Research Article Asymptotic Behavior of Positive ...

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 389638, 9 pages http://dx.doi.org/10.1155/2015/389638

Research Article Asymptotic Behavior of Positive Solutions of a Competitive System Subject to Environmental Noise Huili Xiang,1,2 Zhuang Fang,1 Zuxiong Li,1 and Zhijun Liu1,2 1

Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province, Hubei University for Nationalities, Enshi, Hubei 445000, China 2 Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China Correspondence should be addressed to Zhijun Liu; zhijun [email protected] Received 28 June 2014; Revised 20 September 2014; Accepted 20 September 2014 Academic Editor: Cengiz C ¸ inar Copyright © 2015 Huili Xiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A competitive system subject to environmental noise is established. By using the theory of stochastic differential equations and Lyapunov function, sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solution of the above system are established, respectively. An example together with its corresponding numerical simulations is presented to confirm our analytical results.

1. Introduction Mathematical modelling plays an important role in the mathematical ecology. In the past several years, ecological models based on determinate systems emerged in large numbers (see [1–9]). While the disturbance of environmental noise is unavoidable in the real world, more and more researchers start to pay attention to the study on nonlinear dynamic systems with environmental noise and many valuable results have been obtained (see [10–24]). In [25], Gopalsamy introduced the following competitive system: 𝑐 𝑥 (𝑡) ] 𝑑𝑡, 𝑑𝑥1 (𝑡) = 𝑥1 (𝑡) [𝑟1 − 𝑎1 𝑥1 (𝑡) − 2 2 1 + 𝑥2 (𝑡) 𝑐 𝑥 (𝑡) ] 𝑑𝑡, 𝑑𝑥2 (𝑡) = 𝑥2 (𝑡) [𝑟2 − 𝑎2 𝑥2 (𝑡) − 1 1 1 + 𝑥1 (𝑡)

(1)

where 𝑥𝑖 (𝑡) may represent the densities of species. The coefficients 𝑟𝑖 , 𝑎𝑖 , 𝑏𝑖 , and 𝑐𝑖 are all positive constants. In the absence of interspecific interactions, each species is governed by the logistic equation; however, in the presence of interspecific interactions, each species retains the average growth rate of

the other. In this contribution, we consider the influence of environmental noise and obtain the following form: 𝑑𝑥1 (𝑡) = 𝑥1 (𝑡) [𝑟1 − 𝑎1 𝑥1 (𝑡) −

𝑐2 𝑥2 (𝑡) ] 𝑑𝑡 1 + 𝑥2 (𝑡)

+ 𝜎1 𝑥1 (𝑡) 𝑑𝑤1 (𝑡) , 𝑐 𝑥 (𝑡) 𝑑𝑥2 (𝑡) = 𝑥2 (𝑡) [𝑟2 − 𝑎2 𝑥2 (𝑡) − 1 1 ] 𝑑𝑡 1 + 𝑥1 (𝑡)

(2)

+ 𝜎2 𝑥2 (𝑡) 𝑑𝑤2 (𝑡) , where 𝑥𝑖 (0) > 0, 𝑑𝑤𝑖 (𝑡) is independent white noise with 𝑤𝑖 (0) = 0, 𝑡 ≥ 0, and 𝜎𝑖2 represents the intensity of the noise, 𝑖 = 1, 2. 𝑤𝑖 (𝑡) is standard Brownian motion defined on the complete probability space (Ω, F, {F𝑡 }𝑡≥0 , 𝑃) with a filtration {F𝑡 }𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all 𝑃-null sets). In this paper, we focus on the asymptotic behavior of positive solution of system (2). To the best of our knowledge, there are few published papers concerning system (2). The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. The existence, uniqueness, and stochastic boundedness of positive solution of system (2) are

2

Discrete Dynamics in Nature and Society

discussed in Section 3. The global attraction of system (2) is studied in Section 4. As an application of our main results, we present an example and its numerical simulations to support our theoretical results in Section 5.

3. Existence, Uniqueness, Stochastic Boundedness, and Extinction

2. Preliminaries

Theorem 7. System (2) has a unique positive solution, say (𝑥1 (𝑡), 𝑥2 (𝑡)), on 𝑡 ≥ 0. Furthermore, the solution will remain in 𝑅+2 = {(𝑥1 , 𝑥2 ) ∈ 𝑅2 : 𝑥𝑖 > 0, 𝑖 = 1, 2} with probability one.

In this section, we introduce some definitions and lemmas which are useful for establishing our main results. Definition 1. The solution (𝑥1 (𝑡), 𝑥2 (𝑡)) of system (2) is stochastically bounded if, for any 𝜀𝑖 ∈ (0, 1), there exist positive constants 𝐻𝑖 = 𝐻(𝜀𝑖 ) such that 󵄨 󵄨 lim sup 𝑃 {󵄨󵄨󵄨𝑥𝑖 (𝑡)󵄨󵄨󵄨 > 𝐻𝑖 } < 𝜀𝑖 , 𝑖 = 1, 2. (3) 𝑡 → +∞ Definition 2. Let (𝑥1 (𝑡), 𝑥2 (𝑡)) be a positive solution of system (2). If another positive solution (𝑥1∗ (𝑡), 𝑥2∗ (𝑡)) of system (2) satisfies 󵄨 󵄨 lim 𝐸 (󵄨󵄨󵄨(𝑥1 (𝑡) , 𝑥2 (𝑡)) − (𝑥1∗ (𝑡) , 𝑥2∗ (𝑡))󵄨󵄨󵄨) = 0, (4) 𝑡 → +∞

then (𝑥1 (𝑡), 𝑥2 (𝑡)) is global attractive. Definition 3. Let (𝑥1 (𝑡), 𝑥2 (𝑡)) be a positive solution of system (2). The solution 𝑥(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡)) to system (2) is said to be exponentially extinct with probability one if ln 𝑥𝑖 (𝑡) < 0, 𝑡 𝑡→∞

lim sup

a.s. 𝑖 = 1, 2.

