Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 793761, 13 pages

http://dx.doi.org/10.1155/2014/793761

Research Article

Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

Hongli Li, Long Zhang, Zhidong Teng, and Yaolin Jiang

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 22 November 2013; Accepted 30 April 2014; Published 29 May 2014

Academic Editor: Guang Zhang

Copyright © 2014 Hongli Li 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.

Abstract.

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos.

1. Introduction

As pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both biology and mathematical biology due its universal existence and importance. Though predator-prey models with periodic or almost periodic coefficients have been studied extensively (see [2–14] and the references cited therein), few papers consider the mutual interference between the predators and prey, which was introduced by Hassell in 1971. During his research of the capturing behavior between hosts and parasites, he found that the hosts or parasites had the tendency to leave each other when they met, which interfered with hosts capture effects. Obviously, the mutual interference will be stronger, while the size of parasite becomes larger. From the observation, Hassell introduced the concept of mutual interference constant    and established a Lotka-Volterra predator-prey model with mutual interference as follows:

where and are the sizes of the prey and predator populations, respectively, and is mutual interference constant; see [15, 16] for more details on the biological meaning of mutual interference constant . Some results have appeared for this type of predator-prey system [17–20].

Recently, much more attention has paid on the functional responses. Based on experiments, Holling [21] investigated three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more consistent with the real ecosystem. The Lotka-Volterra system with Holling type II functional response is introduced by Wang and Zhu [22] in the following form:

However, ecological system is often deeply perturbed by nature and human exploit activities (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding, etc.), which are not suitable to be considered continually. To accurately describe the system, impulsive differential equations may be a better candidate than ordinary differential equation. The theory of impulsive differential equations not only is now being recognized to be richer than the corresponding theory of differential equations without impulse, but also represents a more natural framework for mathematical modelling of many real-world phenomena. In recent years, many important and interesting results on the permanence, extinction, global attractivity of systems, the existence and uniqueness of positive periodic solutions, bifurcation and dynamical complexity, and so forth have been proposed in [23–28] and references cited therein. Motivated by above studies, in this paper we will consider the following Holling II predator-prey system with mutual interference and impulses:

where and are the sizes of the prey and predator populations, respectively. and is a positive constant. ,  ,  and     are continuous periodic functions defined on with a common period . ,  ,   are strictly positive. Also   , . Assume that ,     are constants. and , and there exists an integer such that , , . The interpretations of parameters can be seen in Table 1.

Table 1: Parameter values used in the simulations of system (3).

Parameter
Interpretation
Value

Growth rate of the prey

Death rate of the predator

Intracompetition coefficient among the prey
(2πt)

Intracompetition coefficient among the predators

Capture rate of predator on prey

Conversion rate of nutrition

The environment carrying capacity
10

Mutual interference constant
1/3

Impulse perturbations on prey
—

Impulse perturbations on predator
—

The organization of this paper is as follows. In Section 2, we give some definitions and state some lemmas which are essential to study the main theorems. In Section 3, sufficient conditions for the permanence, extinction of system (3), and the existence, uniqueness, and global attractivity of positive periodic solution are obtained. In Section 4, numerical examples are presented to illustrate our results. In the last section, we give a brief discussion.

2. Preliminaries

In this section, we introduce some notations and definitions and state some lemmas which will be useful in the subsequent sections.

Let , . The map is defined by the right hand of the first and second equations of system (3). Let ; then is said to belong to class if

(1)is continuous in and for each , , there exists ;(2) is locally Lipschitzian in .

Definition 1. Let ; then, for , the upper right derivative of with respect to the impulsive differential system (3) is defined as

Definition 2. System (3) is permanent, if, for any positive solution of system (3), there exist positive constants ,  ,  , and such that

Lemma 3 ([29]). Let and is a solution of (3) with initial value . Assume that

where is continuous in and for , , exists and is nondecreasing. Let be the maximal solution of scalar impulsive differential equation

existing on . Then implies that for all .

Similarly, assume that inequality (6) is reversed. Let be the minimal solution of (7) existing on . Then, implies that for all .

Let be continuous -periodic function defined on the ; we introduce the following notation:

Consider the following periodic logistic equation with impulses:

We have the following results.

Lemma 4 (see [30]). ( 1) If

then system (9) has a unique T-periodic solution , which is globally asymptotically stable.

(2) If

then .

Consider a periodic Logistic equation with impulses:

where is a positive constant, and are continuous -periodic functions with and , and  , . We have the following results.

Lemma 5 (see [31]). ( 1) If

then system (10) has a unique -periodic solution , which is globally asymptotically stable.

(2) If

then .

