# Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales

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## Abstract

This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when the time scale Open image in new window is chosen as Open image in new window or Open image in new window , the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.

### Keywords

Periodic Solution Functional Response Jump Operator Continuation Theorem Fredholm Mapping## 1. Introduction

which incorporates mutual interference by predators, where Open image in new window is a Michaelis-Menten type functional response function. Equation (1.1) has been studied by many authors and seen great progress (e.g., see [6, 7, 8, 9, 10, 11]).

In order to consider periodic variations of the environment and the density regulation of the predators though taking into account delay effect and diffusion between patches, more realistic and interesting models of population interactions should take into account comprehensively other than one or two aspects. On the other hand, in order to unify the study of differential and difference equations, people have done a lot of research about dynamic equations on time scales. The principle aim of this paper is to systematically unify the existence of periodic solutions for a delayed ratio-dependent predator-prey system with functional response and diffusion modeled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on Gaines and Mawhin's continuation theorem of coincidence degree theory, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations.

- (H)
Open image in new window are all rd-continuous positive periodic functions with period Open image in new window ; Open image in new window are nonnegative constants.

which is the discrete time ratio-dependent predator-prey diffusive system of three species with time delays and is also a discrete analogue of (1.6).

## 2. Preliminaries

A time scale Open image in new window is an arbitrary nonempty closed subset of the real numbers Open image in new window . Throughout the paper, we assume the time scale Open image in new window is unbounded above and below, such as Open image in new window and Open image in new window . The following definitions and lemmas can be found in [13].

Definition 2.1.

If Open image in new window , then Open image in new window is called right-dense (otherwise: right-scattered), and if Open image in new window , then Open image in new window is called left-dense (otherwise: left-scattered).

If Open image in new window has a left-scattered maximum Open image in new window , then Open image in new window ; otherwise Open image in new window . If Open image in new window has a right-scattered minimum Open image in new window , then Open image in new window ; otherwise Open image in new window .

Definition 2.2.

Definition 2.3.

A function Open image in new window is said to be rd-continuous if it is continuous at right-dense points in Open image in new window and its left-sided limits exists (finite) at left-dense points in Open image in new window . The set of rd-continuous functions Open image in new window will be denoted by Open image in new window .

Definition 2.4.

provided this limit exists, and one says that the improper integral converges in this case.

Definition 2.5 (see [14]).

One says that a time scale Open image in new window is periodic if there exists Open image in new window such that if Open image in new window , then Open image in new window . For Open image in new window , the smallest positive Open image in new window is called the period of the time scale.

Definition 2.6 (see [14]).

Let Open image in new window be a periodic time scale with period Open image in new window . One says that the function Open image in new window is periodic with period Open image in new window if there exists a natural number Open image in new window such that Open image in new window , Open image in new window for all Open image in new window and Open image in new window is the smallest number such that Open image in new window .

If Open image in new window , one says that Open image in new window is periodic with period Open image in new window if Open image in new window is the smallest positive number such that Open image in new window for all Open image in new window .

Lemma 2.7.

Every rd-continuous function has an antiderivative.

Lemma 2.8.

Every continuous function is rd-continuous.

Lemma 2.9.

If Open image in new window and Open image in new window , then

- (a)
- (b)
if Open image in new window for all Open image in new window , then Open image in new window ;

- (c)

Lemma 2.10.

If Open image in new window , then Open image in new window is nondecreasing.

Notation.

where Open image in new window is an Open image in new window -periodic function, that is, Open image in new window for all Open image in new window , Open image in new window .

Notation.

Let Open image in new window be two Banach spaces, let Open image in new window be a linear mapping, and let Open image in new window be a continuous mapping. If Open image in new window is a Fredholm mapping of index zero and there exist continuous projectors Open image in new window and Open image in new window such that Open image in new window , Open image in new window , then the restriction Open image in new window is invertible. Denote the inverse of that map by Open image in new window . If Open image in new window is an open bounded subset of Open image in new window , the mapping Open image in new window will be called Open image in new window -compact on Open image in new window if Open image in new window is bounded and Open image in new window is compact. Since Open image in new window is isomorphic to Open image in new window , there exists an isomorphism Open image in new window .

Lemma 2.11 (Continuation theorem [15]).

- (a)
for each Open image in new window ;

- (b)
for each Open image in new window ;

- (c)

Then the operator equation Open image in new window has at least one solution in Open image in new window .

Lemma 2.12 (see [16]).

## 3. Existence of Periodic Solutions

The fundamental theorem in this paper is stated as follows about the existence of an Open image in new window -periodic solution.

Theorem 3.1.

- (i)
- (ii)
- (iii)
- (iv)

then the system (1.4) has at least one Open image in new window -periodic solution.

Proof.

Take Open image in new window , where Open image in new window is taken sufficiently large such that Open image in new window , and such that each solution Open image in new window of the system Open image in new window satisfies Open image in new window if the system (3.35) has solutions. Now take Open image in new window . Then it is clear that Open image in new window verifies the requirement (a) of Lemma 2.11.

When Open image in new window , Open image in new window is a constant vector in Open image in new window with Open image in new window , from the definition of Open image in new window , we can naturally derive Open image in new window whether the system (3.35) has solutions or not. This shows that the condition (b) of Lemma 2.11 is satisfied.

where Open image in new window is the Brouwer degree. By now we have proved that Open image in new window verifies all requirements of Lemma 2.11. Therefore, (1.4) has at least one Open image in new window -periodic solution in Open image in new window . The proof is complete.

Corollary 3.2.

If the conditions in Theorem 3.1 hold, then both the corresponding continuous model (1.6) and the discrete model (1.7) have at least one Open image in new window -periodic solution.

Remark 3.3.

If Open image in new window and Open image in new window in (1.6), then the system (1.6) reduces to the continuous ratio-dependence predator-prey diffusive system proposed in [17].

Remark 3.4.

where Open image in new window are positive Open image in new window -periodic functions, Open image in new window is nonnegative constant. It is easy to obtain the corresponding conclusions on time scales for the system (3.40).

Corollary 3.5.

Suppose that (i) Open image in new window , (ii) Open image in new window hold, then (3.40) has at least one Open image in new window -periodic solution.

Remark 3.6.

The result in Corollary 3.5 is same as those for the corresponding continuous and discrete systems.

## Notes

### Acknowledgments

The author is very grateful to his supervisor Prof. M. Fan and the anonymous referees for their many valuable comments and suggestions which greatly improved the presentation of this paper. This work is supported by the Foundation for subjects development of Harbin University (no. HXK200716) and by the Foundation for Scientific Research Projects of Education Department of Hei-longjiang Province of China (no. 11513043).

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