(5)

Lemma 4 (𝐶𝑝 inequality). Suppose that 𝑎1 , 𝑎2 , . . . , 𝑎𝑛 are all real numbers; then for any positive real number 𝑝 we have 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨󵄨 󵄨󵄨𝑎1 + 𝑎2 + ⋅ ⋅ ⋅ + 𝑎𝑛 󵄨󵄨󵄨 ≤ 𝐶𝑝 (󵄨󵄨󵄨𝑎1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎2 󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 ) , (6) where 1, 0 < 𝑝 ≤ 1, 𝐶𝑝 = { 𝑝−1 𝑛 , 𝑝 > 1.

We first present the existence and uniqueness of positive solution of system (2).

Proof. The proof of this lemma is rather standard. It is obvious that the coefficients of system (2) are local Lipschitz continuous. Then, for any initial value (𝑥1 (0), 𝑥2 (0)) with 𝑥𝑖 (0) > 0, there exists a unique local solution (𝑥1 (𝑡), 𝑥2 (𝑡)), 𝑡 ∈ [0, 𝜏∗ ), where 𝜏∗ is the explosion time (see [10, 19]). Therefore, to prove that the local solution is also global, we only need to show that 𝜏∗ = +∞ a.s. Let 𝑛0 > 0 be sufficiently large so that every component of (𝑥1 (0), 𝑥2 (0)) lies in [1/𝑛0 , 𝑛0 ]. For each integer 𝑛 ≥ 𝑛0 , we define the stopping time as follows: 1 1 𝜏𝑛 = inf {𝑡 ∈ [0, 𝜏∗ ) : 𝑥1 (𝑡) ∉ ( , 𝑛) or 𝑥2 (𝑡) ∉ ( , 𝑛)} . 𝑛 𝑛 (10) Here we set inf 0 = +∞ (0 denotes the empty set). Obviously, 𝜏𝑛 is increasing as 𝑛 → +∞. Denote 𝜏+∞ = lim𝑛 → +∞ 𝜏𝑛 , whence 𝜏+∞ ≤ 𝜏∗ a.s. We need to show that 𝜏+∞ = +∞ a.s. Otherwise, there exist constants 𝑇 > 0 and 𝜀 ∈ (0, 1) such that 𝑃{𝜏+∞ ≤ 𝑇} > 𝜀. Then, by denoting Ω𝑛 = {𝜏𝑛 ≤ 𝑇}, there exists an integer 𝑛1 ≥ 𝑛0 such that for all 𝑛 ≥ 𝑛1 , 𝑃 (Ω𝑛 ) ≥ 𝜀. We now define a 𝐶2 -function 𝑉 as 𝑉 (𝑥1 , 𝑥2 ) = [𝑥1 − ln 𝑥1 − 1] + [𝑥2 − ln 𝑥2 − 1] ,

(7)

Lemma 5 (see [26]). Let 𝑓(𝑡) be a nonnegative integrable and uniformly continuous function defined on [0, +∞) such that 𝑓(𝑡) is integrable and uniformly continuous on [0, +∞). Then lim𝑡 → +∞ 𝑓(𝑡) = 0. Lemma 6 (see [27, 28]). Suppose that a stochastic process 𝑋(𝑡) on 𝑡 ≥ 0 satisfies the condition 󵄨 󵄨 (8) 𝐸 󵄨󵄨󵄨𝑋 (𝑡) − 𝑋 (𝑠) |𝛼 ≤ 𝑐󵄨󵄨󵄨 𝑡 − 𝑠|1+𝛽 , 0 ≤ 𝑠, 𝑡 < +∞, for some positive constants 𝛼, 𝛽, and 𝑐. Then there exists a ̃ of 𝑋(𝑡), which has the property continuous modification 𝑋(𝑡) that, for every 𝛾 ∈ (0, 𝛽/𝛼), there is a positive random variable ℎ(𝜔) such that 󵄨󵄨 ̃ ̃ (𝑠, 𝜔)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑋 (𝑡, 𝜔) − 𝑋 2 󵄨≥ 𝑃 {𝜔 : sup } 𝛾 1 − 2−𝛾 |𝑡 − 𝑠| 0 0, taking expectations one shows that

≤ 𝑉 (𝑥1 (0) , 𝑥2 (0)) + 𝐻𝐸 (𝜏𝑛 ∧ 𝑇) .

