor-Corrector Schemes for Stochastic Differential Equations R. Zeghdane†, A. Tocino Department of Mathematics, University of Bordj Bou Arreridj, Algeria Department of Mathematics, University of Salamanca, Spain Abstract

Submission Info

Deterministic predictor-corrector schemes are used mainly because of their numerical stability which they inherit from implicit counterparts of their corrector schemes. In principle these advantages carry over to the stochastic case. In this paper a complete study for the linear MS-stability of the oneparameter family of weak order 1.0 predictor-corrector Taylor schemes for scalar stochastic differential equations is given. Figures of the MS-stability regions that confirm the theoretical results are shown. It is also shown that mean-square A-stability is recovered if the parameter is increased.

Communicated by D. Volchenkov Received 7 July 2017 Accepted 15 May 2018 Available online 1 January 2019 Keywords Stochastic Taylor scheme Weak approximations Mean-square stability Multiplicative noise


1 Background Consider a filtered probability space (Ω, Ft , P), a one-dimensional Wiener process {Wt } and a scalar stochastic differential equation (SDE) of Itô type given by dXt = a(Xt )dt + b(Xt )dWt , Xt0 = x0 , t0 ≤ t ≤ T

(1)

where x0 is a constant vector and the coefficients a(x), b(x) are assumed to be measurable functions that satisfy a Lipschitz and a linear growth bound condition with respect to x, conditions that ensure the existence and uniqueness of solution, see [1]. Consider a equidistant discretization t0 < t1 < · · · < tN = T with step-size ∆ = (T − t0 )/N. The one-parameter family of weak order 1.0 predictor-corrector Taylor schemes for computing approximations Yn ≈ Xtn , n = 1, . . . , N is given by (2) Yn+1 = Yn + θ a(Y¯n+1 ) + (1 − θ )a(Yn ) ∆ + b(Yn )∆Wn as corrector, and the weak Euler scheme

Y¯n+1 = Yn + a(Yn )∆ + b(Yn )∆Wn † Corresponding


(3)

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R. Zeghdane, A. Tocino / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 397–401

as predictor, where θ ∈ [0, 1], and ∆Wn can be chosen as √Gaussian N (0, ∆) distributed random variable or as two-point distributed random variable with P(∆Wn = ± ∆) = 21 . Note that if θ = 21 the equations (2)-(3) becomes the modified trapezoidal method of weak order 1.0, see [4]. To study the stability properties of this predictor-corrector method we introduce the multiplicative noise linear test equation dXt = λ Xt dt + µ Xt dW (t), t ≥ 0,

(4)

with λ , µ ∈ R and X0 = x0 6= 0. The equilibrium position of (4) is asymptotically mean-square stable if and only if, see [5–7], 2λ + µ 2 < 0, (5) and the set SSDE = {(λ , µ ) ∈ R × R : 2λ + µ 2 < 0}

(6)

is called the MS-stability domain of the linear test equation (4). To study the linear MS-stability of an explicit one-step method, we apply the scheme to the test problem (4) giving an expression of Yn+1 in terms of the step-size ∆, the parameters of the problem λ , µ , the parameters and random variables of the method, and Yn ; then, taking mean-square norm in the obtained expression we get a recurrence of the form 2 (7) E[Yn+1 ] = R(λ , µ , ∆)E[Yn2 ]. From here, E[Yn2 ] → 0 if and only if R(λ , µ , ∆) < 1; then R(λ , µ , ∆) is called the MS-stability function of the method and the set SSM (∆) = {(λ , µ ) ∈ R × R : R(λ , µ , ∆) < 1} (8)

is called the stability domain of the stochastic method applied with step ∆. Following Higham, see [2, 3], to know if for a given step-size the method preserves the properties of the problem it is natural to compare the MS-stability domains SSM (∆) and SSDE . In particular, the inclusion SSDE ⊂ SSM (∆) for all ∆ > 0 means that whenever the SDE is stable, then so is the stochastic method for any stepsize. This is a generalization of deterministic A-stability property of numerical methods for deterministic equations.

2 Mean square stability analysis Our aim here is to investigate the linear mean-square stability of the method (2)-(3). The work is inspired by [2] and [3], where the author studies mean-square stability of semi-implicit Euler and Milstein θ -methods. Notice in particular that when θ = 0 the predictor-corrector method (2) − (3) becomes the Euler method; it coincides with the stochastic θ -method with θ = 0 in [2]. In that work it is proved that (a) if the test problem is unstable then so is the Euler method for all ∆ > 0, and (b) if the problem is stable then so is the Euler method for ∆ < −(2λ + µ 2 )/λ 2 . In this section we find a similar step-size bound for each predictor-corrector θ -method, i.e. a constant ∆θ > 0 such that, if the problem is stable, then so is the predictor-corrector method for ∆ < ∆θ . Applying the method (2)-(3) to the test problem (4) produces the recurrence Yn+1 = 1 + λ ∆ + θ λ 2 ∆2 + µ (1 + θ λ ∆)∆Wn Yn .

(9)

Taking mean-square norm in (9) yields the difference equation

2 E[Yn+1 ] = Rθ (λ , µ , ∆)E[Yn2 ]

with Rθ (λ , µ , ∆) = (1 + λ ∆ + θ λ 2 ∆2 )2 + µ 2 (1 + θ λ ∆)2 ∆.

(10)


399

A straightforward computation gives that the stability condition Rθ (λ , µ , ∆) < 1 is equivalent to the inequality (2λ + µ 2 )(1 + θ λ ∆) + λ 2 ∆(1 + θ λ ∆ − 2θ ) (1 + θ λ ∆) < 0 (11)

which can be written

θ λ 3 ∆2 + λ (θ µ 2 + λ )∆ + 2λ + µ 2 (1 + θ λ ∆) < 0.

(12)

Notice that the discriminant of the second order polynomial in ∆ which appears in the first factor of (12) is D = λ 2 (θ µ 2 + λ )2 − 4(2λ + µ 2 )θ λ 3

= λ 2 (1 − 8θ )λ 2 + µ 2 θ (µ 2 θ − 2λ ) = λ 2 (λ − µ 2 θ )2 − 8θ λ 2 ;

(13)

then if D ≥ 0, the stability condition (12) becomes

θ λ 3 (∆ − ∆+)(∆ − ∆− )(1 + θ λ ∆) < 0, where ∆ = ±

−(θ µ 2 + λ ) ±

Remark 1. If D ≥ 0, since ∆ = ±

−(θ µ 2 + λ ) ±

p (λ − µ 2 θ )2 − 8θ λ 2 . 2θ λ 2

(14)

(15)

p (θ µ 2 + λ )2 − 4(2λ + µ 2 )θ λ , 2θ λ 2

when (5) holds, we have for any 0 ≤ θ ≤ 1/2 that λ + θ µ 2 < 0 and 0 < ∆− < ∆+ . Theorem 1. If 0 ≤ θ ≤ 1, the θ -predictor-corrector method (2)-(3) applied with constant step-size ∆ > 0 to a stable test problem (4) is stable if ∆ < ∆θ where  2λ + µ 2   for θ = 0 −   λ2    1   ∆− for 0 < θ ≤   8   √   −1 1 1 µ2 8θ − 1 for < θ < < and ∆θ = θ λ 8 2 − λ θ   √   2  − µ 8θ − 1 1 1   and ∆ for < θ < ≥    8 2 −λ θ     1  −1 for ≤ θ ≤ 1 θλ 2

and ∆− is given in (15).

Proof. For θ = 0, see [2]. If 0 < θ ≤ 1 we suppose that the test problem is stable, i.e. that condition (5) holds (in particular λ < 0), and we shall prove that inequality (11) or (12) fulfill for ∆ < ∆θ . Suppose that 0 < θ ≤ 81 . In this case, the second expression in (13) shows that D ≥ 0 and then 0 < ∆− < ∆+ . If ∆ < ∆− then ∆
0 and (14) holds. Finally, suppose that 1/2 ≤ θ < 1 and ∆ < −1/θ λ . Since 1+ θ λ ∆ > 0, the stability condition (11) obviously holds under the assumption (5) because 1 − 2θ ≤ 0. A geometrical representation of MS-stability regions (6) and (8) helps to interpret the bounds found in the above theorem. Following [2] we shall take x = ∆λ , y = ∆ µ 2 , to obtain such representation in the real half-plane {(x, y) ∈ R2 : y ≥ 0}. Notice that, given problem parameters λ , µ ∈ R, varying ∆ correspond to moving along a ray that passes trough the origin and (λ , µ 2 ). The pairs (x, y), y ≥ 0 such that 2x + y < 0 constitute the MS-stability region of the test equation SSDE and correspond to the squared area between the marked ray y = −2x and the negative half-axe x shown in the pictures in Figure 1. For


401

each θ -predictor corrector method, the stability condition (11) becomes (2x + y)(1 + θ x)2 + x2 (1 + θ x − 2θ )(1 + θ x) < 0, which gives as representation of SSM the set (x, y) ∈ R2 : y ≥ 0, y(1 + θ x)2 < −x2 (1 + θ x)2 − 2x(1 + θ x) .

In Figure 1 this region is represented by the shaded area plotted for different values of θ . For each θ the bound ∆θ found in Theorem 1 can be interpreted in the following sense: Giving a stable problem, i.e. a pair (λ , µ ) fulfilling (5) determines the ray y = (µ 2 /λ )x (contained in the squared area); there exists a value ∆θ such that the segment {(x, y) = (λ ∆, µ 2 ∆) : 0 < ∆ < ∆θ } is entirely contained in the shaded region. Notice that for θ = 1/7, since θ ∈ (1/8, 1/2),√ there is a value (represented by the dashed line in the √ left bottom plot of Figure 1 and in this case equal to 8θθ −1 = 56 − 7), such that the extreme of the entirely √ 2 /λ > 7 − 56) or to the θ (when the slope satisfies 0 > µ contained ray belongs to the vertical line y = −1/ √ curve y = −x2 − 2x/(1 + θ x) (when µ 2 /λ < 7 − 56 < −2). References [1] Arnold, L. (1974), Stochastic Differential Equations, Wiley, New York. [2] Higham, D.J. (2000), Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38, 753-769. [3] Higham, D.J. (2000), A-stability and stochastic mean-square stability, BIT , 40, 404-409. [4] Kloeden, P.E. and Platen, E. (1992), Numerical solution of stochastic differential equations, Springer, Berlin. [5] Saito, Y. and Mitsui, T. (1996), Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33, 2254-2267. [6] Saito, Y. and Mitsui, T. (2002), Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30, 551-560. [7] Schurz, H. (1996), Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise, Stoch. Anal. Appl. , 14, 313-354.



Investigation into the Regular and Chaotic States of Twitter Victor Dmitriev, Andrey Dmitriev† National Research University Higher School of Economics, 33 Kirpichnaya Street, Moscow, Russia Submission Info Communicated by D. Volchenkov Received 7 August 2017 Accepted 8 June 2018 Available online 1 January 2019 Keywords Sociophysics Low-dimensional chaos Correlation dimension Twitter

Abstract The present paper is devoted to the investigation into the nonlinear dynamics of Twitter. A new model of Twitter as a thermodynamic non-equilibrium system is suggested. Dynamic variables of such system are represented by the variations of tweet/retweet number and instantaneous diversity between the densities of population on different levels around the equilibrium values. Regular and chaotic states of networks are described. It is pointed out, that the system is in a condition of an asymptotically stable equilibrium when the intensity values of an external information are small (the number of tweets eventually tends to its equilibrium value). If the intensity values of external information exceed the critical value, then the chaotic oscillations of tweets are to be observed. We have made the calculations of the correlation dimension and embedding dimension for the dynamics of the 10 most popular @ (TOP 100 by data of Twitter Counter). The results show, that all observed time series have clearly defined chaotic dynamical nature. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction From the second half of the XX century the general trend of science development is the penetration of the ideas and methods of physics in natural and traditional humanities disciplines. Methods of physical modeling are often used in sciences such as demography, sociology and linguistics. In the middle 1990s there was an interdisciplinary research field, known as Econophysics, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics. The term “econophysics” was coined by H. Eugene Stanley, to describe the large number of papers written by physicists in the problems of (stock and other) markets (for econophysics reviews see refs. [1–3]). Physicists’ interest in the social sciences is not new; Daniel Bernoulli, as an example, was the originator of utility-based preferences. Sociophysics is the study of social phenomena from a physics perspective, often using the human atom, social atom, or human particle perspective (for sociophysics reviews see refs. [3, 4]). The main objective of this new field of natural science consists in the research of objectively measured and formalized laws that define various social processes. The present paper is devoted to the investigation into the nonlinear dynamics of Twitter as a thermodynamic † Corresponding


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Victor Dmitriev, Andrey Dmitriev / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 403–411

non-equilibrium system. Our interest in this problem mainly stems from the hypothesis that the Twitter time series results from an inherently low-dimensional chaotic process. The main objective of the present paper is an investigation into the regular and chaotic states of Twitter. To achieve this goal the following research tasks were defined: 1. Building of the non-equilibrium macroscopic Twitter model in the form of the system of ordinary differential equations of the first order. 2. Investigation into the forming of the regular and chaotic orders of Twitter system operation depending on its control parameters values; definition of essential conditions for the change from one order to another. 3. Determination of the correlation dimension for a supposed chaotic process directly from experimental Twitter time series. There are a number of works in the field of physical modeling of social networks. The main physical models of the social networks are as follows. 1. Ising model [5–7]. 2. Bose-Einstein Condensate model [8, 9]. 3. Quantum walk model [10]. 4. Ground state and community detection [11]. Other relevant work in this area is that of refs. [12–15]. This paper is organized as follows. In section 2 we present the definition of Twitter as a complex dynamic system of the thermodynamic type and relevant background. In section 3 we present the nonlinear dynamic model of Twitter, including the definition of regular and chaotic state of the social network. In section 4 we give the numerical results of correlation dimension and embedding dimension from Twitter time series. In section 5 we conclude this paper.

