An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

Wang, Zhen; Shao, Yuanfu; Fang, Xianjia; Ma, Xiangmin

doi:https://doi.org/10.1155/2016/3897234

Discrete Dynamics in Nature and Society

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Abstract Introduction and Model Formulation Preliminaries Examples Conclusion Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 3897234 | https://doi.org/10.1155/2016/3897234

An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

Zhen Wang,¹Yuanfu Shao,¹Xianjia Fang,¹and Xiangmin Ma¹

Academic Editor: Ryusuke Kon

Received26 Jan 2016

Accepted27 Apr 2016

Published26 May 2016

Abstract

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies.

1. Introduction and Model Formulation

In real world, the study on models of three or more species is very popular, such as food-chain and food webs systems, which have extremely rich dynamics [1, 2]. For predator-prey model, in portrayal of the relationship between predator and prey, a crucial element is the classic definition of a predator’s functional response. In the past few decades, many different functional responses have been extensively investigated [3–7]. For example, Liu et al. [5] gave the following Holling type II functional response which describes the relations of one prey and one predator:where is the density of prey and is the density of predator at time . is the intrinsic growth rate of prey. represents the rate of intraspecific competition or density dependence. is the death rate of predator. is transformation rate for the predator to prey; denotes the Holling type II functional response.

However, in the actual ecosystem, if examining more complicated ecological case, some preys show herd behavior. That is to say, the predator interacts with the prey along the outer corridor of the herd of prey. Hence actual dynamic behaviors of individuals have not been described in detail by the predation term of Holling type II functional response. Ajraldi et al. [8] pointed out that, by using the terms of the square root of the prey population, the response functions of prey that exhibited herd behavior are more properly modeled. In this respect, Braza [9] gave the following predator-prey model: where is the square root functional response. Ma et al. [10] also investigated a predator-prey system with square root functional response. Their results showed that square root functional response brought about large influence to the dynamical behaviors.

On the other hand, few researchers consider the mutual interference between predators, but mutual interference between predators always exists in the actual ecosystem. In 1971, Hassell set about studying the capturing behavior between hosts and parasites; he discovered that hosts or parasites had the tendency to depart from each other when they met, which affected the hosts capturing. If the size of parasite became larger and larger, then the mutual interference would be stronger and stronger. Hence he introduced the mutual interference of predator [11]. Considering the effect from mutual interference between predators, the dynamic behaviors were more complex. For example, He et al. [12] and Zhang et al. [13] investigated the mutual interference of the predator in detail and obtained much different dynamics with those models without mutual interference. Hence, for predator-prey system, it is necessary to consider the mutual interference of predator.

Based on above discussion, we give the following prey-predator system with square root functional response and mutual interference of the predator:where (see [8] for the details).

As is known to all, insects have a profound impact on the survival and development of human beings. Most insects are beneficial to human beings; only a few insects are harmful to human life and agricultural development when they reach a certain amount. Hence it is necessary to kill the harmful pests or control them in a certain quantity. Chemical control and biological control are two most commonly used methods. Chemical control is often applied by spraying pesticides, which are used widely because they can kill pests quickly and reduce economic losses in a short time, but they also produce serious environmental pollution. For less pollution to the environment, by stocking or releasing natural enemies, biological control appears, but the effects are not very great. In order to combine different approaches to control pests at the same time, integrated pest management is given to maximize control efficiency and reduce pollution. During the last two decades, ecological pest control is a complex project [14, 15]. For predator-prey system, pest control strategy has been an important topic for many researchers [16, 17].

The main purpose of this paper is to investigate the dynamical behaviors of an impulsive one-predator two-prey model with mutual interference, square root functional response, and integrated control methods. The model is described by the following differential equations:where , , and are densities of two preys and one predator at time , respectively. is intrinsic increasing rate; is the death rate of predator. represents the mutual interference of the predator: . is transformation rate for the predator to prey. is death rate of prey; represents the percent of prey-predator that dies at time . is the releasing number of predators at . Parameters are competitive effects between two preys, respectively. Parameter is the moment period of impulsive effect. The integer ; is the set of all nonnegative integers. All parameters are positive constants.