(𝑡) 𝑥𝑖 (𝑡) [𝑟𝑖 − 𝑎𝑖 𝑥𝑖 (𝑡) −

+ 𝑝𝜎𝑖 𝑥𝑖 (𝑡) 𝑑𝑤𝑖 (𝑡) .

+ (1 − 𝑥2 (𝑡)−1 ) 𝜎2 𝑥2 (𝑡) 𝑑𝑤2 (𝑡)] . (17)

𝐸 (𝑉 (𝑥1 (𝜏𝑛 ∧ 𝑇) , 𝑥2 (𝜏𝑛 ∧ 𝑇)))

𝑝 (𝑝 − 1) 𝑝−1 𝑥𝑖 (𝑡) 𝑑𝑥𝑖 (𝑡) 𝑑𝑥𝑖 (𝑡) 2

(𝑡) 𝑑𝑥𝑖 (𝑡) +

= {𝑝𝑥𝑖

As a consequence, one has

(23)

Proof. By Itô’s formula, one can show that

−1

+ (1 − 𝑥2 (𝑡) ) 𝜎2 𝑥2 (𝑡) 𝑑𝑤2 (𝑡)]

𝑖 = 1, 2.

𝑡

1 + 𝑥𝑗 (𝑡)

(𝑝 − 1) 2 𝜎𝑖 ]} 𝑑𝑠 2

(25)

𝑝

+ ∫ 𝑝𝜎𝑖 𝑥𝑖 (𝑡) 𝑑𝑤𝑖 (𝑡) . 0

Taking expectations, we obtain that 𝑝

𝑝

𝐸 (𝑥𝑖 (𝑡)) − 𝐸 (𝑥𝑖 (0)) 𝑡

𝑝

= ∫ 𝑝𝐸 {𝑥𝑖 (𝑠) [𝑟𝑖 − 𝑎𝑖 𝑥𝑖 (𝑠) − (20)

𝑐𝑗 𝑥𝑗 (𝑡)

0

+

𝑐𝑗 𝑥𝑗 (𝑠) 1 + 𝑥𝑗 (𝑠)

(𝑝 − 1) 2 𝜎𝑖 ]} 𝑑𝑠. 2

(26)

4


So 𝑝 𝑑𝐸 (𝑥𝑖

On the other hand, by the Chebyshev inequality and Lemma 8, we obtain that 󵄨 󵄨 𝑃 {󵄨󵄨󵄨𝑥𝑖 (𝑡) − 𝐸 (𝑥𝑖 (𝑡))󵄨󵄨󵄨 > 𝛿𝑖 }

(𝑡))

𝑑𝑡 =

𝑝 𝑝𝐸 {𝑥𝑖

𝑐𝑗 𝑥𝑗 (𝑡)

(𝑝 − 1) 2 + 𝜎𝑖 ]} (𝑡) [𝑟𝑖 − 𝑎𝑖 𝑥𝑖 (𝑡) − 1 + 𝑥𝑗 (𝑡) 2 𝑝

𝑝+1

≤ 𝑝𝑟𝑖 𝐸 (𝑥𝑖 (𝑡)) − 𝑝𝑎𝑖 𝐸 (𝑥𝑖 =

𝑝 𝑝𝐸 (𝑥𝑖

(𝑡)) +

(𝑝 − 1) 2 𝑝 𝜎𝑖 𝐸 (𝑥𝑖 (𝑡)) 2

(𝑝 − 1) 2 1/𝑝 𝑝 𝜎𝑖 ] − 𝑎𝑖 [𝐸 (𝑥𝑖 (𝑡))] } . (𝑡)) {[𝑟𝑖 + 2 (27)

2

(36)

𝑟𝑖 + (1/2) 𝜎𝑖2 } ≤ 𝜀𝑖 . √𝜀𝑖 𝑎𝑖

(37)

2 2 Var (𝑥𝑖 (𝑡)) 𝐸 (𝑥𝑖 (𝑡)) (𝑟𝑖 + (1/2) 𝜎𝑖 ) ≤ ≤ ≤ . 𝛿𝑖2 𝛿𝑖2 𝑎𝑖2 𝛿𝑖2

Let 𝛿𝑖 = (𝑟𝑖 + (1/2)𝜎𝑖2 )/√𝜀𝑖 𝑎𝑖 . Then we have 𝑃 {𝑥𝑖 (𝑡) ≥ 𝐸 (𝑥𝑖 (𝑡)) + It follows from Lemma 8 that 𝐸 (𝑥𝑖 (𝑡)) ≤

Let 𝑍𝑖 (𝑡) =

𝑝 𝐸 (𝑥𝑖

(𝑡)) ;

(28)

we have (𝑝 − 1) 2 𝑑𝑍𝑖 (𝑡) 1/𝑝 ≤ 𝑝𝑍𝑖 (𝑡) [𝑟𝑖 + 𝜎𝑖 − 𝑎𝑖 (𝑍𝑖 (𝑡)) ] . 𝑑𝑡 2 As a consequence (𝑝 − 1) 2 𝑑𝑍𝑖 (𝑡) 1/𝑝 ≤ 𝑝 [𝑟𝑖 + 𝜎𝑖 − 𝑎𝑖 (𝑍𝑖 (𝑡)) ] 𝑑𝑡. 𝑍𝑖 (𝑡) 2

(30)