2 Twitter as a thermodynamic system In brief, Twitter is an online social networking service that enables users to send and read short messages called “tweets”. Tweets are publicly visible by default, but senders can restrict message delivery to just their followers. Users may subscribe to other users’ tweets - this is known as “following” and subscribers are known as “followers” or “tweeps”, a portmanteau of Twitter and peeps. Individual tweets can be forwarded by other users to their own feed, a process known as a “retweet”. We suppose that the social network can be considered as such dynamic system of a thermodynamic type [16] that can produce aggregated factors (flows) out of joint activity of individual interests. These flows start to appear on a macro-scale and work according to laws of determined ties and relations. It can be assumed that such network is homeomorphic to the dynamic systems of a hydrodynamic type (from the viewpoint of the generalized macroscopic flows). Therefore, if there are so-called interacting colliding flows in such systems, then as a rule there can appear the phenomenon of generalized turbulence that generates the crisis mode in development of such dynamic systems. At present time some attempts to include the social dynamics within the scope of theoretic approaches which work well enough within the natural sciences are being implemented. Such attempts were multiple. However, owing to specifics of the research subject, different scientists have different understanding of the question on how to implement the applied mathematics to model the sociological reality. It turned out that the direct application of existing mathematical constructs to the social dynamics was ineffective. Moreover, there is no definition of social dynamics, such as one of, for example, electrodynamics. The practice of building of mathematical constructions which could efficiently model the dynamics in various physical systems and processes shows that the appropriate mathematical model becomes adequate (to one extent or another) to an original. This happens when all its characteristic properties are being derived out of the features and structure of that kind of movement, which forms the system dynamics. Correctness of such methodological principle when building new theories was brilliantly shown by J. Maxwell by way of creating


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the classic thermodynamics. Large and effective experience of the preceding physics-mathematical modeling makes the following methodological specification: there are no ready formalisms in mathematics, which could be immediately applied to description of a new “dynamics”. In fact, it turns out that for every new type or class of dynamical structures it is necessary to make a new construction of mathematical formalisms, which could be relevant its matter and peculiarities of its “dynamics”. Further we will concentrate on a special class of complex dynamic systems, which is called thermodynamic systems (TS) [16]. Complex thermodynamic systems are the systems made of the large number of approximately single-type elements (“atoms” or “users” for Twitter). Interaction between such elements occurs under the definite ontogenetic law. Kinematics and dynamics of such systems depend on a “life story” (or individual “life lines”) of every element within such system. There are 3 structural levels of functioning for this system type. The first one is a level of micro-local dynamics S0 , where the local interaction of every element (“user”) ai ∈ Λ with any other system element is being under consideration. This level is an ontogenetic level of a system, which generates all other dynamic effects there. In second, aggregative (“reductive”) information structure S1 follows the S0 -level. It is reasonable to call it the system mesodynamics [17]. It is shown by the averaged kinematics and dynamics of any its element. An intermediate dynamics of such system type is as a rule represented by the kinetic equations of the Boltzmann type in the molecular-kinetic theory. Finally, if there are the functions of the system state that are defined in a phase space of a complex dynamic system, then it is possible to get the macroscopically observed expressions of explored systems by means of averaging them over the individual and meso-stories. Such empirically defined artifacts are usually marked by the term “observed”. Let the macroscopic observable that is numerated by index α ∈ Λ and related to the time point t ∈ T , be xα (t). Let’s unite the set of all simultaneously observed dynamic variables {xα (t)} into a single complex: x (t) = hx1 (t) , x2 (t) , ..., xα (t)i .

(1)

And let’s consider it as a factor characterizing the global macroscopic state of a system under consideration. In particular, dynamic variables of Twitter are represented by the variations of tweet/retweet number and instantaneous diversity between the densities of population on different levels around the equilibrium values. Let’s assume that every individual element ai ∈ Λ, belonging to the base set of system elements Λ, has its own space of “internal states” Ai . Let the eigenstate of ai −element related to the time point tk , be marked as ωik . When all ωik are gathered in a single complex

ω k = hω1k , ω2k , ..., ωNk i,

(2)

we shall get the phase microscopic states of the whole system, related to the time moment tk . In particular, Twitter includes users, which can have just two states: a ground state and an excited state. Those users, who didn’t get sufficient amount of information from the mass media and other sources to be able to send tweets, stay in the ground state. Those users, who got sufficient amount of external information to be able to send tweets, are in the excited state. Then, let’s unite all the possible observable macroscopic system states Xβ into the single aggregate Φ = Xβ , β ∈ B (3) and call it a macroscopic phase space of a system. By analogy, if all possible micro-local system states ω α are united into Ω = {ω α } , α ∈ A, (4)

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then we’ll get a micro-local reference space of a system. It should be taken into account that the space Ω represents an abstract hypothetical mathematical construct. Its states ω α are not observable. However there are some reasonable assumptions that can be brought into the mathematical structure of the space Ω and factors of interaction between the system elements. These assumptions allow the abstract theoretical defining of the stochastic dynamics on the micro-level of the selected system. If such the hypothetical micro-local system dynamics is successfully assumed, than applying some folding techniques to the micro-local dynamic information it is possible to theoretically reconstruct the macroscopic global system dynamics in the language of phase space Φ as well as of related dynamic operator generating the macroscopic system dynamics. Match making process between the hypothetical micro-local dynamics of the Ω -space and the observable macro-global dynamics unfolding in the Φ -space is the main task of the stochastic dynamics of TS. It is determined that any TS being left on her own always tends (both in micro-local and macro-global representations) to its equilibrium state. If a deviation (by observed parameters) from the equilibrium state doesn’t equal 0, than the system will definitely tend to the nearest local equilibrium. Such a transition period is called relaxation [18]. There has been discovered a peculiar phenomenon of self-organization and spontaneous appearance of ordered structures of one or another type in large physicochemical TS during the investigation into the transition processes. There are necessary conditions of appearance of the new valuable dynamic information in a complex dynamic TS. They are the strong non-equilibrium of the related system states and the non-linear character of interaction between the system elements. Discovery of this phenomenon that works universally within the large amount of complex dynamic TS has become a cause of appearance of the interdisciplinary trend, which is called “synergetics”. Thus synergism proves itself in sociology because of the fact that the social systems (like, for example, microblogging social networks) are complex dynamic TS. And the following content of the present paper is aimed to show, how the thermodynamic conceptualism works in the dynamics of the complex social systems. Let’s mark out the essential terms and propositions that we shall use later on. Let’s consider that any system is organized by elements {ai } that are parts of a reference underlying set Λ ≡ {ai }. We shall examine large, but finite systems. These are the systems where the power of set Λ (or the value |Λ| = N) goes to infinity, but never equals it. This proposition is fundamental. Within the TS theory the N → ∞ abstraction is called thermodynamic limit. If N-factor is not large enough, then some of stochastic dynamics effects have a risk not to appear in the system. At the same time, the limit dividing large and small systems is very relative, but we shall not consider this problem at the present paper. When considering any complex dynamic system, the system factors are being in the foreground. Among them are: ontological nature of the system elements (“users”); nature and type of interaction (on the “user” level) between the system elements; nature of “movements” arising in the system (user interaction). Let’s consider as uniting symbol S the totality of algebra-geometric structural properties of interaction V in the system and “internal” state spaces Ai . If the basic defining factors hΛΣ , SΣ ,VΣ i are specified for some particular complex dynamic system Σ, then from the abstract-theoretical point of view such system can be considered as a definite complex Σ ≡ hΛΣ , SΣ ,VΣ i . (5) The modern theory of complex systems has allowed to discover the type of “movement” appearing in a system dynamics. Dynamic system class related to the stochastic dynamics paradigm is huge. Leaving out of consideration such universalism, it is possible to notice that it is important to understand the structural possibilities of applying the methods of stochastic dynamics for solving the definite range of problems in social dynamics. One of the essential peculiarities of stochastic dynamics consists in the fact that it is able to represent a dynamic mode of operation with the definite degree of uncertainty. The quantitative measure of different un-


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certainty types is a “physical” value of entropy. From the other hand, there is also a measure of certainty, or order, in the system. That is information. Thus, the stochastic dynamics can be considered as the informational dynamics. Synergetic conjugacy of these two poles of binary opposition lies as a basis of another binary opposition – atomism and holism. There is a fully developed theory of stochastic dynamics nowadays, and we shall not go deep into its details.

3 Nonlinear dynamic model of Twitter Twitter includes users, which can have just two states: a ground state and an excited state. Those users, who didn’t get sufficient amount of information from the mass media and other sources to be able to send tweets, stay in the ground state. Those users, who got sufficient amount of external information to be able to send tweets, are in the excited state. By sending tweets the network users transfer from the excited state to the ground state. Let’s consider the amount of users being in the ground state at the time point t as ω0 (t) and the amount of users in the excited state as ω1 (t). Let’s also introduce a parameter characterizing an instantaneous diversity between the densities of population on different levels (i.e. inversion): x3 (t) = ω1 (t) − ω0 (t) . (6) This variable will be further considered as one of the dynamic variables of the system. As the second dynamic variable we shall consider the variation of the tweet amount around the equilibrium position x1 (t) = T (t) − T0 , where T0 is a number of tweets in the network that stays in a state of the stable equilibrium. Let’s represent the variation speed of the tweet number as following: x˙1 (t) = −α x1 (t) + β x2 (t) .

(7)

Equation (7) has the following terms: the first term of the right side corresponds to the decrease in the number of tweets in the system due to the system relaxation; the second term correlates to the increase in the number of tweets in the network due to the increase in variation of the number of retweets x2 (t) =R (t) − R0 , where R0 is the number of tweets in a network being in the state of the stable equilibrium; α = 1 τ1 is a relaxation parameter; τ1 is a relaxation time. According to the Le Chatelier-Braun principle [19], if a system deviates from the state of the stable equilibrium, then the forces arise and try to return the system back to the equilibrium state. If |T (t) − T0 | ≪ T0 , then as a first approximation it can be considered that these forces are proportionate to deviation. As the third dynamic variable we shall consider the variation of the number of retweets, which can be represented as follows: Let’s represent the variation speed of the tweet number as following: x˙2 (t) = −γ x2 (t) + cx1 (t) x3 (t) .

(8)

The first term of the right side (8) corresponds to the decrease in the number of tweets due to the system relaxation with the relaxation parameter γ = 1 τ2 . The nature of the second term in the right side (8) can be explained in the following way. The number of retweets (created by the network users in the presence of the tweet flow) is proportionate to the number of tweets and depends on the level at which the user is (i.e. ground or excited level). An average contribution to the speed of the retweet change is proportionate to the product of the number of tweets in a network and the difference (6). The third equation describes the change of difference in a density of population on different layers (9) and looks like the following: (9) x˙3 (t) = ε (I0 − N (t)) − kx1 (t) x2 (t) ,

408


where ε is a population relaxation parameter; I0 characterizes the intensity of the external flow of information incoming to the system. The term x1 (t) x2 (t) corresponds to the power that is spent by the tweet flow for the retweet creation. Let’s introduce some new variables and parameters: √ √ (10) x = kcx1 /γ , y = β kcx2 /(αγ ), z = β cx3 (αγ ) , b = ε γ , σ = α γ , ρ = β cI0 (αγ ). Equation (7)-(8) can be reduced to:

x˙ = σ (y − x) , y˙ = −y + xz, z˙ = b (ρ − z) − xy.

(11)

The system (11) describes a dynamics of a Twitter as a point-dissipative system. When it is considered that by the way of changing the variable w = ρ − z the system (11) can be reduced to the well-known Lorenz system [20], one can assume that the parameter ρ is a control parameter of the system (11). Let’s examine the system behavior depending on the control parameter ρ or on the intensity of the external information flow I0 (ρ ∼ I0 ). The Lorenz system is a well examined dynamic system. The main properties of such system are presented in works [21, 22]. Let’s consider only those properties that will be required to analyze the change of the Twitter state due to the external informational influence I0 . Let’s use the dynamic system to discuss a question of existence of the asymptotically stable Twitter state. In the context of the earlier derived model, there is a condition of existence of the equilibrium (which is not obligatory stable) Twitter state. It is the equality of the tweet amount to its equilibrium value. For the tweet amount variation, this condition is equivalent to the following equality: T (t) = T0 . System (11) has 3 stationary points: p p (12) O(0, 0, 0), E(± b(ρ − 1), ± b(ρ − 1), 0).

If ρ < 1 (i.e. low external intensity of information I0 ), then the null point O is asymptotically stable. At the same time another E-points do not exist. The value of parameter ρ = 1 is a bifurcation value of the supercritical forked bifurcation of a system. If ρ > 1, then the stationary point O is unstable. Therefore, at a small quantity of external information the number of tweets asymptotically goes to its stable equilibrium value T0 . It’s one of the regular state of Twitter. If ρ = ρc , where ρc = σ (σ + b + 3) (σ − b − 1) is a critical value of external information, there arose the limit cycles around the non-zero equilibrium points. These are the fluctuations of the number of tweets; their characteristics do not depend on initial values of a system. When ρ > ρc , the limit cycle fails turning into the chaotic deviations. A chaotic attractor appears. For example, Fig. 1 shows the observed dynamics of tweets, derived with the help of the Twitter analysis service TWITONOMY (https://www.twitonomy.com/). It is obvious, that the dynamics of @realmadrid and @fcbarcelona are chaotic. Existence of the chaotic dynamics is connected with the huge amount of external information entering the microblogging network. In fact, the presented time intervals correspond to the increased interest of football fans to the football events Spanish La Liga and UEFA Champions League. Lorenz attractor, defined by the system (11), has also another, mathematically equivalent representation. By changing of variables hx, zi → hx, ui, where (13) u (x, z) = 2σ z − x2 , z (x, u) = u + x2 2σ and considering that y = x˙ σ + x, y˙ = x¨ σ + x, ˙ the Lorenz equation set (11) can be reduced to the following: x¨ = − (σ + 1) x˙ − x3 2 + σ (ρ − 1) − u 2 x,

(14)


(a)

409

(b)

Fig. 1 Dynamics of tweets: (a) @realmadrid and (b) @fcbarcelona.

u˙ = −bu + (2σ − b) x2 .

(15)

Stationary states of the system (14-15) in coordinates hx, ui can be represented by the three points in the coordinate space xOu: p (16) O (0, 0) , E(± b(ρ − 1), (2σ − b)(ρ − 1))

Equations set (9-10) allows to notice in this dynamics a range of well-known standard factors discovered in a theory of nonlinear non-harmonic oscillations. Let’s introduce the following symbols: γ = σ + 1, k = σ (ρ − 1) , ω = k − u 2 (17) and characteristic potential function of the dynamic system (14-15) V (u, x) = x4 8 − ω x2 2.

(18)

Let’s define the generalized “force” F (u, x), acting in direction of the coordinate Ox according to the rule (19) F (u, x) = −x3 2 + ω x. Considering (14), the equation (9) can be represented in the following form: x = f (γ , ω , x, ˙ x) ,

(20)

f = −γ x + F (u, x) .