We aim to investigate the dynamical behaviors of (4). From the biological point of view, we only consider system (4) in the biological meaningful region: and the initial conditions for system (4) are

The structure of this paper is as follows. In Section 2, we give some definitions, notations, and lemmas. In Section 3, by using techniques of impulsive perturbations, Floquet theory, and comparison theorem, we discuss stability, extinction, and permanence of system (4). We give corollaries for single chemical control in Section 4. Then we give some examples and numerical analysis of system (4) in Section 5. Finally, we conclude this paper with a brief discussion in Section 6.

2. Preliminaries

In this section, some helpful remarks, notations, definitions, and lemmas are introduced which are useful for our main results.

Let be the map defined by the right-hand sides of (4). Solution of (4), denoted by , is continuously differentiable on and . Let , and then is said to belong to class if(i) is continuous on and , and for each , and exist;(ii) is locally Lipschitzian in .

Definition 1. Let For and , the upper right derivative of with respect to the impulsive differential system (4) is defined as

Definition 2. If there exist positive constants and , with each positive solution of (4) satisfying for all sufficiently large, then system (4) is said to be permanent.

Remark 3. The solution of (4) is a piecewise continuous function; , is continuous on and , and exists, where and The smoothness properties of ensure the global existence and peculiarity of solutions of (4) (see [18] for the details).

Lemma 4 (see [18]). Let , and assume thatwhere and is continuous on and for exists, are nondecreasing for all Let be the maximal solution of the scalar impulsive differential equation,existing on Then implies that for all , where is any solution of system (4).

Next, we introduce some fundamental properties about the following subsystem of (4):System (10) is a periodically forced linear system easily used to obtain, / is a positive periodic solution of system (10). Since the solution of (10) with initial value is

Lemma 5 (see [18]). Suppose is a positive periodic solution of (10) and ; then we get as

Lemma 6 (see [19]). Suppose is a solution of (4) and , and hence for all It also has if

Lemma 7 (see [20]). Suppose function satisfies the following inequalities:where and , , and are constants Then for , one has

3. Main Theorems

3.1. Boundedness

Theorem 8. For any solution of system (4), there exists a constant , such that , , and hold for all large enough.

Proof. Let be a solution of (4) with initial value
Define a function
When , take a constant such that ; then by calculating the upper right derivative of along the solution of system (4), we havewhere
Further, at moment and at Then by Lemma 7, for all , we getHence is bounded for sufficiently large . Let ; then , , and are bounded by for sufficiently large . This completes the proof.

3.2. Stability of Prey-Eradication Periodic Solution

Theorem 9. Suppose is any solution of (4); then the prey-eradication periodic solution is globally asymptotically stable provided that

Proof. The local stability of periodic solution can be determined by considering the behavior of small amplitude perturbations of the solution.
Define , , and , and we getwhere is a small perturbation. When , and , (4) can be expanded in a Taylor series. Then, neglecting higher-order terms, the linearized equations readLet be the fundamental matrix of above differential equations; then satisfies is the identity matrix. Then the linearization of system (4) becomesThe stability of periodic solution of (4) is determined by the eigenvalues of the matrix :If each of these eigenvalues of matrix is less than one, then the periodic solution is locally stable. The three eigenvalues of the matrix areFrom Floquet theory of impulsive differential equation [18], if , then () is locally asymptotically stable. Here is already less than one, so we only need to calculate
Actually,Hence, if , then is locally asymptotically stable.
Next, we prove the global attractivity of Choose such thatFrom (4), we get It follows from (26) thatContinuing the iteration technique, we can obtain and as Hence, as Then, as since for By the same way, we can get , as
Now we prove as For sufficiently small, there exists such that and Without loss of generality, we assume that and for all . Then from system (4) we obtainSince , from (28), we have Consider the following two comparison systems:We can obtain that the periodic solution , and periodic solution of (31) is For any solutions and of the above two systems, respectively, we have , and as and From Lemmas 5 and 4, we have Then, for any , there exists constant such that Let , and we have for large enough, which implies as This completes the proof.

