International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 970763, 18 pagesdoi:10.1155/2011/970763

Research Article

Existence of Periodic Solutions in a Discrete Predator-Prey System with Beddington-DeAngelis Functional Responses

Changjin Xu¹ and Maoxin Liao²

¹Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China²School of Mathematics and Physics, Nanhua University, Hengyang 421001, China

Received 8 July 2011; Accepted 27 October 2011

Academic Editor: Aloys Krieg

Copyright © 2011 Changjin Xu and Maoxin Liao. 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 discrete predator-prey model with Holling II and Beddington-DeAngelis functional responses is investigated. With the aid of differential equations with piecewise constant arguments, a discrete version of continuous nonautonomous delayed predator-prey model with Beddington-DeAngelis functional responses is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive solutions of the model are established.

1. Introduction

In population dynamics, the functional response refers to the number of prey eaten per predator per unit time as a function of prey density. Based on a lot of experiments on mammals, Holling [1] proposed three kinds of functional responses as follows:(1), (2), (3),

where represents the density of prey. Functions , , and refer to Holling types I, II, and III, respectively, and is the predation rate of the predator. After that, the dynamical properties of predator-prey systems with functional response have received great attention from both theoretical and mathematical biologists; for example, Liu et al. [2] investigated the coexistence of predators and preys of the following predator-prey system with Holling II functional response: where stands for the density of the prey, and are the densities of the predators, respectively, and , , , , , , , , , and are positive constants. Assuming that one predator consumes prey according to the Holling II functional response and the other predator consumes prey according to the Beddington-DeAngelis functional response, Cantrell et al. [3] proposed the revised version of system (1.1) as follows: By using dynamical system technique and the geometrical singular perturbation theory, Cantrell et al. [3] made a discussion on the coexistence of predators and prey of system (1.2) which occurred along a stable positive equilibrium. Ko and Ryu [4] discussed the existence, stability, and uniqueness of coexistence states and the extinction and permanence of a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response. Chen et al. [5] analyzed the extinction of the predator and the global asymptotic stability of the boundary solution of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. Huo et al. [6] studied the global existence of positive periodic solutions for a delayed predator-prey model with the Beddington-DeAngelis functional response. For more knowledge about this topic, one can see [7, 8].

Recently, Song and Li [9] investigated the following two-prey one-predator model, where two prey are competitive and the predator has Holling functional II functional response: where and stand for the population size of prey (pest) species and is the population size of the predator (natural species) species; are intrinsic rates of increase or decrease, and are parameters representing competitive effects between two prey, , are positive constants, and are the Holling II functional responses. is the rate of conversing prey into predator.

To model mutual interference among predators, Beddington [10] andDeAngelis et al. [11] argued that the well-known Holling type II functional response will be replaced by the Beddington-DeAngelis functional responses which is similar to the Holling type II functional response but has an extra term in the denominator, then system (1.3) may be modified as the following predator-prey system with Beddington-DeAngelis functional responses: where and stand for the population size of prey (pest) species and is the population size of the predator  (natural species) species; are intrinsic rates of increase or decrease, and are parameters representing competitive effects between two prey, , are positive constants, , , , and are the Beddington-DeAngelis functional responses. is the rate of conversing prey into predator.

In the natural word, any biological and environmental parameters are naturally subject to fluctuation in time. The effect of a periodically varying environment is important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, assumptions of periodicity of parameters are a way of incorporating periodicity of the environment, such as seasonal effect of weather, food supplies, and mating habits [12]. Based on the point of view, the modification of (1.4) according to the environmental variation is the nonautonomous differential equations As is known to us, discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations. Therefore, it is reasonable and interesting to study discrete time systems governed by difference equations. Recently, there are some papers which deal with these topics see [13–22]. The principle object of this article is to propose a discrete analogue system (1.5) and explore its dynamics.

The remainder of the paper is organized as follows: in Section 2, with the help of differential equations with piecewise constant arguments, we first propose a discrete analogue of system (1.5), modelling the dynamics of time nonautonomous predator-prey system with Beddington-DeAngelis functional responses, where populations have nonoverlapping generations. In Section 3, based on the coincidence degree and the related continuation theorem, sufficient conditions for the existence of positive solutions of the model are obtained.

2. A Discrete Version of Model (1.5)

There are several different ways of deriving discrete time version of dynamical systems corresponding to continuous time formulations. One of the ways of deriving difference equations modelling the dynamics of populations with nonoverlapping generations that we will use in the following is based on appropriate modifications of models with overlapping generations. For more details about the approach, we refer to [15, 18].