(21)

where Considering (21), (14) turns into the Duffing equation [23, 24]. This equation is well-known and used in the theory of non-linear oscillations. Duffing equation (22) x¨ = −γ x˙ − x3 2 + (k − u (x)) 2 x

represents the dynamics (damped by the term −γ x) ˙ along the coordinate Ox in a double-humped potential well, that is described by the function (18). This function has an essential non-standard feature: ω = (k − u (x)) 2 is not constant. Because of the dependance of this function from x, the complex astable ties are generated between u (x) and x (u). These ties are defined by the equation: u˙ = −bu + (2σ − b) x2 .

(23)

The system (11) that is equivalent to the equation set (14-15) generates in the hu, xi-plane some complex irregular movements, which cause the dynamic chaos at the definite correlation of parameters hσ , b, ρ i.

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Table 1 Calculated correlation dimension and embedding dimension for financial time series. Twitter users

Correlation dimension

Embedding dimension

@katyperry

3.80

4

@justionbieber

4.17

5

@taylorswift13

3.49

4

@BarackObama

4.46

5

@rihanna

3.99

6

@YouTube

3.74

4

@ladygaga

3.58

4

@TheEllenShow

3.53

5

@picazomario

3.95

4

@jtimberlake

4.44

5

4 Correlation dimension of Twitter time series The determination of the correlation dimension [25] for a supposed chaotic process directly from experimental time series is an often used means of gaining information about the nature of the underlying dynamics (see, for example, contributions in ref. [26]; for reviews on dimension measurements see ref. [27]). In particular, such analyses have been used to support the hypothesis that the time series results from an inherently low-dimensional chaotic process [26]. The geometry of chaotic attractors can be complex and difficult to describe. It is therefore useful to have quantitative characterizations of such geometrical objects. One of these characterizations is the correlation dimension D2 . The correlation dimension have several advantages comparing to other dimensional measures: D2 is easy to compute from Twitter experimental data (from Twitter analysis service TWITONOMY); using D2 one can distinguish chaotic dynamical system (D2 is finite) from stochastic system (D2 → ∞); D2 is the lower bound estimate of attractor’s dimension d (d ≥ D2 ). The correlation dimension of the attractor of dynamical system can be estimated using the GrassbergerProcaccia algorithm (GP algorithm) [25]. The implementation of GP Algorithm in this work is written in Mathematica Language for Wolfram Mathematica® 10.2. Table 1 provides summary of calculated correlation dimension and embedding dimension (k) for Twitter time series. We have made the calculations of the correlation dimension and embedding dimension for the dynamics of the 10 most popular @ (TOP 100 by data of Twitter Counter http://twittercounter.com/pages/100). The results show, that all observed time series have clearly defined chaotic dynamical nature. It is also noticeable, that Twitter time series have correlation dimension close to each other and equal to 4. It is not strictly proven, that any Twitter time series has a chaotic dynamical nature. However, we can consider the dynamic system (11) as an approximate model of the Twitter. So, we also suggest, that the estimation of correlation dimension shall be a preliminary step of Twitter time series analysis. Moreover, the implementation of GP algorithm, including all optimizations, is computationally effective.

5 Conclusions The main contributions of this paper are as follows. 1. We present the nonlinear dynamic model of Twitter as a thermodynamic non-equilibrium system, including the definition of regular and chaotic state of the social network. It is pointed out, that the system is in a condition of an asymptotically stable equilibrium when the intensity values of an external information are small (the number of tweets eventually tends to its equilibrium value).


411

2. We report new numerical results of the correlation dimension and embedding dimension from Twitter time series. The results show, that all observed time series have clearly defined chaotic dynamical nature. It is also noticeable, that Twitter time series have correlation dimension close to each other and equal to 4.

References [1] Chakraborti, A., Toke, I., Patriarca, V. and Abergel, F. (2011), Econophysics review: II. Agent-based models, Quantitative Finance, 11, 1013-1041. [2] Richmond, P., Mimkes, J. and Hutzler, S. (2013), Econophysics and Physical Economics, Oxford University Press: United Kingdom. [3] Savoiu, G. (2013), Econophysics. Background and Applications in Economics, Finance, and Sociophysics, Elsevier: Amsterdam. [4] Slovokhotov, Y. (2012), Physics vs. sociophysics. Part 1. Physical grounds of social phenomena. Processes in society and solar forcing. Mechanical movement in a system of living particles, Probl. Upr., 1, 2–20. [5] Grabowski, A. and Kosinski, R.A. (2006), Ising-based model of opinion formation in a complex network of interpersonal interactions, Physica A, 361, 651–664. [6] Dasgupta, S., Pan R.K. and Sinha, S. (2009), Phase of Ising spins on modular networks analogous to social polarization, Physical Review E, 80, 025101-1. [7] Bianconi, G. (2002), Mean field solution of the Ising model on a Barabasi-Albert network, Phys. Lett. A, 303, 166. [8] Bianconi, G. ans Barabási, A.L. (2001), Bose-Einstein Condensation in Complex Networks, Phys. Rev. Lett., 86, 56325635. [9] Albert, R. and Barabasi, A.-L. (2002), Statistical mechanics of complex networks, Rev. Mod. Phys., 74, 47-97. [10] Faccin, M., Johnson, T., Biamonte, J. and Kais, S. (2013), Degree Distribution in Quantum Walks on Complex Networks, Phys. Rev. X, 3, p. 041007. [11] Reichardt, J. and Bornholdt, S. (2006), Statistical mechanics of community detection, Phys. Rev. E, 74, p. 016110. [12] Mendes, V. (2005), Tools for network dynamics, J. Bifurcation Chaos, 15, p. 1185. [13] Ebel, H., Davidsen, J. and Bornholdt, S. (2003), Dynamics of Social Networks, Complexity, 8, 24-27. [14] Toivonen, R., Onnela, J.P., Saramaki, J., Hyvonen, J. and Kaski, K. (2006), A model for social networks, Physica A, 371, 851–860. [15] Toivonen, R., Onnela, J.P., Saramaki, J., Hyvonen, J. and Kaski, K. (2009), A comparative study of social network models: Network evolution models and nodal attribute models, Social Networks, 31, 240–254. [16] Mimkes, J. (2006), A Thermodynamic Formulation of Social Science, Wiley: Germany. [17] Robert, V. and Youguel, G. (2015), The economics of knowledge, innovation and systemic technology policy, Routledge Tailor & Francis Group: USA. [18] Kizel, J. (1987), Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag: New York. [19] Atkins, P.W. (1993), The Elements of Physical Chemistry, Oxford University Press: United Kingdom. [20] Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130-141. [21] Sparrow, C. (1982), The Lorenz equations: bifurcations, chaos and strange attractors, Springer: Germany. [22] Hilborn, R.C. (2000), Chaos and nonlinear dynamics: an introduction for scientists and engineers, Oxford University Press: United Kingdom. [23] Kovacic, I. and Brennan, M.J. (2011), The Duffing equation: nonlinear oscillators and their behavior, John Wiley & Sons: USA. [24] Lakshmanan, M. and Murali, K. (1996), Chaos in nonlinear oscillators: controlling and synchronization, World Scientific, 13, 35-90. [25] Grassberger, P. and Procaccia, I. (1983), Measuring the trangeness of strange attractors, Physica D: Nonlinear Phenomena 9, 189-208. [26] Ding, M., Grebogi, C., Ott, E., Sauer, T. and Yorke, J. (1993), Estimating correlation dimension from a chaotic time series: when does plateau onset occur? Physica D, 69, 404-424. [27] Grassberger, P., Schriber, T. and Schaffrath, C. (1991), Nonlinear time sequence analysis, Int. J. Bifurcation Chaos, 01, 521-547.



The Effect of non-Selective Harvesting in Predator-Prey Model with Modified Leslie-Gower and Holling Type II Schemes I. El Harraki†, R. Yafia, A. Boutoulout, M. A. Aziz-Alaoui Ecole Nationale Supérieure des Mines de Rabat, Morocco Submission Info Communicated by D. Volchenkov Received 8 August 2017 Accepted 9 June 2018 Available online 1 January 2019 Keywords Predator-prey Harvesting Positivity and boundedness Local/Global stability Control optimal

Abstract In this paper, we study the effect of harvesting on the qualitative properties of predator-prey model with modified Leslie-Gower and Holling Type II functional responses. The model is given by a system of two ordinary differential equations with non-selective constants harvesting terms. We investigate the impact of harvesting terms on the boundedness of solutions, on the existence of the attraction set, on the stability of different equilibrium points. A Lyapunov function is used to prove the global stability of the interior equilibrium. We also, discussed the policy of optimal harvest and we got the solution for the interior equilibrium by the Pontryagin maximum criterion. Finally, our theoretical results are illustrated by a numerical simulations. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction It has long been apparent that the predator-prey models have contributed in understanding this path to greater dynamical complexity of ecological systems. The first predator-prey model has been proposed by Lotka (see, [8]), in the same context Volterra has developed a predator -prey competition model (see, [11]). In [9], Odum combined the two last models and present a new model which called Lotka-Volterra model. In the last decades, various dynamics of Lotka-Volterra model has attracted several authors from many fields, biology, economics... with many different functional response. The functional response of predators describes the rate of a predator consumes prey [1,4,5,15] and represent a main element in the research of predator-prey models, the most used one is the Holling type II which is characterized by a decelerating intake rate which comes from the assumption that the consumer is limited by its capacity to process food. In this way the predator-prey model proposed by Leslie-Gower describes the properties of a stochastic model interaction between two species (see, [7]). In the study of predator-prey interaction, some studies that treat population can be extended by considering harvesting. In [6], Clark introduce a mathematical model with harvesting of renewable resources and studied the harvesting problem with a combined ecologically independent and logistically growing fish species. Brauer and Soudack [3] developed a predator-prey model with † Corresponding


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constant rate harvesting in prey and studied its dynamical behavior. They also discussed the relation between the presence of harvesting and the region of asymptotic stability in the predator-prey plane. Zhang et al. [14] treated the problem of combined harvesting of prey-predator model with prey reserve by considering different harvesting effort. Rojas-Palma and Gonzalez-Olivares [10] considered an open access fishery of the predatorprey type where the functional response of the predators takes the form of Holling type-III and the prey growth is affected by the Allee-effect in which both prey and predator species are subjected to non-selective harvesting based on the catch per unit effort. In this work, we consider a predators prey model with non-selective harvesting. The model given by a system of two-species food chain which describes a prey population u which serves as food for a predator v with modified Leslie-Gower and Holling type-II functional responses. The equations of the two populations in the presence of harvesting can be written as follows:  a1 v du   )u − E1 u,  = (r1 − b1 u − dt u + k1 (1) a2 v dv    = (r2 − )v − E2 v, dt u + k2

with u(0) > 0 and v(0) > 0, where u and v represent the population densities at time t; r1 , r2 , a1 , a2 and b1 , are model parameters assuming only positive values. These parameters are defined as follows: r1 is the growth rate of prey u, b1 measures the strength of competition among individuals of species u, al is the maximum value which per capita reduction rate of u can attain, kl (respectively, k2 ) measures the extent to which environment provides protection to prey u (respectively, to predator v), r2 describes the growth rate of v, and a2 has a similar meaning to a1 , E1 the effort applied to harvest the prey, E2 the effort applied to harvest the predator. The model without harvesting is studied in [2] and the model without harvesting and with delay is studied in [12, 13]. The paper is organized as follows: Section 2 is devoted to the positivity and the boundedness of solutions. In section 3, we prove the local stability of the possible equilibrium points. Global stability of the positive equilibrium is studied in section 4. In section 5, we maximize the objective functional form of the studied model. In the last section, we give some some discussions and some numerical simulations in order to illustrate our theoretical results.

2 Positivity and boundedness In this section, we will prove that, under some assumptions the solutions of system (1) which start in IR2+ are ultimately bounded. We denote by IR2+ the nonnegative quadrant and by int(IR2+ ) the positive quadrant. Lemma 1. The positive quadrant int(IR2+ ) is invariant for system (1). Proof. Let h1 (u, v) = (r1 − b1 u − and h2 (u, v) = (r2 −

a1 v ) − E1 u + k1

a2 v ) − E2 . u + k2

Integrating equation (1) with initial conditions u(0) > 0 and v(0) > 0, we have ˆ t u(t) = u0 exp( h1 (u(s), v(s)ds) > 0, 0

ˆ t v(t) = v0 exp( h2 (u(s), v(s)ds) > 0. 0


415

Hence all solutions starting in IR2+ remain in IR2+ for all t > 0. Next, we give the following result: Lemma 2. Let φ be an absolutely-continuous function satisfying the differential inequality where t ≥ 0, (α1 , α2 ) ∈ IR2 and α1 6= 0. Then there exists T such that for all t ≥ T ≥ 0,

φ (t) ≤

d φ (t) dt

+ α1 φ (t) ≤ α2 ,

α2 α2 − ( − φ (T ))e−α1 (t−T ) . α1 α1

Proof. The proof of this Lemma is based on the Gronwall Lemma. Definition 1. A solution φ (t,t0 , x0 , y0 ) of system (1) is said to be ultimately bounded with respect to IR2+ if there exists a compact region B ∈ IR2+ and a finite time T (T = T (t0 , x0 , y0 )) such that, for any (t0 , x0 , y0 ) ∈ IR × IR2+ , φ (t,t0 , x0 , y0 ) ∈ B for all t ≥ T . Theorem 3. Let r1 > E1 and B the set defined by: B = {(u, v) ∈ IR2+ : 0 ≤ u ≤

r1 − E1 , 0 ≤ u + v ≤ L}, b1

where L=

1 (a2 (r1 − E1 )(r1 − E1 + 4) + (r2 − E2 + 1)2 (r1 − E1 + b1 k2 )). 4a2 b1

Then 1. B is positively invariant, 2. all solutions of (1) initiating in IR2+ are ultimately bounded with respect to IR2+ . Proof. 1. If (u(0), v(0)) ∈ B, then from Lemma (1) we have (u(t), v(t)) remain in IR2+ for all t ≥ 0. 1 Now, we have to show that u(t) ≤ r1 −E b1 and u(t) + v(t) ≤ L for all t > 0. Since u > 0 and v > 0 in Int(IR2+ ), every solution φ (t) = (u(t), v(t)) of (1) which starts in Int(IR2+ ). Then, u(t) satisfies the following inequality 0≤

du(t) ≤ (u(t))(r1 − E1 − b1 u(t)). dt

Then, we have u(t) ≤ y(t) where y(t) is the solution of the following equation: dy(t) = y(t)(r1 − b1 y(t)) − E1 y(t) dt y(0) = u(0) > 0, and y(t) =

where c = Therefore

1 u(0)

