Positive Periodic Solutions of a Neutral Impulsive Predator-Prey Model

Abstract

We investigate a ratio-dependent predator-prey model with Holling type III functional response based on system of neutral impulsive differential equations. Sufficient conditions for existence of positive periodic solutions are obtained by applying continuation theorem. Our main results demonstrate that under the suitable periodic impulse perturbations, the neutral impulsive system preserves the periodicity of the corresponding neutral system without impulse. In addition, our results can be applied to the corresponding system without impulsive effect, and thus, extend previous results.

1. Introduction

During the recent three decades, impulsive differential equations were intensively investigated. Some researchers devote themselves to the research on the theory of impulsive differential equations. The detailed introductions of impulsive differential equations were given in [1].

In the real world, there are many man-made and natural factors that always lead to sudden changes of population number at certain moments of time. For example, the births of many species are not continuous but happen at some regular time. These changes can often be characterized mathematically in the form of impulses. If these impulsive factors are incorporated into population dynamical models, the corresponding models should be governed by impulsive differential equations. With the development of impulsive differential equations, many models of impulsive differential equations have been considered extensively. For example, many authors have studied some qualitative properties of the solutions for impulsive differential equations such as periodicity [2–6], stability [7–10], and permanence and extinction [2, 11].

Recently, Du and Feng [12] studied the existence of periodic solutions for the following neutral impulsive predator-prey model with Beddington-DeAngelis functional response:

In 2011, Liu and Yan [13] studied the existence of periodic solutions for the following neutral ratio-dependent predator-prey model with Holling type III functional response:

Now, we will consider the following neutral impulsive ratio-dependent predator-prey system with Holling type III functional response: where and represent the prey and predator densities; is the intrinsic growth rate of the prey; is the density-dependent coefficient of the prey; is the death rate of the predator; are the regular pulses at time of the prey and predator, respectively. In addition, represent the right and left limits of at ; represent the right and left limits of at .

Throughout the paper, we give the hypothesis as follows:), , , are -periodic functions; , ; are fixed points and ;(), ; ; is -periodic function for .

The rest of this paper is organized as follows. We first introduce a lemma and the continuation theorem. Then, in Section 3, we obtain the existence of periodic solutions for system (3). In Section 4, a conclusion is given.

2. Preliminaries

In view of the actual applications of (3), we will only consider the solutions of system (3) with the initial condition: where .

Definition 1. Assume that satisfies the initial condition (4). Moreover, (i) and are absolutely continuous on each interval and ;(ii)for and exist and ;(iii)for , satisfies (3) for almost everywhere in .Then is a positive solution of the initial value problem (3) and (4) on .

Consider the following system: where By and , the above functions are all positive -periodic functions.

Lemma 2. Assume that and hold. If is a positive solution of system (5), then is a positive solution of system (3), where

Proof. It is easy to see that and are positive and absolutely continuous on each interval . For , , Hence, the first equation of system (3) holds. Similarly That is, the second equation of system (3) holds. For , , Hence, for , , So is a positive solution of system (3).

Next, we recall some concepts and results on coincidence degree.

Let be normed vector spaces; let be a linear mapping. Assume that and is closed in ; then the mapping is said to be a Fredholm mapping of index zero.

If is a Fredholm mapping of index zero, then there exist continuous projectors and satisfying . It is obvious that the restriction of to has an inverse function. Denote by .

Let be a continuous mapping. is -compact on ; if is an open bounded subset of , is bounded and is relatively compact.

Lemma 3 (see [14]). Let be an open bounded set; a Fredholm mapping of index zero; and -compact on . If the following conditions hold: (i), for any , ;(ii), for any ;(iii), where is an isomorphism;then has at least one solution in .

3. Main Results

Let be a continuous -periodic function. Denote

In the following, we will present our main results.

Theorem 4. Suppose that , , and the following conditions hold. ()For , , ; let be the inverse function of and , where and .()Consider that , where + .() Consider that and .Then system (3) has at least one positive -periodic solution.

Proof. Construct the system Apparently, if there exists one -periodic solution for system (13), then is a positive -periodic solution of system (5). By and Lemma 2, Theorem 4 holds if we can show that there exists one -periodic solution of system (13).
Let be the real Banach space with the norm Let be the real Banach space , , with the norm
Define and by Therefore, system (13) can be written as Evidently, , is closed in , and . Hence is a Fredholm mapping of index zero.
Let and be The generalized inverse (to ) is Clearly, and are continuous projectors satisfying So and are Obviously, it is easy to check that and are continuous. Furthermore, is bounded; are relatively compact for any open bounded set . Thus is -compact on .
For any , consider ; that is, Let be a solution of (22). Thus By , By (23), By (24), Set be the inverse function of . Clearly, , are all -periodic functions. In addition, . By , Thus By (22), (29), and , Note that , are all -periodic functions. By (29) and , which yields So there exists satisfying By , By (30) and , for , As , By (22) and (36), By , In addition, there exist satisfying By (29) and , By and (40), By (38), By (36) and (42), By (27), there exists satisfying Set , where is an integer, . So Denote Clearly, the function is increasing with respect to , where , . By (39), (43), (45), and (46), By , By (22) and (27), Hence So By (22), By (38), (43), (51), and (52), Consider the algebraic system If is a solution of (54), then there exist such that which yields where . Further, In addition, it is not difficult to see that the algebraic system has a unique solution . Let be a constant satisfying Set . Obviously, is independent of . Set Clearly, the condition in Lemma 3 holds. If , then is a constant vector in with . Thus So the condition in Lemma 3 holds.
Define the homology by Similar to the derivations of (57), for any and , . By (57) and (59), for any and ,
Define the isomorphism by . By homotopy invariance of topological degree, So, the condition (iii) of Lemma 3 holds. By Lemma 3, there exists one -periodic solution of system (13). The proof of Theorem 4 is complete.

Remark 5. From the proof of Theorem 4, we can find that Theorem 4 is also true if , . Therefore, we can obtain the following result.

Corollary 6. Suppose that and the following conditions hold. ()For , let be the inverse function of and , where and .()Consider , where + .() Consider and .Then the following neutral ratio-dependent predator-prey system with Holling type III functional response has at least one positive -periodic solution.

Remark 7. If are all constants, then Corollary 6 matches Theorem  2.1 in [13]. Therefore, our main results extend Theorem  2.1 in [13].

4. Conclusion

In this paper, we construct a ratio-dependent predator-prey model with Holling type III functional response based on system of neutral impulsive differential equations. By constructing suitable homology and applying some analysis techniques and homotopy invariance of topological degree, we give the estimate on a priori bounds of periodic solutions for the corresponding system. Then the existence of positive periodic solutions for this model is obtained by applying coincidence degree theory.

Our main results demonstrate that under the suitable periodic impulse perturbations, the neutral impulsive system preserves the periodicity of the corresponding neutral system without impulse. In addition, our results can be applied to the corresponding neutral ratio-dependent predator-prey model without impulsive effect. Further, our results extend some results in the literature.

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 Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi (no. 20120304), Research Project Supported by Shanxi Scholarship Council of China (no. 2013-019), and Natural Science Foundation of Shanxi Province (no. 2014011005-1).