3.3. Permanence of System (4)

Theorem 10. System (4) is permanent if conditions and hold:(H₁),(H₂), where

Proof. Suppose is any solution of system (4) with . From Theorem 8, we know that there exists constant such that with Noticing that , we consider this impulsive differential equation:Periodic solution of (34) is By Lemmas 4 and 6, we have and for sufficiently large. Now we want to find such that for large enough. We prove it by the following two steps.
Step 1. We conclude that there exist such that Otherwise there will be another three cases.
Case 1. There exists a constant such that , but for all .
Case 2. There exists a constant such that , but for all .
Case 3. Consider , and for all .
For Case 1, choose a sufficiently small constant such that Then we have Consider the following comparison equation:the periodic solution of (37) readsFor any solution of (37), we have By comparison theorem, we can easily obtain as Hence, there exists , when
Then we have and Integrating (39) on , we getContinuing the iteration technique, we can obtain as , which is a contradiction. Similar to Case 1, we can prove Case 2 is contradictory.

Next we consider Case 3. Using the assumption in Case 3, it is easy to get

By the same discussion as Case 1, we can find a contradiction. Hence, from the above three cases, there exist and such that

Step 2. If for all , then our aim is achieved. Otherwise, if for some , then we only need to consider those solutions which leave the region and reenter it again. Let , and then there are two possible cases for .

Case 1. If , then for and Choose such thatwhere Let We claim there must be such that Otherwise we have , considering (37) with , and then we getand Then and for These imply that (39) holds for By the same discussion of Step 1, we obtain

From the first equation of (4), we haveIntegrating (44) on , then and , which is a contradiction with . Hence, holds for Let , and then and for So holds for . If , the same arguments can be expanded since

Case 2. If , then for and

Suppose , and then there are two possible cases for

Case 2.1. holds for all . Similar to Case 1, we can prove that there must exist such that ; we omit it here.

Let , and then holds for Then we have , and for For , the same arguments can be followed since

Case 2.2. There exist such that Let , and then for and When , (44) holds. Integrating (44) on , then Since for , the same arguments can be used; then for all .

Thus in both cases, we conclude that holds for all Similarly, we have for all Let , and then we have The proof is completed.

4. Single Chemical Control

If , system (4) concerns the single chemical control. Then we have the following corollary.

Corollary 11. Assume and .(i)Let be any solution of (4). Then the solution is locally stable if (ii)System (4) is permanent if and

5. Examples and Simulations

In this section, we give some examples and numerical simulations to verify our theoretical results and further study the complexity and variety of system (4).

Firstly, let , , , , , , , , , , , , , , , , , , , and . By verification, these parameters satisfy conditions of Theorem 9; then the prey-eradication periodic solution is global asymptotic stability. With Matlab, by simulation, the asymptotic stability can be illustrated by Figures 1(a), 1(b), and 2. Figures 1(a) and 1(b) show the existence of the prey-eradication periodic solution. Figure 2 shows the attractivity of prey-eradication periodic solution. If we change the mutual interference of the predator and other parameters are the same as Figure 1(a), then we get another dynamical portrait that is very different from Figure 1(a). We give an example for (see Figure 1(c)), which shows that the mutual interference of the predator affects the dynamical behaviors of system (4) to be more complex.

(a)

(b)

(c)

(a)

(b)

Secondly, let , , , , , , , , , , , , , , , , , , , , , , and . By verification, these parameters satisfy conditions of Theorem 10. Then we conclude that system (4) is permanent from Theorem 10. By simulation, we can verify the same results. Figure 3(a) is the time series of the three species in the interval []. In addition, we give the phase portrait of a positive periodic solution of system (4) in Figure 3(b). Similarly, if we change the mutual interference of the predator and other parameters are same as Figure 3(a), then we get another dynamical picture that is very different from Figure 3(a). We give an example for (see Figure 3(c)). It also shows that the parameter affects the dynamical behaviors of system (4).

