Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 574541, 16 pages http://dx.doi.org/10.1155/2013/574541
Research Article Optimal Control Policies of Pests for Hybrid Dynamical Systems Baolin Kang,1,2 Mingfeng He,1 and Bing Liu2 1 2
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China
Correspondence should be addressed to Mingfeng He;
[email protected] Received 12 May 2013; Accepted 29 July 2013 Academic Editor: Sanyi Tang Copyright © 2013 Baolin Kang 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. We improve the traditional integrated pest management (IPM) control strategies and formulate three specific management strategies, which can be described by hybrid dynamical systems. These strategies can not only effectively control pests but also reduce the abuse of pesticides and protect the natural enemies. The aim of this work is to study how the factors, such as natural enemies optimum choice in the two kinds of different pests, timings of natural enemy releases, dosages and timings of insecticide applications, and instantaneous killing rates of pesticides on both pests and natural enemies, can affect the success of IPM control programmes. The results indicate that the pests outbreak period or frequency largely depends on the optimal selective feeding of the natural enemy between one of the pests and the control tactics. Ultimately, we obtain the only pest 𝑥2 needs to be controlled below a certain threshold while not supervising pest 𝑥1 .
1. Introduction A pest is an insect which is detrimental to humans or human concerns (as agriculture or livestock production). In its broadest sense, a pest is a competitor of humanity. Often insects are regarded as pests as they cause damage to agriculture by feeding on crops or parasitizing livestock, such as codling moth on apples or boll weevil on cotton. An animal could also be a pest when it causes damage to a wild ecosystem or carries germs within human habitats. Examples of these include those organisms which are vector-borne human diseases, such as rats and fleas which carry the plague disease, mosquitoes which are vector-borne malaria, and ticks which carry Lyme disease. The most serious pests (in the order of economic importance) are insects. Pesticides are chemicals and other agents (e.g., beneficial microorganisms) that are used to control or protect other organisms from insect pests. To control these insect pests, farmers rely strongly on intervention with chemical pesticides, which remain a significant component of the cost of production and ecological problems from pesticide resistance in key pests. In order to address the issue, researchers are increasingly embracing more components of the integrated pest management (IPM) [1–6] systems approach that is always applied in ecology.
With the rise of interdisciplinary research, the mathematical ecology has also emerged and developed rapidly. A variety of mathematical methods can be used in ecological science. There have been numerous publications [7–15] over the last ten years using ecological mathematical model to research IPM strategy (spraying pesticides and introducing additional natural enemy into a pest-natural enemy system). When we study the dynamic property between the pest and its natural enemy (predator-prey), one of the most important components of the predator-prey relationship is the socalled functional responses. In [7–15], the Holling functional responses and Beddington-DeAngelis functional response are introduced. The Holling type extends the range of values of 𝑥 and 𝑦 over which the feeding term is realistic. However, in some situations, the increase of the feeding rate is not proportional to the increase of the predator density, as a result of mutual interference between predators, which decreases the efficiency of predation [16]. In addition, it is shown that most of the mathematical models on IPM include only one pest and one natural enemy. In [1], Finch and Collier’s study concerning IPM strategies in field vegetable crops focuses on two key pests, the cabbage root fly (Delia radicum) and the carrot fly (Psila rosae), the two major root feeding pests. Thus, we consider a continuous three-level food web model with
