Journal of Applied Mathematics

Volume 2013 (2013), Article ID 963046, 13 pages

http://dx.doi.org/10.1155/2013/963046

Research Article

Positive Periodic Solutions in a Discrete Time Three Species Competition System

Changjin Xu¹ and Yusen Wu²

¹Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China

²Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received 11 December 2012; Accepted 27 January 2013

Academic Editor: Zhijun Liu

Copyright © 2013 Changjin Xu and Yusen Wu. 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 periodic discrete time three species competition system is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous nonautonomous three species competition system is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of the model are obtained.

1. Introduction

During the past decades, the dynamical properties of competitive populations have received great attention from both theoretical and mathematical biologists due to their universal prevalence and importance. Numerous excellent results have been reported for a lot of different continuous or impulsive competitive models. For example, Kuang [1] analyzed the permanent coexistence of the following delayed three species competition system:

where    stands for the density of competing species at time . For the biological meaning of model (1), one can see [1]. Tang et al. [2, 3] presented sufficient conditions for the existence and global attractivity of positive periodic solutions of the following periodic -species Lotka-Volterra competition system with delays

Bohner et al. [4] focused on the existence of periodic solutions in a predator-prey and competition dynamic systems, Pao [5] considered the global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, and Gopalsamy and Weng [6] made a detailed analysis on the global attractivity for a competition system with feedback controls. For more related work, one can see [7–10].

In 2011, Zhu and Lu [11] investigated the following delayed three species competitive system:

where    stands for the density of competing species at time , are -periodic functions (), and is a nonnegative function in with

For more details about the model, one can see [11]. By applying the theory of coincidence degree theory, Zhu and Lu [11] established the existence of positive periodic solution for system (3).

Numerous researchers have argued that 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, a great deal of work has been devoted to this topics; see [12–19]. The principle purpose of this paper is to propose a discrete analogue of system (3) and study the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time three species competition system.

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 (3), modelling the dynamics of time nonautonomous competing system 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 difference equations are given.

2. Discrete Analogue of System (3)

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 [17, 20].

Next, we will discretize the system (3). Assume that the average growth rates in system (3) change at regular intervals of time; then we can obtain the following modified system:

where denotes the integer part of ,   and  . Equations of type (5) 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 (5), we mean a function , which is defined for and has the following properties:

(1) is continuous on ;(2)the derivatives ,  ,   exist at each point with the possible exception of the points , where left-sided derivative exists;(3)the equations in (5) are satisfied on each interval with .

We integrate (5) on any interval of the form , , and obtain for ,

Let ; then (6) reads as

which is a discrete time analogue of system (3), where .

In order to obtain our main results, we assume that the following hold.

(H1) are positive -periodic; that is, and for any , where , a fixed positive integer, denotes the common period of the parameters in system (7).

(H2) Consider

3. Existence of Positive Periodic Solutions

For convenience and simplicity on the following discussion, we always use the notations below throughout the paper:

where is an -periodic sequence of real numbers defined for . In order to explore the existence of positive periodic solutions of (7) and for the reader’s convenience, we will first summarize below a few concepts and results without proof, borrowing from [21].

Let be normed vector spaces, a linear mapping, and 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 1 (see [21] continuation theorem). Let be a Fredholm mapping of index zero and let be -compact on . Suppose

(a)for each , every solution of is such that ;(b) for each and .

Then the equation has at least one solution lying in .

Lemma 2 (see [17]). Let be periodic; that is, ; then for any fixed and any , one has

Lemma 3. is an periodic solution of (7) with strictly positive components if and only if is an periodic solution of

The proofs of Lemma 3 are trivial, so we omitted the details here.

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

Next, we will be ready to establish our result.

Theorem 4. Let , , , , , , , , , and be defined by (33), (37), (49), (57), (61), (73), (81), (86), (94), and (98), respectively, and set

Suppose that , and    hold, then system (7) has at least an periodic solution with positive components.

Proof. Let ,

where ,  , and

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, 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 (22) for a certain ; summing both sides of (22) from 0 to with respect to , respectively, we obtain

It follows from (22), (23), (24), and (25) that

In view of the hypothesis that , there exist such that

By (23), (24), and (25), we have

Thus

In the sequel, we consider six cases.

(a) If , then it follows from (23) that

which leads to

It follows from (29), (31), and Lemma 2 that

By (32), we derive

From (23) and (33), we obtain that

Then

Thus by (29), (35), and Lemma 2, we get

It follows from (36) that

In view of (33), (37), and (23), we get

Then

Thus by (29), (39) and Lemma 2, we get

It follows from (40) that

(b) If , then it follows from (23) that

which leads to

It follows from (29), (43), and Lemma 2 that

By (44), we derive

From (23) and (45), we obtain that

Then

Thus by (29), (47), and Lemma 2, we get

It follows from (48) that

In view of (45), (49), and (23), we get

Then

Thus by (29), (51), and Lemma 2, we get

It follows from (52) that

(c) If , then it follows from (23) that

which leads to

It follows from (29), (55), and Lemma 2 that

By (56), we derive

From (23) and (57), we obtain that

Then

Thus by (29), (59), and Lemma 2, we get

It follows from (60) that

In view of (57), (61), and (23), we get

Then

Thus by (29), (63), and Lemma 2, we get

It follows from (64) that

(d) If , then it follows from (23) that

which leads to

It follows from (29), (67), and Lemma 2 that

By (56), we derive

From (23) and (57), we obtain that

Then

Thus by (29), (71), and Lemma 2, we get

It follows from (72) that

In view of (69), (73), and (23), we get

Then

Thus by (29), (75), and Lemma 2, we get

It follows from (76) that

(e) If , then it follows from (23) that

which leads to

It follows from (29), (79), and Lemma 2 that

By (80), we derive

From (23) and (33), we obtain that

Then

Thus by (29), (83), and Lemma 2, we get

It follows from (83) and (84) that

In view of (83), (84), and (23), we get

Then

Thus by (29), (88), and Lemma 2, we get

It follows from (89) that

(f) If , then it follows from (23) that

which leads to

It follows from (29), (92), and Lemma 2 that

By (93), we derive

From (23) and (94), we obtain that

Then

Thus by (29), (96), and Lemma 2, we get

It follows from (97) that

In view of (94), (98), and (23), we get

Then

Thus by (29), (100), and Lemma 2, we get

It follows from (101) that

Obviously, are independent of . Take , where is taken sufficiently large such that , where is the unique positive solution of (11). Now we have proved that any solution of (22) in satisfies , .

Let , then it is easy to see that is an open, bounded set in and verifies requirement (a) of Lemma 1. When , is a constant vector in with . Then

Now let us consider homotopic , , where

Letting be the identity mapping and by direct calculation, we get

By now, we have proved that verifies all requirements of Lemma 1, then it follows that has at least one solution in ; that is, to say, (11) has at least one periodic solution in say . Let , , and ; then by Lemma 3 we know that is an periodic solution of system (7) with strictly positive components. The proof is complete.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J 2100), the Governor Foundation of Guizhou Province ( 53), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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