Noting that 𝑥𝑖 (0) < (𝑟𝑖 + (1/2)(𝑝 − 1)𝜎𝑖2 )/𝑎𝑖 , 𝑖 = 1, 2, we have 1/𝑝

0 < 𝑎𝑖 (𝑍𝑖 (0))

= 𝑎𝑖 𝑥𝑖 (0) < 𝑟𝑖 +

1 (𝑝 − 1) 𝜎𝑖2 . 2

𝑝 1/𝑝

1/𝑝

= (𝑍𝑖 (𝑡))

≤

Then we can obtain that 𝑝

(33)

where 𝑝

𝑟 + ((𝑝 − 1) /2) 𝜎𝑖2 ] . 𝐾𝑖 (𝑝) = [ 𝑖 𝑎𝑖

𝑟 + (1/2) 𝜎𝑖2 ≤ 𝑃 {𝑥𝑖 (𝑡) ≥ 𝐸 (𝑥𝑖 (𝑡)) + 𝑖 } ≤ 𝜀𝑖 . √𝜀𝑖 𝑎𝑖

(39)

Let 𝐻𝑖 (𝜀𝑖 ) = ((√𝜀𝑖 + 1)𝑟𝑖 + (1/2)𝜎𝑖2 )/𝑎𝑖 √𝜀𝑖 , and noting that 𝑥𝑖 (𝑡) > 0, one shows that 󵄨 󵄨 𝑃 {󵄨󵄨󵄨𝑥𝑖 (𝑡)󵄨󵄨󵄨 ≥ 𝐻𝑖 (𝜀𝑖 )} < 𝜀𝑖 . (40) 󵄨 󵄨 lim sup 𝑃 {󵄨󵄨󵄨𝑥𝑖 (𝑡)󵄨󵄨󵄨 ≥ 𝐻𝑖 (𝜀𝑖 )} < 𝜀𝑖 ,

𝑡→∞

𝑟𝑖 + (1/2) (𝑝 − 1) 𝜎𝑖2 . (32) 𝑎𝑖

𝐸 (𝑥𝑖 (𝑡)) ≤ 𝐾𝑖 (𝑝) ,

𝑟𝑖 𝑟𝑖 + (1/2) 𝜎𝑖2 + } 𝑎𝑖 √𝜀𝑖 𝑎𝑖

Therefore, (31)

Furthermore, using the standard comparison principle, one can show that [𝐸(𝑥𝑖 (𝑡)) ]

(38)

which, together with (37), leads to 𝑃 {𝑥𝑖 (𝑡) ≥

(29)

𝑟𝑖 , 𝑎𝑖

(34)

This completes the proof. Finally, we discuss the stochastic boundedness of the positive solutions of system (2). Theorem 9. If 𝑥𝑖 (0) < 𝑟𝑖 /𝑎𝑖 , 𝑖 = 1, 2, then the solution (𝑥1 (𝑡), 𝑥2 (𝑡)) of system (2) is stochastically bounded. Proof. On one hand, for any positive number 𝛿𝑖 , 𝑖 = 1, 2, one derives that 󵄨 󵄨 𝑃 {󵄨󵄨󵄨𝑥𝑖 (𝑡) − 𝐸 (𝑥𝑖 (𝑡))󵄨󵄨󵄨 ≥ 𝛿𝑖 } ≥ 𝑃 {𝑥𝑖 (𝑡) ≥ 𝐸 (𝑥𝑖 (𝑡)) + 𝛿𝑖 } . (35)

(41)

which implies that the solution of system (2) is stochastically bounded. The proof is complete. Theorem 10. Suppose that all coefficients of system (2) are positive and 𝑟𝑖 < 𝜎𝑖2 /2, 𝑖 = 1, 2. Then the solution (𝑥1 (𝑡), 𝑥2 (𝑡)) of system (2) is exponentially extinct with probability one. Proof. Define, respectively, Lyapunov functions ln 𝑥1 (𝑡) and ln 𝑥2 (𝑡). Then the following conclusions can be obtained by Itô’s formula 𝑐 𝑥 (𝑡) 1 ] 𝑑𝑡 𝑑 (ln 𝑥1 (𝑡)) = [𝑟1 − 𝜎12 − 𝑎1 𝑥1 (𝑡) − 2 2 2 1 + 𝑥2 (𝑡) + 𝜎1 𝑑𝑤1 (𝑡) , 𝑐 𝑥 (𝑡) 1 ] 𝑑𝑡 𝑑 (ln 𝑥2 (𝑡)) = [𝑟2 − 𝜎22 − 𝑎2 𝑥1 (𝑡) − 1 1 2 1 + 𝑥1 (𝑡)

(42)

+ 𝜎2 𝑑𝑤2 (𝑡) . Integrating from 0 to 𝑡, one concludes that 𝑡 1 ln 𝑥1 (𝑡) ≤ ln 𝑥1 (0) + (𝑟1 − 𝜎12 ) 𝑡 + ∫ 𝜎1 𝑑𝑤1 (𝑠) , 2 0 𝑡 1 ln 𝑥2 (𝑡) ≤ ln 𝑥2 (0) + (𝑟2 − 𝜎22 ) 𝑡 + ∫ 𝜎2 𝑑𝑤2 (𝑠) . 2 0

(43)


5

Dividing 𝑡 on both sides of (43), sending 𝑡 → ∞, and employing the strong law of large numbers for local martingales, one acquires that lim sup

𝑡→∞

ln 𝑥𝑖 (𝑡) < 0, 𝑡

a.s. 𝑖 = 1, 2.