In the following, we will discrete the system (1.5). Assume that the average growth rates in system (1.5) change at regular intervals of time, then we can obtain the following modified system: where denotes the integer part of , and . Equations of type (2.1) are known as differential equations with piecewise constant arguments, and these equations occupy a position midway between differential equations and difference equations. By a solution of (2.1), we mean a function , which is defined for and has the following properties:(1) is continuous on ;(2)the derivative exists at each point with the possible exception of the points , where left-sided derivative exists;(3)The equations in (2.1) are satisfied on each interval with .

We integrate (2.1) on any interval of the form , , and obtain for , ,

Let , then (2.2) takes the following form: which is a discrete time analogue of system (1.5), where .

3. Existence of Positive Periodic Solutions

For convenience and simplicity in the following discussion, we always use the notations below throughout the paper: where is an -periodic sequence of real numbers defined for . For system (2.3), we always assume that, , ,, , , , are -periodic.

In order to explore the existence of positive periodic solutions of (2.3) and for the reader’s convenience, we will first summarize below a few concepts and results without proof, borrowing from [23].

Let be normed vector spaces, is a linear mapping, is a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , it follows that is invertible. We denote the inverse of that map by . If is an open bounded subset of , the mapping will be called compact on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms .

Lemma 3.1 (see Continuation Theorem [23]). Let be a Fredholm mapping of index zero, and let be compact on . Suppose that, (a)for each , every solution of is such that ;(b) for each , and ,then the equation has at least one solution lying in .

Lemma 3.2 (see [15]). Let be -periodic, that is, . Then for any fixed and any , one has

Define

For , define . Let denote the subspace of all -periodic sequences equipped with the usual supremum norm , that is, , for any . It is easy to show that is a finite-dimensional Banach space.

Let

Then it follows that and are both closed linear subspaces of and

In the following, we will be ready to establish our result.

Theorem 3.3. Let , , , , , and be defined by (3.30), (3.50), (3.34), (3.46), (3.39), and (3.55), respectively. In addition to the condition (H1), assume further that the following conditions: , hold. Then system (2.3) has at least an -periodic solution.

Proof. Let , , and . Then (2.3) takes the form where Let , where , . Then it is trivial to see that is a bounded linear operator and Then it follows that is a Fredholm mapping of index zero. Define It is not difficult to show that and are continuous projectors such that Furthermore, the generalized inverse (to ) exists and is given by Obviously, and are continuous. Since is a finite-dimensional Banach space, using the Ascoli-Arzela theorem, it is not difficult to show that is compact for any open bounded set . Moreover, is bounded. Thus, is -compact on with any open bounded set .Now we are at the point to search for an appropriate open, bounded subset for the application of the continuation theorem. Corresponding to the operator equation , , we have Suppose that is an arbitrary solution of system (3.14) for a certain . Summing both sides of (3.14) from 0 to with respect to , respectively, we obtain It follows from (3.14) and(3.15) that In view of the hypothesis that , there exist such that Then it is obvious that where denotes the forward difference operator .In view of (3.8), we get Then we have which leads to From (3.16), we have that is, In the sequel, we consider two cases.Case 1. If , then it follows from (3.26) that which leads to In view of Lemma 3.2, (3.16), (3.24), and (3.28), we have It follows from (3.29) that By (3.26), it is easy to obtain that Hence, In view of Lemma 3.2, (3.18), (3.24), and (3.32), we have It follows from (3.33) that Noticing that is an increasing function with respect to , from (3.21), we have Hence, From (3.16), we get Then, where Thus, In view of Lemma 3.2, (3.18), (3.36), and (3.40), we have It follows from (3.41) that Case 2. If , then it follows from (3.26) that which leads to In view of Lemma 3.2, (3.17), (3.24), and (3.44), we have Combining both equation of (3.45), one obtains By (3.26), it is easy to obtain that Hence, By virtue of Lemma 3.2, (3.16), (3.24), and (3.48), we have It follows from (3.49) that Considering that is an increasing function with respect to , from (3.21), we have Therefore, From (3.16), we know that Then, where Thus, In view of Lemma 3.2, (3.18), (3.52), and (3.56), we have It follows from (3.57) that Obviously, , , , , and are independent of . Take , where is taken sufficiently large such that each solution of the following algebraic equations: satisfies .Now we have proved that any solution of (3.6) in satisfies .Let . Then it is easy to see that is an open, bounded set in and verifies requirement (a) of Lemma 3.1. When , is a constant vector in with . Then Define the homotopy by , , where Let be the identity mapping. According to the definition of topology, direct calculation yields where Then, it follows from (3.62) that By now, we have proved that verifies all requirements of Lemma 3.1. Then it follows that has at least one solution in , that is to say, (3.6) has at least one -periodic solution in , say . Let , and . Then we know that is an -periodic solution of system (2.3) with strictly positive components. We complete the proof.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 10961008), the Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560), and Doctoral Foundation of Guizhou College of Finance and Economics (2010).

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