1 b1 + ce−r1t r1 − E1

b1 − r1 −E . 1

u(t) ≤

r1 − E1 b1

(2)

416


for all t ≥ 0. Now we prove that u(t) + v(t) ≤ L for all t ≥ 0. Define the function x(t) = u(t) + v(t), then we have dx du dv a1 v a2 v = + = (r1 − b1 u − )u − E1 u + (r2 − )v − E2 v. dt dt dt u + k1 u + k2 As all parameters are positive and all solutions initiating in IR2+ remain in IR2+ , then a2 v dx ≤ (r1 − b1 u)u − E1 u + (r2 − )v − E2 v. dt u + k2 Therefore max (r1 − E1 − b1 u)u =

u∈IR+

(r1 − E1 )2 . 4b1

Thus, we have a2 v dx(t) (r1 − E1 )2 ≤ − x(t) + u + v + (r2 − E2 − )v dt 4b1 u + k2 and

(r1 − E1 )2 a2 v dx(t) + x(t) ≤ + u + (1 + r2 − E2 − )v. dt 4b1 u + k2

From inequality 2, we get: (r1 − E1 )2 r1 − E1 a2 b1 v dx(t) + x(t) ≤ + + (1 + r2 − E2 − )v. dt 4b1 b1 r1 + b1 k2 and max (1 + r2 − E2 −

v∈IR+

a2 b1 v (r2 − E2 + 1)2 (r1 − E1 + b1 k2 ) )v = r1 − E1 + b1 k2 4a2 b1

Consequently dx(t) + x(t) ≤ L. dt Using Lemma 2, (with α1 = 1 and α2 = L), we find x(t) ≤ L − (L − x(T ))e−(t−T ) for all t ≤ T ≤ 0. Then x(t) ≤ L − (L − x(T ))e−t if T = 0 Since (u(0), v(0)) ∈ B, we have x(t) = u(t) + v(t) ≤ L for all t ≥ 0. 2. Secondly we prove that lim u(t) ≤

t−→∞

r1 − E1 b1

and lim u(t) + v(t) ≤ L.

t−→∞

From (2) and Lemma 2, we obtain lim u(t) ≤

t−→∞

r1 − E1 , b1

(3)


417

since the solutions of the following initial value problem du = (r1 − b1 u)u − E1 u, dt u(0) ≥ 0, satisfies lim u(t) ≤

t−→∞

(4)

r1 − E1 . b1

ε For the second inequality. Let ε > 0, then there exists T1 > 0 such that for all t ≤ T1 we have u(t) < 1 + . 2 From equation (19) with T = T1 , we get x(t) = u(t) + v(t) ≤ L − (L − x(T1))e(T1 −t)

(5)

x(t) ≤ L − (LeT1 − (u(T1 ) + v(T1 )eT1 )e−t ≤ L − (L − (u(T1 ) + v(T1 )eT1 )e−t .

(6)

and for all t ≥ T1 ≥ 0, we have

Then x(t) ≤ L + Let T2 ≥ T1 such that

ε ε − (L + − (u(T1 ) + v(T1 )eT1 )e−t ∀t ≥ T1 ≥ 0. 2 2

| (L +

(7)

ε ε − (u(T1 ) + v(T1 )eT1 )e−t |≤ . 2 2

Then u(t) + v(t) ≤ L +

ε ∀t > T2 . 2

Hence lim x(t) ≤ L, we deduce the result. t−→∞

3 Local stability In this section, we prove the local stability of the possible equilibrium points of system (1). It’s easy to show that system (1) has three trivial equilibria (belonging to the boundary of IR2+ ; i.e. at which (r2 −E2 )k2 1 . one or more of populations has zero density) P0 = (0, 0) , P1 (u, ˜ 0) and P2 (0, v) ˜ where u˜ = r1 −E b1 and v˜ = a2 The following result gives the existence of the non trivial equilibrium point. Proposition 4. Let us assume the following condition: a1 ((r2 − E2 )k2 ) < a2 ((r1 − E1 )k1 ) and r2 > E2 . Then system (1) has a unique interior equilibrium P3 (u∗ , v∗ ). Proof. From system (1), a steady state satisfies (r1 − E1 − b1 u∗ )(u∗ + k1 ) = a1 v∗

(8)

r2 (u∗ + k2 ) − E2 (u∗ + k2 ) . a2

(9)

and v∗ = We get

a2 b1 u∗2 + (a1 (r2 − E2 ) − a2 (r1 − E1 ) + a2 b1 k1 )u∗ + a1 (r2 − E2 )k2 − a2 (r1 − E1 )k1 = 0.

418


Thus u∗ =

1 (−(a1 (r2 − E2 ) − a2 (r1 − E1 + a2 b1 k1 )) ± ∆1/2 ), 2a2 b1 v∗ =

(r2 − E2 )(u∗ + k2 ) . a2

Where ∆ = (a1 (r2 − E2 ) − a2 (r1 − E1 ) + a2 b1 k1 )2 − 4a2 b1 (a1 (r2 − E2 )k2 − a2 (r1 − E1 )k1 ). If a1 ((r2 − E2 )k2 ) < a2 ((r1 − E1 )k1 )

holds, then ∆ > 0 and we also obtain that u∗+ > 0 and u∗− < 0. Therefore, system (1) possesses a unique positive interior equilibrium P3 (u∗ , v∗ ) given by u∗ =

1 (−(a1 (r2 − E2 ) − a2 (r1 − E1 + a2 b1 k1 )) + ∆1/2 ), 2a2 b1 v∗ =

(r2 − E2 )(u∗ + k2 ) . a2

Next, we prove the local stability for different equilibrium points under some parameter assumptions. To do that,one need to compute the Jacobian matrix M at an equilibrium point (u, v)   −a1 u a1 v(u + k1 ) − a1 uv r − E − 2b u − 1 1  1 (u + k1 )2 u + k1   (10) M= 2  a2 v 2a2 v  r − E − 2 2 (u + k2 )2 u + k2

Then, we have the following results:

Proposition 5. i) The equilibrium point P0 is stable only if E1 > r1 and E2 > r2 . ii) The equilibrium point P1 is stable if r1 > E1 and E2 > r2 . iii)The equilibrium point P2 is stable if is stable if E2 0 and P∗ (u∗ , v∗ ) is the global minimum of L. Since the solutions of the system are bounded and the set B which is positively invariant, we restrict our study on this set. Computing the time derivative of L along the solution of the system (1) and using (8) and 9, we have a1 u∗ a1 v a1 (u∗ + k2 )(v − v∗ ) a2 v∗ a2 v dL = (u∗ + k1 )(u − u∗ )(−b1 (u − u∗ ) + ∗ − )+ ( ∗ − ) dt u + k1 u + k1 a2 u + k2 u + k2 a1 v∗ (u + k1 ) − a1 v(u∗ + k1 ) = (u∗ + k1 )(u − u∗ )(−b1 (u − u∗ ) + (u + k1 )(u∗ + k1 ) ∗ ∗ v (u + k2 ) − v(u + k2 ) +a1 (u∗ + k2 )(v − v∗ ) (u∗ + k2 )(u + k2) −a1 k1 (v − v∗ ) − a1 u(v − v∗ ) + a1 v(u − u∗ ) = (u∗ + k1 )(u − u∗ )(−b1 (u − u∗ ) + (u + k1)(u∗ + k1 ) ∗ ∗ −k2 (v − v ) − u(v − v ) + v(u − u∗ ) +a1 (u∗ + k2 )(v − v∗ ) . (u∗ + k2 )(u + k2 ) From expression of

dL dt ,

we have

a1 v a1 v dL = (−b1 (u∗ + k1 ) + )(u − u∗ )2 + (−a1 + )(u − u∗ )(v − v∗ ) − a1 (v − v∗ )2 . dt u + k1 u + k2

(16)

The above equation can be written as dL = −(u − u∗ , v − v∗ ) H dt

u − u∗ v − v∗

.

(17)


421

Where H=

− f1 (u, v) − f2 (u, v) − f2 (u, v) a1

f1 (u, v) = −b1 (u∗ + k1 ) +

,

a1 v , u + k1

1 a1 v f2 (u, v) = (−a1 + ). 2 u + k2 Using Lemma 6 and Lemma 7, we get 1. f1 (u, v) < 0, 2. g(u, v) = −a1 f1 (u, v) − f22 (u, v) < 0. Since a1 > 0, we have H is a positive-definite matrix by Sylvester’s criterion. Then, in the first quadrant except (u∗ , v∗ ). So P∗ is globally asymptotically stable.

∂L ∂t

< 0 along all trajectories

5 Optimal harvesting policy Let us define the following objective functional: ˆ ∞ e−δ t (p1 u − c1 )E1 (t) + (p2 v − c2 )E2 (t)dt, J(E1 , E2 ) = 0

where • c1 fishing cost per unit effort for prey species, • c2 fishing cost per unit effort for predator species, • p1 price per unit biomass of the prey, • p2 price per unit biomass of the predator. Our objective is to maximize the form of the harvesting model with the instantaneous annual rate of discount δ subject to the state constraints of the system (1) and the control constraints 0 ≤ Ei ≤ Eimax (i = 1, 2). Hamiltonian for the problem is given by H =e−δ t [(p1 u − c1 )E1 (t) + (p2 v − c2 )E2 (t)] a1 v + λ1 [(r1 − b1 u − )u − E1 u] u + k1 a2 v + λ2 [(r2 − )v − E2 v]. u + k2 where λ1 and λ2 are adjoint variables. We have ∂H = e−δ t (p1 u − c1 ) − λ1 u = s1 (t), ∂ E1

∂H = e−δ t (p2 v − c2 ) − λ2 v = s2 (t). ∂ E2

(18)

(19) (20)

422


Hence the optimal control policy is  max if si (t) > 0,   Ei Ei (t) = 0 if si (t) < 0,   ∗ Ei if si (t) = 0.

for i = 1, 2 and singular control si (t) (i = 1, 2). From (19) and (20) we get c1 λ1 = e−δ t (p1 − ), u c2 −δ t λ2 = e (p2 − ). v By Pontryagin’s maximum principle, the adjoint equations are d λ1 ∂H =− , dt ∂u d λ2 ∂H =− . dt ∂v

(21) (22)

(23) (24)

From (18) and (23), we have d λ1 a1 v(u + k1 ) − a1 vu a2 v2 ( λ = −[e−δ t E1 p1 + λ1 (r1 − 2b1 u − ) + )] 2 dt (u + k1 )2 (u + k2 )2

(25)

From (18) and (24), we get a1 u a2 v d λ2 = −[e−δ t E2 p2 + λ1 (− ) + λ2 (r2 − 2 )]. dt (u + k1 ) (u + k2 )

(26)

Using equilibrium conditions (25) and (26), we find a1 vu a2 v2 d λ1 = −[e−δ t E1 p1 + λ1 (−b1 u + ) + )], ( λ 2 dt (u + k1 )2 (u + k2 )2 d λ2 a1 u a2 v = −[e−δ t E2 p2 + λ1 (− ) + λ2 (− )]. dt (u + k1 ) (u + k2 ) From (23) and (24) and by integrating (27) and (28)

λ1 =

(27) (28)

c1 a1 vu c2 a2 v2 1 −δ t ) + (p − )( )], e [E1 p1 + (p1 − )(−b1 u + 2 δ u (u + k1 )2 v (u + k2 )2

(29)

c1 a1 u c2 a2 v 1 −δ t ) + (p2 − )(− )]. e [E2 p2 + (p1 − )(− δ u (u + k1 ) v (u + k2 )

(30)

λ2 = From (21) and (22), we get e−δ t (p1 −

c1 1 c1 a1 vu c2 a2 v2 ) = e−δ t [E1 p1 + (p1 − )(−b1 u + ) + (p − )( )], 2 u δ u (u + k1 )2 v (u + k2 )2

c2 1 c1 a1 u c2 a2 v ) = e−δ t [E2 p2 + (p1 − )(− ) + (p2 − )(− )]. v δ u (u + k1 ) v (u + k2 ) Equations (31) and (32) give the optimal harvesting efforts as e−δ t (p2 −

E1 =

(31) (32)

1 c1 δ c1 a1 vu c2 a2 v2 [δ p1 − − (p1 − )(−b1 u + ) − (p − )( )], 2 p1 u u (u + k1 )2 v (u + k2 )2

(33)

c1 a1 u c2 a2 v δ c2 1 − (p1 − )(− ) − (p2 − )(− )]. [δ p2 − p2 v u (u + k1 ) v (u + k2 )

(34)

E2 =


0.9

0.8

0.8

0.7

0.7

0.6

0.6 Population Sizes

0.9

Predator

0.5 0.4 0.3

0.5 0.4 0.3

0.2

0.2

0.1

0.1

0

0

−0.1 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

423

−0.1

0

10

20

Prey

30

40

50

Time, t

12

12

10

10

8

8 Population Sizes

Predator

Fig. 1 Local stability for P0 for E1 < E2 , r1 = 1.8; r2 = 2.1; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 2.5; E2 = 3.

6

4

6

4

2

2

0

0

−2

0

5

10 Prey

15

−2

0

10

20

30

40

50

Time, t

Fig. 2 Stability for P1 for E1 < E2 , r1 = 1.8; r2 = 2; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 1.2; E2 = 3.

6 Numerical simulations and discussions In this section, some numerical simulations are given to illustrate the results presented in this paper, we examine the three cases of harvesting E1 < E2 , E1 > E2 and E1 = E2 (in left we plot u vs v and in the right (t, u) and (t, v)) spaces. In this paper, we have investigated the effect of harvesting for a modified Leslie-Gower predator-prey model. A qualitative analysis show the impact of harvesting in terms of boundedness of solutions and the existence of an attraction set. According to different parameters of the model, the local stability of different equilibrium point is studied. We prove the global stability for the interior equilibrium using a Lyapunov function. Finally, by the Hamiltonian and the Pontryagin’s maximum principle, the optimal harvesting are established and some numerical simulations are given to illustarte our results.

424


1.8

3.5

1.6

3

1.4

Population Sizes

Predator

2.5

2

1.5

1.2 1 0.8 0.6

1 0.4 0.5

0

0.2

0

1

2

3

4

0

5

0

10

20

Prey

30

40

50

Time, t

Fig. 3 Stability for P2 for E1 < E2 , r1 = 2; r2 = 2.5; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 1.5; E2 = 2.