Without loss of generality, we assume that 𝑝 > 2. Using the moment inequality (see [10]) to stochastic integral (45), we can obtain that 󵄨󵄨𝑝 󵄨󵄨 𝑡2 󵄨 󵄨 𝐸󵄨󵄨󵄨󵄨∫ ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑤1 (𝑠)󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑡1

(44)

This completes the proof.

⩽[

4. Global Attraction In this section, we first introduce Lemma 11 before we show the global attraction of system (2).

𝑝 (𝑝 − 1) ] 2

𝑝/2

𝑝/2

(𝑡2 − 𝑡1 )

(48)

𝑡

2 󵄨 󵄨𝑝 × ∫ 𝐸󵄨󵄨󵄨ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 𝑑𝑠, 𝑡 1

Lemma 11. If 𝑐𝑖 ≤ 𝑎𝑖 , 𝑥𝑖 (0) < 𝑟𝑖 /𝑎𝑖 , 𝑖 = 1, 2, then almost every sample path of the solution (𝑥1 (𝑡), 𝑥2 (𝑡)) of system (2) is uniformly continuous on 𝑡 ≥ 0.

where 0 ⩽ 𝑡1 < 𝑡2 < +∞ and 𝑝 > 2. We further let 1 1 + = 1; 𝑝 𝑞

𝑡2 − 𝑡1 < 1,

Proof. It follows from system (2) that 𝑡

𝑥1 (𝑡) = 𝑥1 (0) + ∫ 𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑠 0

(45)

𝑡

+ ∫ ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑤1 (𝑠) ,

then by (47), (48), and Lemma 4, one yields that 󵄨 󵄨𝑝 𝐸󵄨󵄨󵄨𝑥1 (𝑡2 ) − 𝑥1 (𝑡1 )󵄨󵄨󵄨

0

where 𝑐 𝑥 (𝑠) ], 𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) = 𝑥1 (𝑠) [𝑟1 − 𝑎1 𝑥1 (𝑠) − 2 2 1 + 𝑥2 (𝑠)

𝑡2

= 𝐸 [∫ 𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑠 𝑡1

𝑡1

(46) Applying Lemmas 4 and 8, for any 𝑝 > 1, one derives that 󵄨 󵄨𝑝 𝐸 (󵄨󵄨󵄨𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 ) 󵄨󵄨 𝑐 𝑥 (𝑠) 󵄨󵄨󵄨󵄨𝑝 𝑝 󵄨 = 𝐸 (𝑥1 (𝑠) 󵄨󵄨󵄨𝑟1 − 𝑎1 𝑥1 (𝑠) − 2 2 󵄨 ) 1 + 𝑥2 (𝑠) 󵄨󵄨󵄨 󵄨󵄨 𝑐 𝑥 (𝑠) 2𝑝 1 1 2𝑝 ≤ 𝐸 (𝑥1 (𝑠)) + 𝐸 ((𝑟1 − 𝑎1 𝑥1 (𝑠) − 2 2 ) ) 2 2 1 + 𝑥2 (𝑠) 1 1 2𝑝 󵄨2𝑝 󵄨 󵄨2𝑝 󵄨 ≤ 𝐸 (𝑥1 (𝑠)) + 𝐸 [32𝑝−1 󵄨󵄨󵄨𝑟1 󵄨󵄨󵄨 + 32𝑝−1 󵄨󵄨󵄨𝑎1 𝑥1 (𝑠)󵄨󵄨󵄨 2 2 󵄨2𝑝 󵄨 +32𝑝−1 󵄨󵄨󵄨𝑐2 𝑥2 (𝑠)󵄨󵄨󵄨 ]

𝑝

𝑡2

≤ 𝐸 [2𝑝−1 (∫ 𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑠) 𝑡1

𝑝

𝑡2

+ 2𝑝−1 (∫ ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑤1 (𝑠)) ] 𝑡1

𝑡

2 󵄨 󵄨𝑝 ≤ 2𝑝−1 𝐸 (∫ 󵄨󵄨󵄨𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 𝑑𝑠) 𝑡 1

𝑡

2 󵄨 󵄨𝑝 + 2𝑝−1 𝐸 (∫ 󵄨󵄨󵄨ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 𝑑𝑤1 (𝑠)) 𝑡 1

𝑡2

𝑝/𝑞

≤ 2𝑝−1 (∫ 1𝑞 𝑑𝑠)

1 1 2𝑝 2𝑝 󵄨 󵄨2𝑝 1 󵄨 󵄨2𝑝 = 𝐸 (𝑥1 (𝑠)) + 32𝑝−1 󵄨󵄨󵄨𝑟1 󵄨󵄨󵄨 + 32𝑝−1 󵄨󵄨󵄨𝑎1 󵄨󵄨󵄨 𝐸 (𝑥1 (𝑠)) 2 2 2 1 2𝑝 + 32𝑝−1 |𝑐2 |2𝑝−1 𝐸 (𝑥2 (𝑠)) 2