Lemma 6 (see [31]). Let be the unique -periodic positive solution of system (9) and the unique -periodic positive solution of the following system:

where is the continuous T-periodic function with and . Then

Furthermore, we consider the following periodic logistic equation with impulses:

By Lemma 5, we have the following Corollary.

Corollary 7. If

then system (15) has a unique T-periodic solution , which is globally asymptotically stable.

Next, one considers the following periodic logistic equation with impulses:

By Lemma 5, we have the following Corollary.

Corollary 8. If

then system (16) has a unique T-periodic solution , which is globally asymptotically stable.

3. Main Results

In this section, we will study the permanence, extinction, and global attractivity of system (3) and the existence and uniqueness of positive periodic solution.

Lemma 9. For any solution of system (3), if , , then and for all .

Proof. From the first equation of (3), one has

where

thus, from , we have

Similarly, it follows from the second equation of system (3) that . This completes the proof of Lemma 9.

Lemma 10. Assuming that and hold, for any positive solution of system (3), there exist constants    such that

Proof. First, we show that is ultimate upper bounded. It follows from (3) that

Consider the following auxiliary system:

By Lemma 3, we have , where is the solution of (22) with . By , from Lemma 4 we get that system (22) has a unique -periodic solution , which is globally stable. Therefore, for any being small enough, there exists a such that

for any . Let ; we have

for any . Let ; we obtain

Next, we prove that is ultimately upper bounded. Define . From system (3), we have

For any and , we have

Consider the following auxiliary system:

By Lemma 3, for any , we have , where is the solution of (28) with . By , from Lemma 5, system (28) has a unique -periodic solution , which is globally stable. Therefore, letting , from Lemma 6, we have . Letting , for any being small enough, there exists , and we have

for any . Let ; we have

This completes the proof of Lemma 10.

Theorem 11. Assume that and hold, and

where is the unique T-periodic solution of system (16) and . Then species and are permanent.

Proof. From Lemma 9, there exist constants such that

From the proof of Lemma 10, for above being small enough, there exists such that

for any . From the first equation of system (3), for and , we have

Consider the following auxiliary system:

where is the solution of (34) with . By Lemma 3, for any , we have . From condition , we have

From inequality (35) and Lemma 4, system (34) has a unique -periodic solution , which is globally stable. Since condition holds, for the above being small enough, we have ; letting , from Lemma 6, we have . Letting , for the above being small enough, there exists such that , for any . Let ; we have . From system (3), for and , we have

Consider the following auxiliary system:

where is the solution of (37) with . By Lemma 3, for all , we have , for the above being small enough. By condition , we obtain

From inequality (38), Lemma 4, system (37) has a unique -periodic solution , which is globally asymptotically stable. Since condition holds, for the above being small enough, such that , let , and from Lemma 6, we have , where is the unique -periodic globally asymptotically stable solution of the following system:

Letting , for the above being small enough, there exists such that , for any . Let ; we have .

Theorem 12. Assume that

Then, for any positive solution of system (3), one has

where and are unique T-periodic solutions of systems (9) and (39), respectively.

Proof. From system (3), for and , we have

Consider the following auxiliary system:

From and Lemma 4, we have

Therefore, for being small enough, there exists such that for any . Since holds, we obtain

Thus, from the first equation of system (3), we have

for any . Consider the following auxiliary system

From and Lemma 4, system (46) has unique -periodic solutions and , respectively, which are globally asymptotically stable. By Lemma 3, for any being small enough, there exists such that for any ; letting , we have ; since is arbitrarily small, we have .

Theorem 13. Suppose that holds and

Then species and are extinct; that is,

Proof. From and Lemma 4, system (9) has a unique -periodic solution , and . By Lemma 3, we have ; therefore, .

Theorem 14. Suppose that all the conditions of Theorem 11 hold and there exist positive constants and such that

Then system (3) has a unique positive T-periodic solution which is globally asymptotically stable.

Proof. Let and be any two positive solutions of system (3); for any sufficiently small , from Theorem 11, there is a large enough such that for any , we have

Define the Lyapunov function as follows:

By the mean value theorem and (48), for any , , and , we have

where is suited between and , is suited between and , and  . For ,     and the above small enough ; from conditions and , there exist positive constants and such that

Calculating the upper right derivative of , we obtain

In view of

substituting (53) and (54) into (52), we have

For all and   , we have

Hence, is continuous for all . The above analysis shows that, for all ,

and we further have

It is obvious that as ; that is,

We further have

Choose constants and . By Theorem 11, we have

Now let us consider the , , and . It is compact in the domain , since and for all . Let be a limit point of this sequence ; then we have . In fact, since and as , we have

The sequence has a unique limit point; otherwise, let the sequence have two limit points and . Then, taking into account (60) and , we have

and hence ; the solution is the unique periodic solution of system (3). By (60), it is globally asymptotically stable. This completes the proof of Theorem 14.