(a)

(b)

(c)

Thirdly, by numerical analysis, we aim to investigate the bifurcation diagrams of impulsive period . Let , , and , and , , , , , , , , , , , , , , , , , , and . The bifurcation diagrams of , , and with respect to parameter in range [] are shown in Figure 4. Figures 4(a)–4(c) are bifurcation diagrams of , , and , respectively. From these diagrams, we can see that impulsive period heavily affects the dynamical behaviors. For example, Figure 5 shows the complex dynamic behaviors of . Figure 5 is the magnified parts of Figure 4(c). From Figure 5(a), we can see the following: when impulsive period , it is stable; if , 142.4, 148, and 149.1, then bifurcation appears, respectively. That is to say, when , there is a cascade of period-doubling bifurcations leading to -period solution (Figure 5(a)); when , 4-period solution appears (Figure 5(a)); when , 8-period solution appears (Figure 5(b)); when , 16-period solution appears (Figure 5(b)). As increase beyond 149.5, the phenomenon of “crisis” emerges. When , there is a typical chaotic oscillation (Figure 5(c)). When is near 153.5, we can see in the neighborhood of , after the period-doubling bifurcations, the symmetry-breaking bifurcations appear (Figure 5(d)), which are specially simple bifurcations that come into being multiplicity of steady states [5]. It implies that when , that is, spraying pesticides and releasing natural enemies are frequent, the solution of this system is stable in this situation; when , periodic behaviors of prey and the predator will appear; if , then dynamical properties of this system are complex and the development of this system may be unpredictable.

(a)

(b)

(c)

(a)

(b)

(c)

(d)

Fourthly, let , , and and , , , , , , , and . By verification, these parameters satisfy conditions of Corollary 11; then we can get the globally asymptotic stability of prey-eradication periodic solution and permanence for single chemical control (see Figures 6(a) and 6(b)).

(a)

(b)

6. Conclusion

In this paper, an impulsive two-prey and one-predator system with square root functional responses, mutual interference, and integrated pest management is constructed. Numerical simulations are given to portray the complex behaviors of this system. From Theorem 9, the existence and global asymptotic stability of prey-eradication periodic solution of (4) are obtained. Some methods and sufficient conditions are given to prove the permanence of system (4) in Theorem 10.

From Theorems 9 and 10 and simulations, we know dynamical properties of system (4) are very complex which depend on impulsive period , the releasing amount of predator , the mutual interference of predator , and the parameter of pests or predator which dies from the chemical control. Figures 1(c) and 3(c) show that different values of mutual interference of predator have different dynamical properties for system (4). Figures 1(a), 1(b), and 2 show the existence and attractivity of prey-eradication periodic solution. Figures 3(a) and 3(b) show the existence of positive periodic solution. Figure 4 implies that impulsive period heavily influences the dynamic behavior of system (4). As changes, periodic behaviors, bifurcations, “crisis” phenomenon, chaotic phenomenon, chaotic oscillation, and symmetry-breaking bifurcations appear, respectively. Hence, we can choose moderate value of for some different control strategies.

In this paper, by considering mutual interference and square root functional responses, our constructed model is new and complex, which more rationally reflects the real world. Furthermore, we also give the corresponding results on strategies of integrated pest management and classical chemical control. By our obtained results, if combining biological control and chemical control to eradicate preys, we can choose moderate impulsive period and moderate parameters and to effectively eliminate preys and reduce environmental pollution. In particular, we analyze the influence from impulsive period . These theories have some guidance to our real life and the natural balance.

Competing Interests

None of the authors has any competing interests in the paper.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (11161015), Natural Science Foundation of Guangxi (2013GXNSFAA019003), Applied Statistics Key Laboratory (Guijiaokeyan () number 6), and Statistics Guangxi Key Discipline (Guijiaokeyan () number 13) and is partially supported by the National High Technology Research and Development Program 863 under Grant no. 2013AA12A402.

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Copyright © 2016 Zhen Wang 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.

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