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Beddington-DeAngelis functional response, which consists of two competing pests (𝑥1 and 𝑥2 ) and a natural enemy (𝑦), and it can be represented as follows [17]: 𝑥 𝑐𝑥 𝑑𝑥1 = 𝑎1 𝑥1 (1 − 1 − 1 2 ) − 𝐹1 (𝑥1 , 𝑥2 , 𝑦) 𝑦, 𝑑𝑡 𝐾1 𝐾1 𝑥 𝑐𝑥 𝑑𝑥2 = 𝑎2 𝑥2 (1 − 2 − 2 1 ) − 𝐹2 (𝑥1 , 𝑥2 , 𝑦) 𝑦, 𝑑𝑡 𝐾2 𝐾2
(1)
𝑑𝑦 = 𝜔1 𝐹1 (𝑥1 , 𝑥2 , 𝑦) 𝑦 + 𝜔2 𝐹2 (𝑥1 , 𝑥2 , 𝑦) 𝑦 − 𝑑𝑦, 𝑑𝑡 𝑥𝑖 (0) ≥ 0,
𝑦 (0) ≥ 0,
𝑖 = 1, 2,
where 𝐹𝑖 (𝑥1 , 𝑥2 , 𝑦) = 𝛼𝑖 𝑥𝑖 /(𝑟 + 𝑥1 + 𝑏𝑥2 + 𝑐𝑦), 𝑖 = 1, 2. 𝑎𝑖 , 𝐾𝑖 , 𝜔𝑖 , 𝑐𝑖 , and 𝑑 are positive (𝑖 = 1, 2). The prey 𝑥𝑖 grows with intrinsic growth rate 𝑎𝑖 and carries capacity 𝐾𝑖 in the absence of the predator. The constant 𝑐𝑖 is the interspecific competition rate between the two prey species. The predator 𝑦 consumes the prey 𝑥𝑖 with a functional response of Beddington-DeAngelis type 𝐹𝑖 (𝑥1 , 𝑥2 , 𝑦) and contributes to its growth with rate 𝜔𝑖 𝐹𝑖 (𝑥1 , 𝑥2 , 𝑦). The constant 𝑑 is the death rate of the predator, and the term, 𝑐𝑦, measures the mutual interference between predators. The constants 𝛼𝑖 (𝑖 = 1, 2) show the predation capacity of predators, and 𝑟 is the saturating functional response parameters. The parameter 𝑏 is the predator’s relative preference on 𝑥2 with respect to 𝑥1 . There are numbers of diagnostic tools to detect the qualitative behavior of the system (1). We give a simple investigation of the dynamics of the system (1) through constructing a bifurcation diagram using the software XPPAUT [18]. To construct the bifurcation diagram, we use a numerical integration with varying a key parameter and keeping other parameters fixed; our choice of parameters is guided by two assumptions: first, the system has to be biologically feasible, and second, natural enemies have food preference phenomenon [19]; that is, 𝛼1 > 𝛼2 or 𝛼1 < 𝛼2 in this paper. The bifurcation diagram as a function of 𝛼1 in the range 0.3 < 𝛼1 < 0.7 is drawn in Figure 1. As shown in Figure 1, the thick black lines represent the stable limit cycle, the black solid curves represent the stable equilibrium state, and the black dashed curves represent the unstable equilibrium state. When 𝛼1 = 0.4601, system (1) undergoes a Hopf bifurcation, where the stable focus becomes unstable and a stable limit cycle emanates from it. When the parameter 𝛼1 increases further, system (1) undergoes a Hopf bifurcation again at 𝛼1 = 0.5644, where an unstable focus becomes stable and the other unstable focus is still unstable. When 𝛼1 = 0.5698, a transcritical bifurcation appears, where the stability of two focuses occurs in exchange. In Figure 2, we fixed the parameter 𝛼1 = 0.6000, and the other parameters are the same as those in Figure 1. As we can see in Figure 2, the pest 𝑥1 tends to be extinct eventually due to the impact of the food preference phenomenon of natural enemies before recurrence of the next generation of pests, while the pest 𝑥2 and natural enemies can coexist through stabilizing the boundary equilibrium. In order to manage pests through spraying pesticides and introducing additional natural enemies (IPM strategy),
we apply the impulsive differential equations (IDE) [20– 22] to integrate system (1). In [13], the authors constructed a predator-prey impulsive system to show the process of releasing natural enemies periodically and spraying pesticides twice at different fixed times in a period. The authors, obviously, avoid the side effects of pesticides on natural enemies existing in [7–9]. But only two pulses in a period cannot effectively control pests. And most of the research is single species of pests while few papers have discussed multispecies of pests. However, most real pests communities are more complex than the community previously analysed by them. In the present paper, we make the following improvements: (a) in the modeling, we introduce a onenatural enemy and two-pest model, and the natural enemy shows optimal foraging between pest 𝑥1 and pest 𝑥2 , which is a well-known behavior of many predators [23]; (b) in the control strategy, we can control two pests more selectively by controlling pulse frequencies appropriately.