4. Numerical Simulations

In system (3), we take , , which means that system (3) is a one-period system with one impulse perturbation during one period.

In case 1, – are, respectively, equivalent to the following inequalities:

Taking , , conditions and are, respectively, equivalent to the following inequalities:

By calculation, conditions –, , and hold; from Theorem 11 and Theorem 14, system (3) is globally asymptotically stable. Numerical simulations of these results can be seen in Figure 1.

Figure 1: Time series of and .

Figure 1 shows the dynamical behavior of the solutions with seven group initial values , , , , , , and , respectively. From Figure 1, we can obtain that there exists a unique positive periodic solution of system (3) such that any solution of system (3) with initial value tends to the positive periodic solution as .

In case 2, conditions and are, respectively, equivalent to the following inequalities:

By calculation, conditions and hold. From Theorem 12, we have that species will be extinct and species will be globally asymptotically stable. Numerical simulations of these results are plotted in Figure 2.

Figure 2: Time series of and .

Figure 2 shows the dynamical behavior of the solutions with seven group initial values , , , , , , and , respectively. From Figure 2, we can see that species will be extinct and species will be globally asymptotically stable (Table 2).

Table 2: The effects of and on the dynamical behavior of system (3).

Case

Figure

1
0.2
0.03
Permanent
Permanent
Figure 1

2
−0.2
−0.6
Permanent
Extinct
Figure 2

Furthermore, we investigate the effects of impulse coefficients and on the dynamical behavior of system (3); we keep the parameter fixed at . From Table 3, it is found that the impulse coefficient has an important influence on the sizes of prey and predator populations. Therefore, we plot the bifurcation diagram of system (3) when ; see Figure 3(a); it is very clear that the number of prey populations increases with increasing parameter .

Table 3: The effects of on the dynamical behavior of system (3).

Case

Figure

—
Bifurcation parameter
0.3
Figure 3(a)

1
−0.4
0.3
Figure 3(b)

2
0
0.3
Figure 3(c)

3
0.6
0.3
Figure 3(d)

(a)

(b)

(c)

(d)

Figure 3: (a) Bifurcation diagram of when is fixed at 0.3; (b)–(d) phase portraits.

(a)

(b)

(c)

(d)

Figure 4: (a) Bifurcation of when is fixed at 0.4; (b)–(d) phase portraits.

(a)

(b)

(c)

(d)

Figure 5: Dynamical behavior of system (3); here, we take initial value .

On the other hand, we keep the parameter fixed at . For , the effects of on the dynamical behavior of system (3) have been also investigated. From Table 4, it is found that the impulse coefficient has an important influence on the sizes of prey and predator populations.

Table 4: The effects of on the dynamical behavior of system (3).

Case

Figure

—
0.4
Bifurcation parameter
Figure 4(a)

1
0.4
0.1
Figure 4(b)

2
0.4
0.3
Figure 4(c)

3
0.4
0.4
Figure 4(d)

Moreover, on increasing the period of impulse perturbations, some interesting phenomena happen; the details are given in Table 5.

Table 5: The effects of impulse period on the dynamical behavior of system (3).

Case

Figure

1
1
Periodic
Periodic
Figures 5(a) and 5(b)

2
1.002
Almost periodic
Almost periodic
Figure 5(c)

3
1.014
Chaotic
Chaotic
Figure 5(d)

5. Conclusion

In this paper, a class of Holling type II predator-prey systems with mutual interference and impulses is studied. Some sufficient conditions for the permanence and extinction of the system and existence and uniqueness of positive periodic solution are obtained. These obtained results are new and the complement to previously known results. From the view point of biology, the mathematical results are full of biological meanings and can be used to provide reliable foundation for making control strategy. The conditions of Theorem 13 show that human activities may save the extinct species living in a week environment.

From the numerical simulations, we find that the behavior of impulse perturbations of species has an important influence on the system. Besides, the dynamics of species are very sensitive with the period matching between species' intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, the dynamics behavior of species periodically changes. If they mismatch each other, the dynamic behaviors would change from period to period until there is chaos. It is very consistent with real ecosystems, which helps us interpret the diversity and many phenomena of ecosystems.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11361059, 10901130, 11271312, and 11371287), the Natural Science Foundation of Xinjiang (Grant no. 2012211B07), and the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2013I03).

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