15

3.5

3

2.5

Predator

Population Sizes

10

2

1.5

5 1

0.5

0

0

0.5

1

1.5

2

2.5

3

0

3.5

0

10

20

Prey

30

40

50

Time, t

Fig. 4 Stability for P3 for E1 < E2 , r1 = 2; r2 = 4; b1 = 0.6; a1 = 1; a2 = 1; k1 = 10; k2 = 1; E1 = 1.5; E2 = 2.

0.9

0.9

0.8

0.8 0.7

0.7

0.6 Population Sizes

Predator

0.6 0.5 0.4 0.3

0.4 0.3 0.2

0.2

0.1

0.1 0 −0.1

0.5

0 0

0.1

0.2

0.3

0.4 Prey

0.5

0.6

0.7

0.8

−0.1

0

10

20

30

40

50

Time, t

Fig. 5 Stability for P0 for E1 > E2 , r1 = 1.8; r2 = 2.1; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E2 = 2.5; E1 = 3.


12

425

5

10

4

Population Sizes

Predator

8

6

4

3

2

1 2 0

0

−2

0

5

10

15 Prey

20

25

−1

30

0

10

20

30

40

50

Time, t

Fig. 6 Stability for P1 for E1 > E2 , r1 = 3; r2 = 1; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 1.5; E2 = 2.

10

5

9 4

8

Population Sizes

7

Predator

6 5 4

3

2

1

3 2

0

1 0 −2

0

2

4

6

8

10

12

14

−1

16

0

10

20

Prey

30

40

50

Time, t

Fig. 7 Stability for P2 for E1 > E2 , r1 = 3; r2 = 1.5; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 3; E1 = 2; E2 = 1.

15

6 5.5 5 4.5

Predator

Population Sizes

10

5

4 3.5 3 2.5 2 1.5

0 0.5

1

1.5

2 Prey

2.5

3

3.5

1

0

10

20

30

40

50

Time, t

Fig. 8 Stability for P3 for E1 > E2 , r1 = 4; r2 = 4; b1 = 0.6; a1 = 1; a2 = 1; k1 = 15; k2 = 1; E1 = 3; E2 = 2.

426


0.9

9

0.8

8 7

0.7

6 Population Sizes

Predator

0.6 0.5 0.4 0.3

5 4 3 2

0.2

1

0.1

0

0 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−1

0.8

0

10

20

Prey

30

40

50

Time, t

Fig. 9 Stability for P0 for E1 = E2 , r1 = 1.8; r2 = 2.1; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 2.5; E2 = 2.5.

12

18 16

10

14 12 Population Sizes

Predator

8

6

4

10 8 6 4 2

2

0 0

0

2

4

6

8

10

12

14

16

−2

18

0

10

20

Prey

30

40

50

Time, t

Fig. 10 Stability for P1 for E1 = E2 , r1 = 4; r2 = 2.5; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 1; E1 = 3; E2 = 3.

10

14

9

12

8

10 Population Sizes

Predator

7 6 5

8 6 4

4 2

3

0

2 1 −0.5

0

0.5

1

1.5

2 Prey

2.5

3

3.5

4

−2

0

10

20

30

40

50

Time, t

Fig. 11 Stability for P2 for E1 = E2 , r1 = 1; r2 = 2.5; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 1; k2 = 3; E1 = 1; E2 = 1.


35

427

30

30 25

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25

20

15

20

15

10 10 5

0

0

5

10

15

20 Prey

25

30

35

5

0

10

20

30

40

50

Time, t

Fig. 12 Stability for P3 for E1 = E2 r1 = 3; r2 = 1.5; b1 = 0.06; a1 = 0.7; a2 = 0.6; k1 = 10; k2 = 3; E1 = 1; E2 = 1.

References [1] M.A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population, Chaos Sol. and Fractals 14 (8), 1275- 1293, (2002). [2] M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified LeslieGower and Holling-type IIschemes, Appl. Math. Lett. 16 (2003) 1069-1075. [3] Brauer, F., Soudack, A.C.: Stability regions in predator-prey systems with constant-rate prey harvesting. J. Math. Biol. 8, 55-71 (1979). [4] L. Chen, F. Chen, L. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl. 11 (1) (2010). [5] S. Chen, J. Shi, J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos 22(03) (2012). 246-252. [6] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976). [7] P.H. Leslie, J.C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika 47 (1960) 219-231. [8] A.J. Lotka, Elements of Physical Biology, Waverly Press, Williams Wilkins Company, Baltimore, MD,USA, 1925. [9] E.P. Odum, Fundamentals of Ecology, Saunders, Philadelphia, 1971. [10] A. Rojas-Palma, E. Gonzalez-Olivares, Optimal harvesting in a predator-prey model with Allee effect and sigmoid functional response. Appl. Math. Model. 36, 1864-1874 (2012). [11] V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES J. Mar. Sci. 3 (1928) 3-51. [12] R. Yafia, F. El Adnani and H. Talibi Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II scheme, Nonlinear Analysis: Real World Applications Vol.9, (2008), pp: 2055-2067. [13] R. Yafia, F. El Adnani and H. Talibi, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Applied Mathematical Sciences, Vol. 1, no. 3, (2007), pp: 119-131. [14] R. Zhang, J. Sun, H. Yang, Analysis of a prey-predator fishery model with prey researve. Appl. Math. Sci. 1, 2481-2492 (2007). [15] Z. Zeng, M. Fan, Study on a non-autonomous predator-prey system with Beddington-DeAngelis functional response, Math. Comput. Modelling 48(11) (2008) 1755-1764.



Ant Colony Optimization Algorithm for Lesion Border Detection in Dermoscopic Images Asmae Ennaji†, Abdellah Aarab† LESSI laboratory, Faculty of Sciences Dhar el Mahraz, Fes, USMBA, Morocco Submission Info Communicated by D. Volchenkov Received 9 August 2017 Accepted 9 June 2018 Available online 1 January 2019 Keywords Ant colony optimization Computer aided diagnosis Dermoscopic images Image segmentation Skin cancer

Abstract Medical image segmentation plays a crucial role in computer aided diagnosis system that have a significant potential for early detection of skin cancer. The aim of segmentation process in this field is to facilitate the characterization and the visualization of the lesion in dermoscopic images. This paper proposes a new method for improving the lesion border detection in dermoscopic images, based on the ant colony optimization algorithm. Our experiments show that the proposed method achieved a significant improvement in image segmentation when compared to the deterministic canny procedure.


1 Introduction Due to the proven benefits of applying digital imaging to dermatology, image processing research has directed a strong effort to develop Computer Aided Diagnosis (CAD) tools to assist physicians in their task of analyzing pigmented lesions, especially because a dermatologist is not always the physician that analyzes them in a first trial. Computer aided diagnosis (CAD) of melanoma is generally assured by five steps: • Image acquisition • Preprocessing phase • Segmentation process • Features extraction • Classification step † Corresponding

author. Email address: [email protected](Asmae Ennaji), aarab [email protected](Abdellah Aarab) ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.12.007

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Asmae Ennaji, Abdellah Aarab / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 429–436

In computer aided diagnosis of skin cancer, segmentation plays a crucial stage, because it aim to detect, with precision, the lesion edge, that will be used in the next phases of the diagnosis process. Image segmentation is the process of dividing an image into multiple parts, in order to detect the regions of interest. This is typically used to identify objects, or other relevant information, in the input image. Several methods have been proposed for the segmentation of skin lesion, in dermoscopic images. These algorithms can be broadly classified into three classes: Thresholding algorithms, Edge based algorithms and Region based algorithms [1]. In [2] an example of thresholding algorithm can be found, where a fusion of global thresholding, adaptive thresholding, and clustering is used. When there is good contrast between the lesion and the skin, thresholding methods achieve good results, thus the corresponding image histogram is bimodal, but usually fails when the modes from the two regions overlap. Edge-based approaches were used in [3] where the segmentation is based on the zero-crossings of the Laplacian of Gaussian (LOG) and in several active contour methods like the gradient vector flow (GVF) used in [4] and the geodesic active contour model (GAC) and the geodesic edge tracing described in [5]. Edge-based approaches perform poorly when the boundaries are not well defined, for instance when the transition between skin and lesion is smooth. Region-based approaches have also been used. Some examples include the multi scale region growing described in [6], the modified fuzzy c-means algorithm which is proposed in [7] and the morphological flooding used in [8]. Region-based approaches have difficulties when the lesion or the skin region are textured or have different colors present, which leads to an over-segmentation. In this paper we propose an automatic segmentation process that aims to detect the lesion border in dermoscopic images, based on the ant colony optimization algorithm. Additionally, we compare the obtained results with the deterministic canny procedure. This article is structured as follows; the first section is made up of the present introduction. The second section describes the dermoscopic images and Dermoscopy technique used in image acquisition phase; The third section describes the principal of the basic ACO algorithm and the fourth section present the proposed ACO based algorithm for lesion border detection; The fifth section shows the experimental results obtained, and the final section presents the conclusion of the work.

2 Dermoscopic Images Dermoscopy refers to the examination of the skin using skin surface microscopy, and is also called ‘epiluminoscopy’ and ‘epiluminescent microscopy’. It is a non-invasive technique, to assist dermatologists in the diagnosis of skin cancer. That used to evaluate pigmented lesions in order to distinguish malignant skin lesions, such as melanoma and pigmented basal cell carcinoma, from benign melanocytic naevi and seborrhoeic keratoses [9]. Allowing a better visualization of surface and subsurface structures, this diagnostic tool permits the recognition of morphologic structures, not visible by the naked eye, thus opening a new dimension of the clinical morphologic features of pigmented skin lesions. Dermoscopic images are a special original images obtained after the acquisition phase provided by the Dermoscopy technique and the segmentation of the lesion in this kind of images has a crucial role in the computer aided diagnosis system of skin cancer. Several studies have demonstrated that dermoscopy is useful in the identification of melanoma, when used by experts [9]. • It may be up to 35% more accurate than clinical diagnosis. • It may reduce the number of benign lesions excised. • In primary care, it may result in the referral of more suspicious lesions and fewer banal ones.


431

Fig. 1 The handy scope digital dermascope accessory for

Fig. 1 The handy scope digital dermascope accessory for iPhone.zde.

Fig. 2 Example Exampleofofdermoscopic dermoscopicimage imageofofskin, skin,obtained obtained

Fig. 2 Example of dermoscopic image of skin, obtained from the department of dermatology in CHU Hassan II Fes.

3 Ant colony optimization Ant colony optimization (ACO) is a nature-inspired optimization algorithm, inspired by the natural behavior of real ants’ observations. ACO was introduced, for the first time, by Marco Dorigo in 1992 [10], motivated by the natural phenomenon, that ants deposit pheromone on the ground, in order to mark some favorable path, that should be followed by other members of ants in the colony. The basic ACO algorithm, called the ant system, was proposed by Dorigo et al. [11]. Since then, a number of improved versions algorithms of ACO have been presented [12], such as the Max-Min ant system and the ant colony system [13, 14]. The purpose of apply the ACO algorithm, in image segmentation, is to extract the border information in the dermoscopic images. The algorithm exploits a number of ants, which move on the two dimensional image, driven by local variation of the image’s intensity values, so as to establish a pheromone matrix which represents the border information in each pixel location of the image. The purpose of ACO is to iteratively find the optimal solution of the target problem, through a guided search over the solution space, by constructing the pheromone information. Suppose totally K ants are applied to find the optimal solution in a space χ that consists of M1 × M2 nodes, the procedure of ACO can be summarized as follows. ACO process is based on two fundamental issues: the establishment of the probabilistic transition matrix p(n) and the update of the pheromone matrix τ (n), each of which, used in our work, is presented in section 4.

432


·

Initialize the positions of totally K ants, as well as the pheromone matrix τ (0).

·

For the construction-step index n = 1 : N, o

For the ant index k = 1 : K,

Ø Consecutively move the kth ant for L steps, according to a probabilistic transition matrix p(n) (with a size of M1M2 ×M1M2). o ·

Update the pheromone matrix τ(n).

Make the solution decision according to the final pheromone matrix τ (N).

Fig. 3 ACO algorithm.

4 The proposed ACO algorithm for lesion border detection A set of ants move on the 2D image for constructing a pheromone matrix. Each ant represents the edge information at each pixel location of the image. The movements of the ants are steered by the local variation of the image’s intensity values. Step of the proposed approach: 1. Initialization process 2. Construction process of the pheromone matrix 3. Update process 4. Decision process 4.1

ሺ଴ሻ

Initialization process

In the first, K ants are randomly assigned on the image IM1∗M2 , such as the pixels of the image presents the nodes in the ACO optimization and we initialize the value of each component of the pheromone matrix τ init. 4.2

Construction process of the pheromone matrix

At the nth construction-step, one ant is randomly selected ሺ୬ିଵሻ ஑ from ஒthe above-mentioned total K ants and this ant will ቀ ቁ ሺ୧ǡ୨ ሻ ୧ǡ୨ ሺ୬ሻ for L movement-steps. consecutively move on the image This ant moves from the node (l, m) to its neighboring ሺ୪ǡ୫ሻǡሺ୧ǡ୨ሻ ൌ ሺͶሻ ሺ୬ିଵሻ ஑ ஒ node (i, j) according to a transition probability is σሺ୧ǡ୨ሻ‫ א‬that ሻ as: ሺ defined ሺ୧ǡ୨ ሻ ሺౢǡౣሻ ୧ǡ୨ ሺ୬ିଵሻ ୧ǡ୨

(n) p(l,m),(i, j)

=

β i, j ) (n−1) α ) (ni, j )β i, j

(n−1) α ) (n

(τi, j

∑(i, j)∈Ω(l,m) (τ

(1)

where: (n−1)

• τi, j

is the pheromone value of the node (i, j)

• Ω(l, m) is the neighborhood nodes of the node (l, m) • ηi, j represents the heuristic information at the node (i, j)

ୡ ሺ ୧ǡ୨ ሻ ୑ଵ σ୧ୀଵ σ୑ଶ ୡ ሺ influence the ୧ǡ୨ ሻ pheromone ୨ୀଵof

ൌ

• α and β are two constants, represent the respectively. ୧୨

matrix and the heuristic matrix,


433

In the construction process there are two crucial issues. The first issue is the determination of the heuristic information ηi, j in (4) by:

ηi j =

Vc (Ii, j ) M1 M2 ∑i=1 ∑ j=1 Vc (Ii, j )

where • Ii, j : is the intensity value at the pixel (i, j) • Vc : is the function of a local group of pixels c that represents the clique pixel such as:

where

Vc (Ii, j ) = f ( Ii−2, j−1 − Ii+2, j+1 + Ii−2, j+1 − Ii+2, j−1 + Ii−1, j−2 − Ii+1, j+2 + Ii−1, j−1 − Ii+1, j+1 + Ii−1, j − Ii+1, j + Ii−1, j+1 − Ii−1, j−1 + Ii−1, j+2 − Ii−1, j−2 + Ii, j−1 − Ii, j+1 )   sin π x λ f (x) =  0

if 0 ≤ x ≤ λ else;

λ : Parameter that adjusts the function’s shape. The second issue is to determine the permissible range of the ant’s movement (i.e., Ω(l, m) in (4)) at the position (l, m). It is proposed to be either the 4-connectivity neighborhood or the 8-connectivity neighborhood. 4.3

Update process

The update process is carried out by two steps: • after the movement of each ant by the following equation: (n−1)

τi, j

=

 th  (1 − ρ ).τ (n−1) + ρ ∆(k) i, j , if the position (i, j) is visited by the k current ant (i, j) 

(n−1)

τ(i, j) , otherwise

ρ : is the evaporation rate (k)

(k)

∆i, j is determined by the heuristic matrix; that is, ∆i, j = ηi, j • After the movement of the all k ants the pheromone matrix is update by:

τ (n) = (1 − Ψ) · τ (n−1) · τ (0) where Ψ is the pheromone decay coefficient. 4.4

Decision process

The final pheromone matrix is used to classify each pixel either as an edge or a not. The decision is made by applying a threshold on the final pheromone matrix. The threshold value is computed based on the method described in [15], also known as the Otsu thresholding technique.