𝑡1

+ 2𝑝−1 [

𝑡

2 󵄨 󵄨𝑝 𝐸 (∫ 󵄨󵄨󵄨𝑔1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 𝑑𝑠) 𝑡 1

𝑝 (𝑝 − 1) ] 2

𝑝/2

(𝑝−2)/2

(𝑡2 − 𝑡1 )

𝑡

2 󵄨 󵄨𝑝 × ∫ 𝐸󵄨󵄨󵄨ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 𝑑𝑠 𝑡

1 1 󵄨 󵄨2𝑝 1 󵄨 󵄨2𝑝 ≤ 𝐾1 (2𝑝) + 32𝑝−1 󵄨󵄨󵄨𝑟1 󵄨󵄨󵄨 + 32𝑝−1 󵄨󵄨󵄨𝑎1 󵄨󵄨󵄨 𝐾1 (2𝑝) 2 2 2

1

𝑡2

𝑝/𝑞

≤ 2𝑝−1 (∫ 1𝑞 𝑑𝑠)

1 󵄨 󵄨2𝑝−1 + 32𝑝−1 󵄨󵄨󵄨𝑐2 󵄨󵄨󵄨 𝐾2 (2𝑝) ≜ 𝐿 1 (𝑝) , 2 󵄨𝑝 󵄨 𝐸 (󵄨󵄨󵄨ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠))󵄨󵄨󵄨 ) 𝑝

𝑝

𝑡2

+ ∫ ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) 𝑑𝑤1 (𝑠)]

ℎ1 (𝑠, 𝑥1 (𝑠) , 𝑥2 (𝑠)) = 𝜎1 𝑥1 (𝑠) .

𝑝 𝑝

(49)

𝑡1

+ 2𝑝−1 [

𝑝

= 𝐸 (𝜎1 𝑥1 (𝑠)) ≤ 𝜎1 𝐸 (𝑥1 (𝑠)) ≜ 𝑀1 (𝑝) . (47)

𝑡2

𝐸 (∫ 𝐿 1 (𝑝) 𝑑𝑠)

𝑝 (𝑝 − 1) ] 2

𝑡1

𝑝/2

= 2𝑝−1 𝐿 1 (𝑝) (𝑡2 − 𝑡1 )

(𝑝−2)/2

(𝑡2 − 𝑡1 ) 𝑝/2

𝑡2

∫ 𝑀1 (𝑝) 𝑑𝑠 𝑡1

6

Discrete Dynamics in Nature and Society 𝑝−1

+2

𝑝 (𝑝 − 1) [ ] 2 (𝑝−2)/2

≤ 2𝑝−1 (𝑡2 − 𝑡1 )

𝑝/2

(𝑝−2)/2

(𝑡2 − 𝑡1 )

[𝐿 1 (𝑝) + [

+ sgn (𝑥2 (𝑡) − 𝑥2∗ (𝑡))

𝑀1 (𝑝)

𝑝 (𝑝 − 1) ] 2

× {[(𝑟2 − 𝑎2 𝑥2 (𝑡) −

𝑝/2

𝑀1 (𝑝)] . + 𝜎2 𝑑𝑤2 (𝑡) ]

(50) It follows from Lemma 6 that almost every sample path of 𝑥1 (𝑡) is uniformly continuous on 𝑡 ≥ 0. Similarly, we can show that almost every sample path of 𝑥2 (𝑡) is uniformly continuous on 𝑡 ≥ 0. Therefore, (𝑥1 (𝑡), 𝑥2 (𝑡)) is uniformly continuous on 𝑡 ≥ 0, a.s. This completes the proof.

− [(𝑟2 − 𝑎2 𝑥2∗ (𝑡) −

= sgn (𝑥1 (𝑡) − 𝑥1∗ (𝑡))

Theorem 12. If 𝑐𝑖 ≤ 𝑎𝑖 , 𝑥𝑖 (0) < 𝑟𝑖 /𝑎𝑖 , 𝑖 = 1, 2, then system (2) has a unique global attractive positive solution, denoted by (𝑥1 (𝑡), 𝑥2 (𝑡)), on 𝑡 ≥ 0.

× [𝑎1 (𝑥1∗ (𝑡) − 𝑥1 (𝑡)) + 𝑐2 (

Proof. It follows from Theorem 7 that system (2) has a unique positive solution (𝑥1 (𝑡), 𝑥2 (𝑡)). Assume that (𝑥1∗ (𝑡), 𝑥2∗ (𝑡)) is another positive solution of system (2). Consider a Lyapunov function 𝑉(𝑡) defined by

× [𝑎2 (𝑥2∗ (𝑡) − 𝑥2 (𝑡)) + 𝑐1 (

Applying Itô’s formula, a calculation of the right differential 𝐷+ 𝑉(𝑡) of 𝑉(𝑡) along the solution, one yields that 𝐷 𝑉 (𝑡) = sgn (𝑥1 (𝑡) − 𝑥1∗ (𝑡)) {[

𝑑𝑥1 (𝑡) (𝑑𝑥1 (𝑡)) − ] 𝑥1 (𝑡) 2𝑥12 (𝑡)

(52)