2. Model Formulation and Auxiliary Lemmas Considering the previous factors, firstly, two models of different control strategies are discussed as follows. Case 1. We suppose pesticides are sprayed several times in a release period, and the kill rates of pesticides to pests (𝛿𝑘𝑖 , 𝑖 = 1, 2) and natural enemies (𝛿𝑘3 ) are different in different impulsive moments in a release period. That is, we consider the following model: 𝑥 𝑐𝑥 𝑑𝑥1 = 𝑎1 𝑥1 (1 − 1 − 1 2 ) 𝑑𝑡 𝐾1 𝐾1 −
𝛼1 𝑥1 𝑦 , 𝑟 + 𝑥1 + 𝑏𝑥2 + 𝑐𝑦
𝑥 𝑐𝑥 𝑑𝑥2 = 𝑎2 𝑥2 (1 − 2 − 2 1 ) 𝑑𝑡 𝐾2 𝐾2 −
𝛼2 𝑥2 𝑦 , 𝑟 + 𝑥1 + 𝑏𝑥2 + 𝑐𝑦
𝑑𝑦 𝜔1 𝛼1 𝑥1 𝑦 = 𝑑𝑡 𝑟 + 𝑥1 + 𝑏𝑥2 + 𝑐𝑦 +
𝜔2 𝛼2 𝑥2 𝑦 − 𝑑𝑦, 𝑟 + 𝑥1 + 𝑏𝑥2 + 𝑐𝑦 𝑡 ≠ 𝜏𝑛𝑘 ,
𝑡 ≠ 𝑛𝑇,
+ 𝑥1 (𝜏𝑛𝑘 ) = (1 − 𝛿𝑘1 ) 𝑥1 (𝜏𝑛𝑘 ) , + ) = (1 − 𝛿𝑘2 ) 𝑥2 (𝜏𝑛𝑘 ) , 𝑥2 (𝜏𝑛𝑘 + ) = (1 − 𝛿𝑘3 ) 𝑦 (𝜏𝑛𝑘 ) , 𝑦 (𝜏𝑛𝑘
𝑡 = 𝜏𝑛𝑘 , 𝑥1 (𝑛𝑇+ ) = 0, 𝑥2 (𝑛𝑇+ ) = 0,
Abstract and Applied Analysis
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Figure 1: Bifurcation diagram of system (1) for 0.3 < 𝛼1 < 0.7, where 𝑋 = 𝑥1 , 𝑌 = 𝑥2 , 𝑍 = 𝑦, and 𝐴1 = 𝛼1 and the initial value 𝑥1 = 1.3; 𝑥2 = 1.5; and 𝑦 = 5 with parameters as follows: 𝑎1 = 0.9, 𝑎2 = 0.8, 𝛼2 = 0.5, 𝑐1 = 0.55, 𝐾1 = 20, 𝑟 = 1.2, 𝑏 = 0.25, 𝑐 = 0.3, 𝛼2 = 0.5, 𝐾2 = 30, 𝑐2 = 0.55, 𝑊1 = 0.85, 𝑊2 = 0.75, and 𝑑 = 0.25.
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Figure 2: Time series and phase portrait of system (1) with 𝛼1 = 0.6000 and other parameters the same as Figure 1.
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Abstract and Applied Analysis 𝑦 (𝑛𝑇+ ) = 𝑦 (𝑛𝑇) + 𝜃, 𝑡 = 𝑛𝑇, (2)
where 𝑛𝑇 = 𝜏𝑛0 < 𝜏𝑛1 < 𝜏𝑛2 < ⋅ ⋅ ⋅ < 𝜏𝑛𝑞 < 𝜏𝑛(𝑞+1) = (𝑛 + 1)𝑇, 𝑛, 𝑞 ∈ 𝑍+ ; 𝑘 = 0, 1, 2, . . . , 𝑞; 𝜏𝑛(𝑘+1) − 𝜏𝑛𝑘 = 𝑇/(𝑞 + 1); 0 ≤ 𝛿𝑘𝑖 < 1 (𝑖 = 1, 2, 3) represents an effective kill rate to pest 𝑥1 , pest 𝑥2 , and natural enemy 𝑦 at time 𝑡 = 𝜏𝑛𝑘 , respectively. 𝜃 represents natural enemy number of additional release at 𝑡 = 𝑛𝑇; 𝑇 is a release period; 𝜏𝑛𝑘 is the 𝑘th spraying moment in the 𝑛th release period. Other parameters are the same as those in system (1). Case 2. We suppose that the natural enemies are released several times in a spraying period. The release number of natural enemies (𝜃𝑘 ) and an effective kill rate to natural enemies (𝜇𝑘 ) are different in different impulsive moments in a spraying period. That is, we consider the following model:
𝛼1 𝑥1 𝑦 , 𝑟 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + 𝑐𝑦
𝑚 (𝑡) ≤ (≥) 𝑝 (𝑡) 𝑚 (𝑡) + 𝑞 (𝑡) ,
𝑥 𝑐𝑥 𝑑𝑥2 = 𝑎2 𝑥2 (1 − 2 − 2 1 ) 𝑑𝑡 𝐾2 𝐾2 −
Lemma 1. Assume that x(𝑡) is a solution of system (2) with x(0+ ) ≥ 0, then x(𝑡) ≥ 0 for 𝑡 ≥ 0. Lemma 2 (see Lakshmikantham et al. [22]). Consider the following impulsive differential inequalities:
𝑥 𝑐𝑥 𝑑𝑥1 = 𝑎1 𝑥1 (1 − 1 − 1 2 ) 𝑑𝑡 𝐾1 𝐾1 −
𝜃𝑘 represents natural enemy number of additional release at 𝑡 = 𝜆 𝑛𝑘 ; 𝑇1 is a spraying period; 𝜆 𝑛𝑘 is the 𝑘th releasing moment in the spraying 𝑛th period. Other parameters are the same as those in system (1). Furthermore, some essential notations, definitions and lemmas are given as follows. Let 𝑅+ = [0, ∝) and 𝑅+3 ={x = (𝑥1 (𝑡), 𝑥2 (𝑡), 𝑦(𝑡) ∈ 𝑅3 : 𝑥1 (𝑡), 𝑥2 (𝑡), 𝑦(𝑡) ≥ 0)}. Denote f = (𝑓1 , 𝑓2 , 𝑓3 ) the map defined by the right-hand side of the first, second, and third equations of system (2). The solution of system (2), denoted by x : 𝑅+ → 𝑅+3 , is piecewise continuous, and it is continuous + )) = on ((𝑛 − 1)𝑇, 𝜏𝑛1 ], (𝜏𝑛1 , 𝜏𝑛2 ], . . .,(𝜏𝑛𝑞 , 𝑛𝑇]. Note that x((𝜏𝑛𝑘 lim𝑡 → 𝜏𝑛𝑘+ 𝑥(𝑡) and 𝑥(𝑛𝑇+ ) = lim𝑡 → 𝑛𝑇+ 𝑥(𝑡) exist. The existence and uniqueness of solutions of system (2) are guaranteed by the smoothness of f (see [22]).