434


Fig. 4 Some dermoscopic images of skin used in this work.

a

b

C

Fig. 5 Lesion border detection results for dermoscopic images: (a) original image; (b) canny results; (c) proposed approach result. border detection results for dermoscopic images: (a) original

5 Experimental results The proposed approach was implemented and tested on some dermatologic images obtained from the department of dermatology in the Hassan II university hospital of fez, and from dermoscopyatlas [16]. Figure 2 shows some border detection results for dermoscopic images: (a) original


Original image

Image result with Canny parameters: [0.2 0.8]


435


Fig.6. results of canny detector with different parameters.

Fig. 6 Results of canny detector with different parameters.

dermoscopic images used in this work and the final results are shown is figure 3. The figure shows that the proposed algorithm is able to achieve comforting results for edge detection in many different dermoscopic images used for benign and malignant skin lesion. When compared with canny detector, knowing that the appropriate parameters are chosen for canny detector in each image, the results demonstrate that the proposed ACO based algorithm is an effective solution for border lesion detection in dermoscopic images. The comparison with canny detector was achieved in such a way that the appropriate parameters are chosen for canny detector in each image as showing in figure 6.

6 Conclusions In this paper, we have carried out an automatic algorithm involving ants as a mechanism for edge detection, applied in the segmentation of pigmented skin lesions in dermoscopic images; that is a primordial step in computer aided diagnosis of malignant melanoma. The method has been implemented and tested in medical images of skin cancer, which are obtained from the department of dermatology of the Hassan II university hospital fez and from dermoscopyatlas website [16]. The proposed method provides very competitive and motivated results when it is compared with canny detector and it will be used, in our future work for the detection of lesions in an integrated system of malignant melanoma recognition.

References [1] Luccheseyz, L. (2001), Color image segmentation: a state-of-the-art survey, Dept. of Electrical and Computer Eng., University of California, Santa Barbara; and S.K. Mitray, Dept. of Electronics and Informatics, University of Padua, Italy. [2] Ganster, H., Pinz, P., Rohrer, R., Wildling, E., Binder, M., and Kittler, H.(2001), Automated melanoma recognition, IEEE Trans. Med. Imag., 20, 233-239. [3] Rubegni, P., Ferrari, A., Cevenini, G., Piccolo, D., Burroni, M., Perotti, R., Peris, K., Taddeucci, P., Biagioli, M., Dell’Eva, G., Chimenti, S., and Andreassi, L. (2001), Differentiation between pigmented spitz naevus and melanoma by digital dermoscopy and stepwise logistic discriminant analysis, Melanoma Res., 11(1), 37-44. [4] Erkol, B., Moss, R.H., Stanley, R.J., Stoecker, W.V., and Hva-tum, E. (2005), Automatic lesion boundary detection in dermoscopy images using gradient vector flow snakes, Skin Res. & Technol., 11, 17-26. [5] Chung, D.H. and Sapiro, G. (2000), Segmenting skin lesions with partial-differential-equations-based image processing algorithms, IEEE Trans. Med. Imag., 19, 763-767. [6] Celebi, M., Aslandogan, Y., and Bergstresser, P. (2005), Unsupervised border detection of skin lesion images, in Int. Conf. Information Technology: Coding and Computing (ITCC 2005), 2, 123-128.

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[7] Schmid, P. (1999), Segmentation of digitized dermatoscopic images by twodimensional color clustering, IEEE Trans. Med. Imag., 18, 164-171. [8] Schmid, P. (1999), Lesion detection in dermatoscopic images using anisotropic diffusion and morphological flooding, in Proc. Int. Conf. Image Processing, 3, 449-453. [9] http://www.dermnetnz.org. [10] Dorigo, M. and Stützle, T. (2004), Ant Colony Optimization, Cambridge: MIT Press. [11] Dorigo, M., Maniezzo, V., and Colorni, A. (1996), Ant system: Optimization by a colony of cooperating agents, IEEE Trans. on Systems, Man and Cybernetics, Part B, 26, 29-41. [12] Dorigo, M., Birattari, M., and Stutzle, T. (2006), Ant colony optimization, IEEE Computational Intelligence Magazine, 1, 28-39. [13] Stutzle, T. and Holger, H.H. (2000), Max-Min ant system, Future Generation Computer Systems, 16, 889-914. [14] Dorigo, M. and Gambardella, L.M. (1997), Ant colony system: A cooperative learning approach to the traveling salesman problem, IEEE Trans. On Evolutionary Computation, 1, 53-66. [15] Otsu, N. (1979), A Threshold Selection Method from Gray-level Histograms, IEEE Transactions on Systems, Man and Cybernetics, 9(1), 62-66. [16] http://www.dermoscopyatlas.com



Bifurcation Diagrams and Fomenko’s Surgery on Liouville Tori of a Rigid Body in the Goryachev-Chaplygin Case on Sokolov Terms Jaouad Kharbach1†, Mohammed Benkhali1 , Walid Chatar1 , Ahmed Sali1 , Abdellah Rezzouk1 , Mohammed Ouazzani-Jamil2 1

Department of Physics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P.1796 Fez-Atlas, 30003, Morocco 2 Private University of Fez, Lot. Quaraouiyine Route Ain Chkef, Fez, 30000, Morocco Submission Info Communicated by D. Volchenkov Received 9 August 2017 Accepted 10 June 2018 Available online 1 January 2019 Keywords Goryachev-Chaplygin Top Hamiltonian system Integrability Liouville tori Bifurcation Poincaré section

Abstract In this paper, by taking advantage of the richness of Fomenko’s theory of surgery on (bifurcations of) Liouville tori, we give a complete description of the topology and bifurcations of the invariant level sets of a heavy rigid body around a fixed point corresponding to the Goryachev-Chaplygin case on sokolov terms. In particular, for all values of the parameters of the system, the bifurcation diagrams of the momentum mapping are constructed, bifurcations of the common level sets of the first integrals are described, explicit periodic solutions were determined, the topology of the invariant manifolds and all generic bifurcations are illustrated numerically.


1 Introduction The Goryachev-Chaplygin top is a rigid body rotating about a fixed point O and moves in a uniform gravity field. The weight of the body is mg and the distance from the fixed point to the centre of gravity is equal to r. Suppose OXY Z is a fixed system of coordinates, the OZ axis of which is directed vertically upwards. Another system of coordinates Oxyz is rigidly connected with the moving body, its axes Ox, Oy and Oz are directed along the principal axes of inertia of the body for the point O, the corresponding principal moments of inertia are equal to A1 , A2 , A3 satisfying A1 = A2 = 4A3 and with center of mass lying in the equatorial plane (through the fixed point) going with the moments A1 and A2 . The Euler-Poisson equations, describing the motion of a heavy rigid body around a fixed point, have the form: Aω˙ = Aω ∧ ω + μ r ∧ γ (1) γ˙ = γ ∧ ω †

Corresponding author. Email address: [email protected] ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.12.008

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Jaouad Kharbach et al. / Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 437–449

where A = (A1 ,A2 ,A3 ) is an inertia operator, γ = (γ1 , γ2 , γ3 ) is a unit vector in the vertical direction, ω = (ω1 , ω2 , ω3 ) is the angular velocity vector of the body, and r = (r1 ,r2 ,r3 ) the distance from the fixed point to the centre of gravity of the body. A and r are fixed, the motion is determined by the two vectors ω and γ . This system always has, for all values of the parameters A and r, three time-independent first integrals namely an energy integral H, an area integral I2 , and a geometric integral I3 : 1 (Aω ).ω + r.γ , 2 I2 = (Aω ).γ ,

H=

(2)

I3 = γ .γ = 1. For a general n-dimensional system, we will see that (n − 1) first integrals are needed to integrate the system completely. In the problem of the rigid body motion, therefore, only one extra first integral is needed. The general system has only three first integrals and a fourth integral has only been found in four particular cases, so the Euler-Poisson equations are Hamiltonian on the four-dimensional invariant manifolds M4 = {(ω , γ ) ∈ R6 ∶ ⟨Aω , γ ⟩ = cte,

⟨γ , γ ⟩ = 1} ⊂ R4 .

(3)

Using the projections of the momentum vector M = Aω in the same axes, equations (1) can be presented with Poisson bracket of variables M, γ in the Hamiltonian form M˙ i = {Mi ,H} ,

γ˙i = {γi ,H}, i = 1,2,3,

(4)

the dot denotes derivative with respect to time. Let us consider a generalization of the classical Goryachev-Chaplygin case of Euler-Poisson equations, In the vector form, equation (4) can be written as

∂H ∂H ∂H +γ × , γ˙ = γ × M˙ = M × ∂M ∂γ ∂γ 1 a 1 H = (M12 + M22 + 4M32 ) + λ M3 + μγ1 + 2 2 2 γ3

(5) (6)

where λ , μ and a are constants. Under the condition of the area integral I2 = M.γ = 0 the system has the additional integral: F = (M3 +

a λ )(M12 + M22 + 2 ) − μ M1 γ3 . 2 γ3

(7)

L. N. Sretensky [1] showed the generalization (6) for λ ≠ 0 and a = 0, and in other side on zero value surface of gyrostatic moment λ = 0 it was shown by D. N. Goryachev [2]. The complete form of generalization (6) was considered by I. V. Komarov and V. B. Kuznetsov [3]. Sokolov and Tsyganov [4] present a new generalization of particular integrable case (6) with the Hamiltonian containing the quadratic cross terms with respect to M, γ . The most general form of the integrable family in this case is presented as the following Hamiltonian:

ε 1 H = (M12 + M22 + 4M32 + 2 ) + λ M3 + μ1 γ1 + μ2 γ2 + a1 (2M3 γ1 − M1 γ3 ) + a2 (2M3 γ2 − M2 γ3 ). 2 γ3

(8)

The additional integral is F = (M3 + a1 γ1 + a2 γ2 +

λ ε )(M12 + M22 + 2 ) − (γ1 M1 + γ2 M2 )γ3 . 2 γ3

(9)


439

The purpose of this work is to present a detailed description of the real phase space topology on the bifurcation diagram for noncritical values of the constants of motion H and F of the Hamiltonian system corresponding the motion of a heavy rigid body around a fixed point in the Goryachev-Chaplygin case on sokolov terms. By using Fomenko’s surger, we describe all generic bifurcations of Liouville tori, corresponding to critical values of the constants of motion H and F. We give an explicit periodic solutions for singular common-level sets of the constants of motion. Finally this study is completed by numerical investigation via Poincaré surfaces of section.

2 Topological analysis The Hamiltonian of Goryachev-Chaplygin top in Sokolov and Tsyganov generalization is given by:

ε 1 H = (M12 + M22 + 4M32 + 2 ) + λ M3 + μ1 γ1 + μ2 γ2 + a1 (2M3 γ1 − M1 γ3 ) + a2 (2M3 γ2 − M2 γ3 ) 2 γ3

(10)

at ε = 0 we can separate variables in Hamiltonian (10), we don’t know such separation at ε ≠ 0. The Hamilton Jacobi equation admits a separation of variables. Indeed, let us perform a canonical change of variables [5]: px sin (x) + py sin (y) x−y √ M1 = 2 px py sin ( ), γ1 = , 2 px + py px cos (x)+py cos (y) x−y √ ), γ2 = , M2 = 2 px py cos( (11) 2 px + py √ 2 px py x+y ). γ3 = − cos ( M3 = px − py , px + py 2 The Hamiltonian function becomes:

H=

2(p3x + p3y ) + λ (p2x − p2y ) + μ1 (px sin (x) + py sin (y)) + μ2 (px cos (x)+py cos (y)) + 2a1 (p2x sin (x) − p2y sin (y)+ 2a2 (p2x cos (x) − p2y cos (y)) px + py

(12)

and the corresponding equations of motions reads: x˙ =

2 2 μ1 py (sin (x) − sin (y) μ2 py (cos (x) − cos (y)) ∂ H 4px + 2px py − 2py = +λ + + ∂ px px + py (px + py )2 (px + py )2

+ +

2a1 p2x sin (x) + 4a1 px py sin (x) + 2a1 p2y sin (y) (px + py )2 2a2 p2x cos (x)+4a2 px py cos (x) + 2a2 p2y cos (y) (px + py )2

∂ H 4py + 2px py − 2py μ1 px (sin (y) − sin (x) μ2 px (cos (y) − cos (x)) = −λ + + ∂ py px + py (px + py )2 (px + py )2 2

y˙ =

,

− −

2

2a1 p2y sin (y) + 4a1 px py sin (y) + 2a1 p2x sin (x)

(13)

(px + py )2 2a2 p2y cos (y)+4a2 px py cos (y) + 2a2 p2y cos (x) (px + py )2

,

440


p˙x = −

∂ H −μ1 px cos (x) + μ2 px sin (x) − 2a2 p2x cos (x) + 2a2 p2x sin (x) = , ∂x (px + py )

p˙y = −

∂ H −μ1 py cos (y) + μ2 py sin (x) − 2a2 py cos (y) + 2a2 py sin (y) = . ∂y (px + py ) 2

2

Equating expression (12) to H = h = cte and multiplying by (px + py ), we see that it separates: hpx − 2p3x − λ p2x − μ1 px sin (x) −μ2 px cos (x) − 2a1 p2x sin (x) − 2a2 p2x cos (x) = − hpy + 2p3y − λ p2y + μ1 py sin (y) + μ2 py cos (y) − 2a1 p2y sin (y) − 2a2 p2y cos (y),

(14)

we put: F = f = −2p3x − λ p2x − μ1 px sin (x) −μ2 px cos (x) − 2a1 p2x sin (x) − 2a2 p2x cos (x) + hpx = 2p3y − λ p2y + μ1 py sin (y) + μ2 py cos (y) − 2a1 p2y sin (y) − 2a2 p2y cos (y) − hpy .