𝐸 (𝑉 (𝑡) − 𝑉 (0)) 2

+ sgn (𝑥2 (𝑡) −

𝑡

= 𝐸 {∫ sgn (𝑥1 (𝑠) − 𝑥1∗ (𝑠)) 0

× [𝑎1 (𝑥1∗ (𝑠) − 𝑥1 (𝑠))

2

𝑑𝑥 (𝑡) (𝑑𝑥2 (𝑡)) − ] (𝑡)) {[ 2 𝑥2 (𝑡) 2𝑥22 (𝑡)

+ 𝑐2 (

2

−[

𝑑𝑥2∗ (𝑡) (𝑑𝑥2∗ (𝑡)) − ]} 𝑥2∗ (𝑡) 2𝑥2∗ (𝑡)2

𝑡

0

× [𝑎2 (𝑥2∗ (𝑠) − 𝑥2 (𝑠))

𝜎2 𝑐 𝑥 (𝑡) − 1 ) 𝑑𝑡 × {[(𝑟1 − 𝑎1 𝑥1 (𝑡) − 2 2 1 + 𝑥2 (𝑡) 2

+ 𝑐1 (

+ 𝜎1 𝑑𝑤1 (𝑡) ] − [(𝑟1 −

𝜎2 𝑐 𝑥∗ (𝑡) − 1 ) 𝑑𝑡 (𝑡) − 2 2 ∗ 1 + 𝑥2 (𝑡) 2

+ 𝜎1 𝑑𝑤1 (𝑡) ] }

𝑥2∗ (𝑠) 𝑥2 (𝑠) − )] 𝑑𝑠 1 + 𝑥2∗ (𝑠) 1 + 𝑥2 (𝑠)

+ ∫ sgn (𝑥2 (𝑠) − 𝑥2∗ (𝑠))

= sgn (𝑥1 (𝑡) − 𝑥1∗ (𝑡))

𝑎1 𝑥1∗

𝑥1∗ (𝑡) 𝑥1 (𝑡) − )] 𝑑𝑡. 1 + 𝑥1∗ (𝑡) 1 + 𝑥1 (𝑡)

Integrating from 0 to 𝑡 and taking expectations one can show that

2

𝑑𝑥∗ (𝑡) (𝑑𝑥1∗ (𝑡)) − ]} − [ ∗1 𝑥1 (𝑡) 2𝑥1∗ (𝑡)2 𝑥2∗

𝑥2∗ (𝑡) 𝑥2 (𝑡) − )] 𝑑𝑡 1 + 𝑥2∗ (𝑡) 1 + 𝑥2 (𝑡)

+ sgn (𝑥2 (𝑡) − 𝑥2∗ (𝑡))

𝑡 ≥ 0. (51)

+

𝜎2 𝑐1 𝑥1∗ (𝑡) − 2 ) 𝑑𝑡 ∗ 1 + 𝑥1 (𝑡) 2

+ 𝜎2 𝑑𝑤2 (𝑡) ]}

We can now present the result on global attraction of system (2).

󵄨 󵄨 󵄨 󵄨 𝑉 (𝑡) = 󵄨󵄨󵄨ln 𝑥1 (𝑡) − ln 𝑥1∗ (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨ln 𝑥2 (𝑡) − ln 𝑥2∗ (𝑡)󵄨󵄨󵄨 ,

𝜎2 𝑐1 𝑥1 (𝑡) − 2 ) 𝑑𝑡 1 + 𝑥1 (𝑡) 2

𝑥1∗ (𝑠) 𝑥1 (𝑠) − )] 𝑑𝑠} . ∗ 1 + 𝑥1 (𝑠) 1 + 𝑥1 (𝑠)

Thus 𝑑𝐸 (𝑉 (𝑡)) 𝑑𝑡 = 𝐸 { sgn (𝑥1 (𝑡) − 𝑥1∗ (𝑡))

(53)


7 1

× [𝑎1 (𝑥1∗ (𝑡) − 𝑥1 (𝑡))

0.9

𝑥∗ (𝑡) 𝑥2 (𝑡) + 𝑐2 ( 2 ∗ − )] 1 + 𝑥2 (𝑡) 1 + 𝑥2 (𝑡)

Population sizes

0.8

+ sgn (𝑥2 (𝑡) − 𝑥2∗ (𝑡)) ×

[𝑎2 (𝑥2∗

(𝑡) − 𝑥2 (𝑡))

𝑥1∗ (𝑡) 𝑥1 (𝑡) − )]} 1 + 𝑥1∗ (𝑡) 1 + 𝑥1 (𝑡) 󵄨 󵄨 󵄨 󵄨 ≤ −𝑎1 𝐸 (󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨) + 𝑐2 𝐸 (󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨) 󵄨 󵄨 󵄨 󵄨 − 𝑎2 𝐸 (󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨) + 𝑐1 𝐸 (󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨) 󵄨 󵄨 = (𝑐1 − 𝑎1 ) 𝐸 (󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨) 󵄨 󵄨 + (𝑐2 − 𝑎2 ) 𝐸 (󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨) ,

0.7 0.6 0.5 0.4

+ 𝑐1 (

0.3 0.2

0

20

40

60

80

100

Time t The sample path of x1 The sample path of x1∗

(54)

Figure 1: The sample paths of 𝑥1 (𝑡) and 𝑥1∗ (𝑡).