𝑚 (𝑡𝑘+ ) ≤ (≥) 𝑑𝑘 𝑚 (𝑡𝑘 ) + 𝑏𝑘 ,
𝛼2 𝑥2 𝑦 , 𝑟 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + 𝑐𝑦
𝑡 ≠ 𝜆 𝑛𝑘 ,
𝑡 ≠ 𝑛𝑇1 ,
(4)
𝑡 = 𝑡𝑘 , 𝑘 ∈ 𝑁,
where 𝑝(𝑡), 𝑞(𝑡) ∈ 𝐶(𝑅+ , 𝑅), 𝑑𝑘 ≥ 0, and 𝑏𝑘 are constants. Assume that:
𝜔1 𝛼1 𝑥1 𝑦 𝑑𝑦 = 𝑑𝑡 𝑟 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + 𝑒𝑦 𝜔2 𝛼2 𝑥2 𝑦 + − 𝑑𝑦, 𝑟 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + 𝑒𝑦
𝑡 ≠ 𝑡𝑘 ,
(A0 ) the sequence {𝑡𝑘 } satisfies 0 ≤ 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ , with lim𝑡 → ∝ 𝑡𝑘 =∝; (3)
𝑥1 (𝜆+𝑛𝑘 ) = 0,
(A1 ) 𝑚(𝑡) ∈ 𝑃𝐶 (𝑅+ , 𝑅) and 𝑚(𝑡) is left continuous at 𝑡𝑘 , 𝑘 = 1, 2, . . .; (A2 ) for 𝑘 = 1, 2, . . . , 𝑡 ≥ 𝑡0 .
𝑥2 (𝜆+𝑛𝑘 ) = 0, 𝑦 (𝜆+𝑛𝑘 ) = (1 − 𝜇𝑘 ) 𝑦 (𝜆 𝑛𝑘 ) + 𝜃𝑘 ,
Then,
𝑡 = 𝜆 𝑛𝑘 , 𝑥1 (𝑛𝑇1+ ) = (1 − 𝛿11 ) 𝑥1 (𝑛𝑇1 ) , 𝑥2 (𝑛𝑇1+ )
= (1 −
𝑦 (𝑛𝑇1+ )
𝛿12 ) 𝑥2
= (1 −
𝑡
𝑚 (𝑡) ≤ (≥) 𝑚 (𝑡0 ) ∏ 𝑑𝑘 exp (∫ 𝑝 (𝑠) 𝑑𝑠)
(𝑛𝑇1 ) ,
𝛿13 ) 𝑦 (𝑛𝑇1 ) , 𝑡 = 𝑛𝑇1 ,
where 𝑛𝑇1 = 𝜆 𝑛0 < 𝜆 𝑛1 < 𝜆 𝑛2 < ⋅ ⋅ ⋅ < 𝜆 𝑛𝑝 < 𝜆 𝑛(𝑝+1) = (𝑛 + 1)𝑇1 , 𝑛, 𝑝 ∈ 𝑍+ ; 𝑘 = 0, 1, 2, . . . , 𝑝; 𝜆 𝑛(𝑘+1) − 𝜆 𝑛𝑘 = 𝑇1 /(𝑝 + 1); 0 ≤ 𝛿1𝑖 < 1 (𝑖 = 1, 2, 3) represents an effective kill rate to pest 𝑥1 , pest 𝑥2 and natural enemy 𝑦 at time 𝑡 = 𝜆 𝑛𝑘 , respectively. 𝜇𝑘 represents reduced proportion of natural enemies owing to the delay effect of pesticides and eating the deleterious pests;
𝑡0
𝑡0