(15)

Function F = f = cte is a first integral of the equations of motion (13). From (12) and (13) taking account of (15), we obtain the following system of differential equations √ φ (px ) p˙x = ± , px + py

√ φ (py ) p˙y = ± . px + py

(16)

Where φ (z) is a polynomial of degree 8:

φ (z) = [z(μ1 + 2a1 z)2 + z(μ2 + 2a2 z)2 ]2 − [(hz − 2z3 − λ z2 − f )(μ1 + 2a1 z)]2 .

(17)

We can express the solution of these equations from the hyperelliptic functions of time on the complexified manifolds: (18) A⊄ = {(x,y, px , py ) ∈ ⊄4 ∶ F = f = cte,H = h = cte} ⊂ ⊄4 . 2.1

Topology of regular level sets

In this section, we will give the topology of the real level sets ie the topology of the real phase space: AR = {(x,y, px , py ) ∈ R4 ∶ F = f = cte,H = h = cte} ⊂ R4 .

(19)

To do this we find the bifurcation diagram B which represents the set of critical values of the invariants H and F (like in Hénon-Heiles [6], Kolossoff [7], Hydrogen Atom in van der Waals Potential [8], Fokker-Planck system [9], The study on the Phase Structure of the Paul Trap System [10] and the phase topology of a special case of Goryachev integrability in rigid body dynamics [11]). That B is exactly the discriminant locus of the polynomial φ (z) defined in (16) can be written in the form:

φ (z) = φ1 (z) φ2 (z) Where

φ1 (z) = z(μ1 + 2a1 z)2 + z(μ2 + 2a2 z)2 − ((hz − 2z3 − λ z2 − f )(μ1 + 2a1 z)), φ2 (z) = z(μ1 + 2a1 z)2 + z(μ2 + 2a2 z)2 + ((hz − 2z3 − λ z2 − f )(μ1 + 2a1 z)). Thus the bifurcation diagram B is: B = B1 ∪ B2 .

(20)


441

Fig. 1 Bifurcation diagram B ∩ {a 1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 }.

Where B1 and B2 are respectively the bifurcation diagrams of the polynomials φ1 (z) and φ2 (z). B1 = (x,y, px , py ) ∈ R4 ∶ a1 = c1 ,a2 = c2 , μ1 = c3 , μ2 = c4 , λ = c5 discriminant (φ1 (z)) = 0 ⊂ R2 , B2 = (x,y, px , py ) ∈ R4 ∶ a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5

(21)

discriminant (φ2 (z)) = 0 ⊂ R2 . where c1 , c2 , c3 , c4 and c5 are constants. In order to plot the bifurcation diagram, the polynomials must have real roots, so it is necessary that constants a1 , a2 , μ1 and μ2 check the equation (μ2 + a2 )sin x = (μ1 + a1 )cos x. Then, we find: ⎧ (x,y, px , py ) ∈ R4 ∶ a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎬, B1 = ⎨ 7837 1 17h 2 3 ⎪ ⎪ ⎪ ⎪ − ± f = − 3906250 + 281250h + 6750h + 54h ⎪ ⎪ ⎩ ⎭ 6 108 27 4 ⎧ ⎫ (x,y, p , p ) ∈ R ∶ a = c , a = c , μ = c , μ = c , λ = c ⎪ x y 1 1 2 2 3 4 5⎪ 1 2 ⎪ ⎪ ⎪ ⎪ √ ⎬. B2 = ⎨ 93 1 3h 2 3 ⎪ ⎪ ⎪ ⎪ + ± f= 48000 + 7200h + 360h + 6h ⎪ ⎪ ⎩ ⎭ 2 4 9

(22)

As shown in Fig.1, the set {R4 /B}∩{a1 = c1 ,a2 = c2 , μ1 = c3 , λ = c5 , μ2 = c4 } consists of 9 connected components, in each one of them, the level set AR has the same topological type and it may change only as h and f passes through B ∩ {a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 }. Theorem. The set AR /B consists of 9 connected and nonintersecting with each other domains. The topological type of AR is a disjoint union of two-dimensional two-tori 2T2 , two-dimensional tori T 2 and the empty set φ . Proof. Consider the comlexified system A⊄ = {(x,y, px , py ) ∈ ⊄4 ∶ H = h = cte, F = f = cte}.

(23)

Consider also the hyperelliptic curves

Γ1 ∶ {ω12 = φ1 (z)}

and

Γ2 ∶ {ω22 = φ2 (z)}

(24)

and the corresponding Riemann surfaces R1 and R2 of the same genus j1 = j2 = 2. We obtain the explicit solutions of the initial problem (16) by solving the Jacobi inversion problem [12]. Thus x,y, px , py can be expressed in

442


Table 1 Real roots of the polynomials φ 1 (z), φ2 (z) and φ (z) for (h, f ,a 1 ,a2 , μ1 , μ2 , λ ) ∈ {R4 /B} ⋂{a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 }. DOMAIN

φ 1 (z) z = ui (i = 1, 2, 3)

φ 2 (z) z = ui (i = 1, 2 , 3 )

φ (z)

1

u2 < 0

v1 < 0

u2 < v1 < 0

2

u2 < 0

v1 < 0 < v2 < v3

u2 < v1 < 0 < v2 < v3

3

u1 < 0 < u2 < u3

v1 < 0 < v2 < v3

u1 < v1 < 0 < v2 < u2 < u3 < v3

4

u1 < u2 < 0 < u3

v1 < v2 < 0 < v3

u1 < v1 < v2 < u2 < 0 < u3 < v3

5

u1 < u2 < 0 < u3

v3 > 0

u1 < u2 < 0 < u3 < v3

6

u3 > 0

v3 > 0

0 < u3 < v3

7

u1 < u2 < u3 < 0

v2 < 0

u1 < u2 < v2 < u3 < 0

8

u1 < u2 < u3 < 0

v1 < 0 < v2 < v3

u1 < u2 < v1 < u3 < 0 < v2 < v3

9

u1 < u2 < 0 < u3

0< v1 < v2 < v3

u1 < u2 < 0< u3 < v1 < v2 < v3

terms of hyperelliptic functions living in the Jacobi variety Γ = Γ1 ⊗ Γ2 (where ⊗ is the symmetric product). These functions however are not single valued as can be seen from Eq. (11). Indeed, to each point on the symmetric product Γ1 ⊗ Γ2 there correspond two values of (x,y, px , py ). Thus we define the natural projection (25) A⊄ → Γ1 ⊗ Γ2 . Corresponding to the involution i i ∶ (x,y, px , py ) → (x,y,−px ,−py ) ,

(26)

the real level sets AR = Re(A⊄ ) is the set of fixed points of the complex conjugation on A⊄ :

η ∶ (x,y, px , py ) → (x, ¯ y, ¯ p¯x , p¯y ).

(27)

Consider also the natural projection ξ on the Riemann surface R = R1 ⊗ R2 given in u, v coordinates by:

ξ ∶ (u,v) → (u, ¯ v). ¯

(28)

It induces an involution on the Jacobi variety and hence on A⊄ by the natural projection π . Equations (11) imply that this involution ξ coincides with the complex conjugation (27) on A⊄ the upshot is that in order to describe AR it is enough to study the projection:

π ∶ A⊄ → Jac(R) = Γ1 ⊗ Γ2 .

(29)

Definition 1. A connected component of the set of fixed points of η on the curve Γ1 and Γ2 is called an oval. To determine the ovals of Γ1 and Γ2 it suffices to study the real roots of the polynomials φ1 (z) and φ2 (z) for different values of h and f as shown in Table 1. Using Eq. (11) and the condition that (x,y, px , py ) ∈ R4 , we find exactly two admissible ovals whose projections on the u-plane and the v-plane are given by Δ1 and Δ2 (see Table 2). The product of the admissible ovals in Γ1 ⊗ Γ2 and the projection π of AR such as, AR = π −1 (Γ1 ⊗ Γ2 ) = Δ1 ×Δ2 , gives: We obtain that AR is an empty set if (h, f ) belongs to domain 1, 6 and 7, on domains 2, 3, 4, 5, 8 and 9, AR is a torus or a disjoint union of two tori as shown in Table 2. 2.2

Topology of singular level sets

In this section, we shall give the description of all generic bifurcations of the topological type of AR by using the Fomenko surgery of bifurcation on Liouville tori [13].


443

Table 2 Admissible ovals and topological type of A R . DOMAIN

Δ1

Δ2

AR = Δ 1 ∗ Δ 2

1

∅

[u2 , v1 ]

∅

2

[v2 , v3 ]

[u2 , v1 ]

T2

3

[v2 , u2 ] ∪ [u3 , v3 ]

[u1 , v1 ]

2T2

4

[u3 , v3 ]

[u1 , v1 ] ∪ [v2 , u2 ]

T2 2T

5

[u3 , v3 ]

[u1 , u2 ]

T2

6

[u3 , v3 ]

∅

∅

7

∅

[u1 , u2 ] ∪ [v2 , u3 ]

∅

8

[v2 , v3 ]

[u1 , u2 ] ∪ [v1 , u3 ]

T2 2T

9

[u3 , v1 ] ∪ [v2 , v3 ]

[u1 , u2 ]

T2 2T

Table 3 Generic bifurcations of the level set A R passing from domain i to domain j. 2→1

2→3

5→6

5→4

3→4

2→8 5→9 5→8

2

2

T →∅

T → 2T

2

8→1 8→7

8→9 2

2T → 2T

9→7 2

2T 2 → ∅

This means that we must describe how the topological type of AR change when constants h and f passes through the bifurcation diagram B ∩ {a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 }. When a bifurcation of Liouville tori occurs, from the polynomial φ (z), the bifurcation sets is obtained, it is due to the bifurcation of the polynomial φ (z) roots. We have the following types of bifurcation: 1) A torus shrinks to a circle corresponding to the periodic solution, and then vanishes. 2) A torus splits into two tori (or conversely, two tori glue together). 3) A symmetric bifurcation of two tori into two tori. 4) Two tori are contracted to two circles corresponding to two periodic solutions, and then vanishes. To prove that, it suffices to look at the bifurcation of roots of the polynomial φ (z) (table 4). The first one from is domain (2) to domain (1) passing through curve C1 . Denote this sequence by: 2 → C1 → 1. The topological type of AR in domain (2) is a two dimensional torus T2 . On the curve C1 , this torus T 2 is contracted to the axial circle S1 and then vanishes. Denote this bifurcation as: T 2 → S1 → ∅. In the same way on the curve C5 :

5 → C5 → 6, T 2 → S1 → ∅,

[u3 ,v3 ] ∪ [u1 ,u2 ] → [u3 ,v3 ] ∪ {u1 = u2 } → [u3 ,v3 ] × ∅.

The others bifurcations are proved in a similar way. Namely 3 → C2 → 2,

444


2T 2 → S1 × (S1 ∨ S1 ) → T 2 , [v2 ,u2 ] ∪ [u3 ,v3 ] × [u1 ,v1 ] → [v2 ,u2 = u3 ] ∪ [u3 = u2 ,v3 ] × [u2 ,v1 ] → [v2 ,v3 ] × [u2 ,v1 ].

In the same way for curves C4 , C8 and C10 , where S1 ∨ S1 is a union of two circles having exactly one common point. The bifurcation on the curve C3 : 3 → C3 → 4, 2T 2 → T 2 ∪ S1 → 2T 2 , [v2 ,u2 ] ∪ [u3 ,v3 ] × [u1 ,v1 ] → v2 = u2 ∪ [u3 ,v3 ] × [u1 ,v1 ] → [u3 ,v3 ] × [u1 ,v1 ] ∪ [v2 ,u2 ]. The bifurcation on the curve C9 :

8 → C9 → 7 2T 2 → 2S1 → ∅

[v2 ,v3 ] × [u1 ,u2 ] ∪ [v1 ,u3 ] → v2 = v3 × [u1 ,u2 ] ∪ [v3 ,u3 ] → ∅ × u1 ,u2 ∪ [v2 ,u3 ]

3 Periodic solutions The level set AR degenerates when was the bifurcation of Liouville Tori. Then we can have exceptional families of periodic solutions. We are interested in periodic solutions where AR contains a single isolated circle. This is located on the bifurcation diagram B, more precisely when the first integrals h and f placed on C1 and C5 curves (see Table 4). Consider a fixed periodic solution on the curve C1 , py takes values in the admissible oval [u2 ,v1 ] and px is equal to the double root of the polynomial φ1 (z), px = v2 = v3 = α (see Table 4). From Eq. (16), one can calculate the periodic solution py (py + α )d py , √ ∣py + α ∣ (β − py )(py − γ )(p2y + γ py + (μ12 + μ22 + μ1 h − 2a1 f ) + γ 2 )

(30)

d py d py = ±2 √ . dt = ± √ Q(py ) (β − py )(py − γ )(p2y + γ py + μ 21 + μ22 + μ1 h − 2a1 f + γ 2 )

(31)

dt = ±

where γ = u2 and β = v1 ( py ∈ [γ , β ]). The polynomial Q(py ) has four distinct roots as: p0 = γ , where

p1 = β , p2 = a0 + ib0 , p3 = a0 − ib0 ,

1 1√ 2 3γ + 4(μ12 + μ22 + μ1 h − 2a1 f ), a0 = − γ , b0 = 2 2


445

Table 4 Topological type of A R for (h, f ) ∈ B ⋂{a1 = c1 , a2 = c2 , μ 1 = c3 , μ 2 = c4 , λ = c5 }. Curves

Δ1

Δ2

C1

v2 = v3

[u2 , v1 ]

C2

[v2 , u2 = u3 ] ∪ [u3 = u2 , v3 ]

[u3 , v1 ]

C3

{v2 = u2 } ∪ [u3 , v3 ]

[u1 , v1 ]

AR = Δ1 × Δ2 S1 1

S × (SS 1 ∨ S 1 ) T 2 ∪ S1 1

C4

[u3 , v3 ]

[u1 , v1 = v2 ] ∪ [v2 = v1 , u2 ]

S × (SS 1 ∨ S 1 )

C5

[u3 , v3 ]

u1 = u2

S1

C6

{u3 = v3 = 0}

[u1 , u2 ]

S1

C7

[v2 , v3 ]

[u1 , u2 ] ∪ v1 = u3

T 2 ∪ S1 1

C8

[v2 , v3 ]

[u1 , u2 = v1 ] ∪ [v1 = u2 , u3 ]

S × (SS 1 ∨ S 1 )

C9

{v2 = v3 }

[u1 , u2 ] ∪ [v3 , u3 ]

2SS1

C 10

[u3 , v1 = v2 ] ∪ [v2 = v1 , v3 ]

[u1 , u2 ]

S 1 × (SS 1 ∨ S 1 )

Fig. 2 Correspondence between bifurcations of Liouville tori and polynomials roots.