and hence integrating from 0 to 𝑡 one derives that mentioned in Higham [29], we can derive the following discrete version of system (2):

𝑡

󵄨 󵄨 𝐸 (𝑉 (𝑡)) ≤ ∫ (𝑐1 − 𝑎1 ) 𝐸 (󵄨󵄨󵄨𝑥1 (𝑠) − 𝑥1∗ (𝑠)󵄨󵄨󵄨) 𝑑𝑠 0

𝑡

󵄨 󵄨 + ∫ (𝑐2 − 𝑎2 ) 𝐸 (󵄨󵄨󵄨𝑥2 (𝑠) − 𝑥2∗ (𝑠)󵄨󵄨󵄨) 𝑑𝑠 + 𝑉 (0) , 0

(55)

which implies that 󵄨 󵄨 𝐸 (󵄨󵄨󵄨(𝑥1 (𝑡) , 𝑥2 (𝑡)) − (𝑥1∗ (𝑡) , 𝑥2∗ (𝑡))󵄨󵄨󵄨) 󵄨 󵄨2 󵄨 󵄨2 1/2 = 𝐸 {[󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨 ] } 󵄨 󵄨 ≤ 𝐸 (󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨)

(56)

󵄨 󵄨 ≤ 𝐸 (󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨)

𝑥2 (𝑘 + 1) − 𝑥2 (𝑘)

(59)

𝑐1 𝑥1 (𝑘) ] Δ𝑡 1 + 𝑥1 (𝑘)

+ 𝜎2 𝑥2 (𝑘) √Δ𝑡𝑁2 (𝑘) 1 + 𝜎22 𝑥2 (𝑘) (𝑁22 (𝑘) − 1) Δ𝑡, 2 (57)

󵄨 󵄨 + 𝐸 (󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨) ∈ 𝐿1 [0, ∞) , which, together with Lemmas 5 and 11, leads to 󵄨 󵄨 lim 𝐸 (󵄨󵄨󵄨(𝑥1 (𝑡) , 𝑥2 (𝑡)) − (𝑥1∗ (𝑡) , 𝑥2∗ (𝑡))󵄨󵄨󵄨) = 0 𝑡→∞

𝑐2 𝑥2 (𝑘) ] Δ𝑡 1 + 𝑥2 (𝑘)

1 + 𝜎12 𝑥1 (𝑘) (𝑁12 (𝑘) − 1) Δ𝑡, 2

= 𝑥2 (𝑘) [𝑟2 − 𝑎2 𝑥2 (𝑘) −

󵄨 󵄨 𝐸 (󵄨󵄨󵄨(𝑥1 (𝑡) , 𝑥2 (𝑡)) − (𝑥1∗ (𝑡) , 𝑥2∗ (𝑡))󵄨󵄨󵄨) 󵄨 󵄨2 󵄨 󵄨2 1/2 = 𝐸 {[󵄨󵄨󵄨𝑥1 (𝑡) − 𝑥1∗ (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨 ] }

= 𝑥1 (𝑘) [𝑟1 − 𝑎1 𝑥1 (𝑘) − + 𝜎1 𝑥1 (𝑘) √Δ𝑡𝑁1 (𝑘)

󵄨 󵄨 + 𝐸 (󵄨󵄨󵄨𝑥2 (𝑡) − 𝑥2∗ (𝑡)󵄨󵄨󵄨) ∈ 𝐿1 [0, +∞) . So

𝑥1 (𝑘 + 1) − 𝑥1 (𝑘)

(58)

and hence (𝑥1 (𝑡), 𝑥2 (𝑡)) is global attractive on 𝑡 ≥ 0.

5. An Example In this section, we first give an example to verify the feasibilities of Theorems 9 and 12. Using the Milsten method

where 𝑁1 (𝑘) and 𝑁2 (𝑘) are Gaussian random variables which follow 𝑁(0, 1). Let us choose 𝑟1 = 0.5, 𝑎1 = 0.4, 𝑐1 = 0.1, 𝜎1 = 0.1, 𝑟2 = 0.6, 𝑎2 = 0.5, 𝑐2 = 0.3, 𝜎2 = 0.1, Δ𝑡 = 0.001, and (𝑥1 (0), 𝑥2 (0)) = (0.8, 1.1), (𝑥1∗ (0), 𝑥2∗ (0)) = (0.3, 0.4). A calculation shows that the conditions of Theorems 9 and 12 are satisfied. Figures 1 and 2 show that the positive solution of system (59) is stochastically bounded and global attractive on 𝑡 ≥ 0. Recalling the whole paper, we have derived sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solutions of system (2). However, there are still some limitations in our work which need to be improved. We only especially consider the white noise which is an idealized situation. In fact, the effect

8

Discrete Dynamics in Nature and Society 1.8 1.6

Population sizes

1.4 1.2 1 0.8 0.6 0.4

20

0

40

60

80

100

Time t The sample path of x2 The sample path of x2∗

Figure 2: The sample paths of 𝑥2 (𝑡) and 𝑥2∗ (𝑡).

of colorful noise on system (2) is more general in line with the actual situation, and we leave it for our future work.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 212111), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1323), and the Key Subject of Hubei Province (Mathematics, Forestry), and the Project of Education Department of Hubei Province (B20111909).

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