μ1 , μ2 , a1 are fixed, and on the curve C1 the first integrals h and f are related by f =−

17h 7837 1 √ − ± 3906250 + 281250h + 6750h2 + 54h3 . 6 108 27

By an inversion of the elliptic integral Eq. (30), one explicitly obtains the expression the periodic solution py (t): t

∫ dt = ∫ 0

p

γ

√

d py (py − p0 )(py − p1 )(py − p2 ) (py − p3 )

,

t = n0Cn−1 (cos ϕ ,k0 ).

(33)

where 1 , n0 = √ A0 B0

cos ϕ =

(c − py )B0 − (py − γ )A0 , (β − py )B0 + (py − γ )A0

(32)

k0 =

A0 2 = β 2 + a0 2 , B0 2 = b0 2 + a0 2

β 2 − (A0 − B0 )2 , 4A0 B0

446


Cn−1 (cos ϕ ,k0 ) = F(ϕ ,k0 ) being the incomplete elliptic integral of first kind. The final expression of py (t) is such: √ β B0 {Cn( 2t A0 B0 ) − 1} . (34) √ py (t) = (A0 +B0 ) + (A0 −B0 )Cn( 2t A0 B0 ) The period Tpy associated with the solution py (t) is obtained by calculating the elliptic integral Eq. (30) over the totality of the admissible oval for py : β d py d py Tpy = ∮ dt = ∮ √ . = 2∫ √ γ Q(py ) (β − py )(py − γ )(p2y + γ py + μ 21 + μ22 + μ1 h − 2a1 f + γ 2 )

It’s sufficient to replace the upper bound in the integral by β , in these conditions cos ϕ = −1, one obtain: Tpy (h) = 2n0Cn−1 (−1,k0 ) = 2n0 F(π ,k0 ),

(35)

where Cn−1 (−1,k0 ) = F(π ,k0 ) is the complete elliptic integral of the first kind. The periodic solutions can be determined on the curves C5 , C6 and C9 by the same method. 4 Numerical illustration Using a surface of the section map, we give numerical investigation of the topological analysis studied in Section 2. For fixed values of h and f , the Liouville tori contained in the level set H = h and F = f change their topological type. The surfaces of section map shown in Fig. 3 gives an illustration of the sequence of generic bifurcations of Liouville tori. This map is constructed using a clever method introduced by Poincaré and extended by Hénon [14].

5 Conclusion In this paper we have investigated the classical behavior of the dynamics of the rigid body, in particular the integrable case related to the motion of a heavy rigid body around a fixed point in the Goryachev-Chaplygin case on sokolov terms. By means of a canonical transformation the system is reduced to a separable Hamiltonian system with two degrees of freedom. The system is characterized by a two polynomials whose coefficients depend on the first integrals of motion H and F. The different results obtained show the capacity of the method used to provide precise information on this Hamiltonian system and to bring additional confirmations on the regular behavior of the problem by the search for analytical and numerical solutions. The very important question that we have studied is the topological analysis of the real invariant manifolds of the system. Fomenko’s theory on surgery and bifurcations of the Liouville tori has been combined with that of the algebraic structure to give a rigorous and detailed description of the topology of the invariant manifolds. For noncritical values of H and F on the bifurcation diagram, the topological type of AR is (diffeomorphic to) a two-dimensional tori, to a disjoint union of two-dimensional two-tori, or it is the empty set. In the same way for singular values of first integrals, we have shown how the periodic orbits can be found, how the period of solutions is determined, and how explicit formulas can be established. Finally, we have corroborate the analytic results by a numerical processing, for a regular and singular values of the first integrals of motion H and F on the bifurcation diagrams.


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

447

Fig. 3 Surfaces of section map for different values of h, f and fixed values of a 1 ,a2 , μ 1 , μ2 and λ , where (q 1 , p1 ,q2 , p2 ) = (x, px ,y, py ): (a), (b) domain 2 (h = −10.16, f = 16.41) A R ∼ T 2 (c), (d) domain 3 (h = 97.95, f = 15.62) A R ∼ 2T 2 ; (e), (f) domain 4 (h = 3.42.05, f = −9.744) A R ∼ 2T 2 ; (g), (h) domain 5 (h = −20.26, f = −16.15) A R ∼ T 2 ; (i), (j) domain 8 (h = −13.59, f = 3.342) A R ∼ 2T ; (k), (l) domain 9 (h = −16.15, f = −1.464) A R ∼ T 2 ; (m), (n) Bifurcation T → ∅, domain 5 → domain 6, (h = −10, f = −16.15) (green), (h = −20.26, f = −16.15) (pink), (h = −25.13, f = −16.15) (blue), (h = −25.9, f = −16.15) (red), (h = −25.915, f = 16.15) (black) A R ∼ S; (o), (p) Bifurcation 2T 2 → T 2 , domain 4 → domain 5, (h = 32.56, f = −14.1) (blue and pink), (h = 19.59, f = −14.1) (red and black) A R ∼ S1 × (S1 ∨ S1 ); (q), (r) Bifurcation 2T 2 → ∅, domain 8 → domain 7, (h = −14.8, f = 3.342) (red and black) A R ∼ 2S1 ; (s), (t) Bifurcation 2T 2 → 2T 2 , domain 3 → domain 4, (h = 46.62, f = 1.795) (red and black) A R ∼ T 2 ∪ S1 .

448


(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t) Fig. 3. Continued.


449

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Approximate Controllability of Impulsive Neutral Functional Integrodifferential Systems with Nonlocal Conditions A. Yasotha1†, K. Kanagarajan2 1 2

Department of Mathematics, United Institute of Technology, Coimbatore, India Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, India Submission Info Communicated by A. Luo Received 24 April 2017 Accepted 30 May 2018 Available online 1 January 2019 Keywords Approximate controllability Impulsive integrodifferential equations Analytic semigroup Fractional power operator

Abstract In this paper, we study the approximate controllability of impulsive neutral functional integrodifferential systems with finite delay. The fractional power theory and α -norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability of the impulsive integrodifferential equation are formulated and proved. An example is provided to illustrate the theory.


1 Introduction Many systems in physics and biology exhibit impulsive dynamical behaviour due to sudden jumps at certain instants in the evolution process. Differential equations involving impulse effects occur in many applications such as pharmacokinatics, the radiation of electromagnetic waves, population dynamics, biological systems etc., [1, 2]. Abada et.al [3] discussed the existence and controllability of nondensely defined impulsive semilinear differential systems with inclusion in Banach space. Recently, Chang [4] studied the exact controllability of impulsive functional differential systems with infinite delay in Banach space. Fu and Mei [5] discussed the approximate controllability of semilinear partial functional differential systems by using the α -norm and resolvent operator theory. Approximate controllability of semilinear control systems have been discussed by many authors [6–9]. Naito [10] showed that under a range condition on the control action operator the semilinear control system is approximate controllable. Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay have been studied in [11]. Recently, Subalakshmi and Balachandran [12] discussed the approximate controllability of nonlinear impulsive integrodifferential systems in Hilbert space. Li et.al [13] established sufficient conditions for the exact † Corresponding


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controllability of impulsive functional differential systems using Schaefer’s fixed point theorem. Based on the semigroup theory and fixed point approach, the approximate controllability of nonlinear impulsive differential systems have been studied by Sakthivel et.al [14]. Zhou [15] studied approximate controllability of an abstract semilinear control system by assuming certain inequality conditions that are dependent on the properties of the system components. Dauer and Mahmudov [6] studied the approximate controllability and complete controllability of the semilinear abstract control system by using the Schauder fixed point theorem and the resolvent condition. In [16], using the methods of functional analysis and semigroup theory, Fu and Zhang studied the approximate controllability of systems represented in the following semilinear neutral functional differential systems with state-dependent delay:   d [x(t) + F(t, x )] = −Ax(t) + Bu(t) + G(t, x t ρ (t,xt ) ), t ∈ [0, T ], dt (1)  x(0) = φ ∈ Bα ,

whereas the approximate controllability the second order semilinear stochastic syetem with nonlocal conditions is studied in [17]. Approximate controllability of nonlinear differential neutral and state dependent delay were studied in [18, 19]. The paper is organised as follows. In Section 2, we recall basic notation, some concepts and the results about the analytical semigroup. In Section 3, we investigate the approximate controllability of the impulsive integrodifferential equation. Finally, in Section 4, an example is provided to illustrate applications of the obtained results.

2 Preliminaries In this paper, we consider the approximate controllability of systems represented in the following semilinear impulsive neutral functional differential systems: ˆ t  d   h(t, s, xs )ds), [x(t) + F(t, xt )] = −Ax(t) + Bu(t) + G(t, xt ,   dt  0  t ∈ [0, T ] − t1 ,t2 , ...,tm , (2)   − − +  k = 1, 2, ..., m,  ∆x|t=tk = x(tk ) − x(tk ) = Ik (x(tk )),   x(0) = x0 − g(x),

where x(tk− ) and x(tk+ ) represent the left and right limits of x(t) at t = tk , respectively, 0 < t1 < t2 < ... < tm < T and Ik (k = 1, 2, ..., m) are given functions and the state variable x(·) takes values in a Hilbert space X and the control function u(·) is given in the Banach space L2 ([0, T ];U ) of admissible control functions, U is a Hilbert space. B is a bounded linear operator from U into X . The (unbounded) linear operators −A generates an analytic semigroup and F : [0, T ]×Cα → X , G : [0, T ]×Cα × X → X , h : [0, T ]× [0, T ]×Cα → X are appropriate functions to be specified later. φ ∈ Cα = PC([−r, 0], Xα ), Xα ⊂ X and for any function x(·) ∈ PC([0, T ], X ), the histories xt are defined by xt (θ ) = x(t + θ ) for θ ∈ [−r, 0]. Throughout this paper, the operator −A : D(A) → X is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators (S(t))t≥0 . Let 0 ∈ ρ (A). Then it is possible to define the fractional power Aα , 0 < α ≤ 1, as a closed linear operator on its domain D(Aα ). Furthermore, the subspace D(Aα ) is dense in X and the expression kxkα = kAα xk, x ∈ D(Aα ),

defines the norm on D(Aα ). Hereafter, we denote by Xα the Banach space D(Aα ) normed with kxkα . Then for each 0 < α ≤ 1, Xα is a Banach space, and Xα ֒→ Xβ for 0 < β < α ≤ 1 and the imbedding is compact whenever the resolvent operator of A is compact.

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For the semigroup (S(t))t≥0 , the following properties will be used: (a) there exist M1 ≥ 1 and ω ∈ R such that kS(t)k ≤ M1 eω t for all t ≥ 0; (b) for any 0 ≤ α ≤ 1, there exists a positive constant Cα such that kAα S(t)k ≤

Cα ω t e , t > 0. tα

In particular, one has that, for some M ≥ 1, Mα > 0, and t ∈ [0, T ], kS(t)k ≤ M, Mα kAα S(t))k ≤ α . t For the theory of operator semigroup, we refer the reader to [20]. Denote by Cα the Banach space of continuous functions C([−r, 0]; Xα ) with the norm

(3) (4)

kφ kCα = sup kAα φ (θ )k. −r≤θ ≤0

Definition 1. A function x(.; x0 , u) ∈ PC([−r, T ], Xα ) is said to be a mild solution to the Equation (1) with initial value φ ∈ Cα (under control u(t)) if on [−r, T ] it satisfies ˆ t    AS(t − s)F(s, xs )ds S(t)[x0 − g(x) + F(0, x0 )] − F(t, xt ) +    0  ˆ s ˆ t  (5) x(t) = h(s, τ , xτ )d τ )]ds S(t − s)[Bu(s) + G(s, xs , +   0 0     t ∈ [0, T ], + ∑ S(t − tk )Ik (x(tk− )),  0 0 such that kF(t1 , x) − F(t2 , y)kα +β ≤ L(|t1 − t2 | + kx − ykCα ),

(7)

for any 0 ≤ t1 ,t2 ≤ T an x, y ∈ Cα . Moreover, there exists a positive function f (·) ∈ L1 (R) such that the inequality sup kF(t, φ )kα +β ≤ f (ρ ) kφ kCα ≤ ρ

holds for any φ ∈ Cα .

(H3 ) All Ik ∈ PC(Xα , Xα ), k = 1, . . . m, are bounded, i.e., there exist constants dk , k = 1, . . . , m such that kIk (x)kα ≤ dk , x ∈ Xα . (H4 ) kh(t, s, x)k ≤ m0 (t, s)Ω0 (kxkBh ) for almost t ∈ J and all x ∈ Bh ,where m0 ∈ L1 (J, R+ ) and Ω0 : R+ ×R+ → (0, ∞) is a continuous function. (H5 ) kG(t, x, y)k ≤ m1 (t)Ω1 (kxkBh + kykBh ) for almost all t ∈ J and all x ∈ Bh where m1 ∈ L1 (J, R+ ) and Ω1 : R+ → (0, ∞) is continuous function. For any given xT ∈ X and λ ∈ (0, 1], we take the control function uλ (t), simply denoted by u(t), as follows: u(t) = B∗ S∗ (T − s)R(λ , ΓT ){xT − S(T )[x0 − g(x) + F(0, x0 )] + F(T, xT ) ˆ T ˆ T ˆ s − AS(T − τ )F(τ , xτ )d τ − S(T − τ )G(τ , xτ , h(s, τ , xτ )d τ )d τ 0

−

0

∑

0