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Advances in Difference Equations

, 2009:141589 | Cite as

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

  • Zhenjie Liu
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

The traditional predator-prey model has received great attention from both theoretical and mathematical biologists and has been studied extensively (e.g., see [1, 2, 3, 4] and references therein). Based on growing biological and physiological evidences, some biologists have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be ratio-dependent, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Starting from this argument and the traditional prey-dependent-only mode, Arditi and Ginzburg [5] first proposed the following ratio-dependent predator-prey model:

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]).

Xu and Chen [11] studied a delayed two-predator-one-prey model in two patches which is described by the following differential equations:
In view of periodicity of the actual environment, Huo and Li [12] investigated a more general delayed ratio-dependent predator-prey model with periodic coefficients of the form

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.

Therefore, it is interesting and important to study the following model on time scales Open image in new window :
with the initial conditions
where Open image in new window . In (1.4), Open image in new window represents the prey population in the Open image in new window th patch Open image in new window , and Open image in new window represents the predator population. Open image in new window is the prey for Open image in new window , and Open image in new window is the prey for Open image in new window so that they form a food chain. Open image in new window denotes the dispersal rate of the prey in the Open image in new window th patch Open image in new window . For the sake of generality and convenience, we always make the following fundamental assumptions for system (1.4):
  1. (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.

     
In (1.4), set Open image in new window . If Open image in new window , then (1.4) reduces to the ratio-dependent predator-prey diffusive system of three species with time delays governed by the ordinary differential equations
If Open image in new window , then (1.4) is reformulated as

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.

The forward jump operator Open image in new window , the backward jump operator Open image in new window , and the graininess Open image in new window are defined, respectively, by

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.

Assume Open image in new window is a function and let Open image in new window . Then one defines Open image in new window to be the number (provided it exists) with the property that given any Open image in new window , there is a neighborhood Open image in new window of Open image in new window such that
In this case, Open image in new window is called the delta (or Hilger) derivative of Open image in new window at Open image in new window . Moreover, Open image in new window is said to be delta or Hilger differentiable on Open image in new window if Open image in new window exists for all Open image in new window . A function Open image in new window is called an antiderivative of Open image in new window provided Open image in new window for all Open image in new window . Then one defines

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.

If Open image in new window , Open image in new window , and Open image in new window is rd-continuous on Open image in new window , then one defines the improper integral by

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

Lemma 2.10.

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

Notation.

To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let Open image in new window be Open image in new window -periodic, that is, Open image in new window implies Open image in new window ,

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]).

Let Open image in new window be two Banach spaces, and let Open image in new window be a Fredholm mapping of index zero. Assume that Open image in new window is Open image in new window -compact on Open image in new window with Open image in new window is open bounded in Open image in new window . Furthermore assume the following:

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.

Suppose that (H) holds. Furthermore assume the following:

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

Proof.

Consider vector equation
where Open image in new window is the Euclidean norm. Then Open image in new window and Open image in new window are both Banach spaces with the above norm Open image in new window . Let Open image in new window . Then
and Open image in new window . Since Open image in new window is closed in Open image in new window , then Open image in new window is a Fredholm mapping of index zero. It is easy to show that Open image in new window are continuous projectors such that Open image in new window . Furthermore, the generalized inverse (to Open image in new window ) Open image in new window exists and is given by Open image in new window , thus
Obviously, Open image in new window are continuous. Since Open image in new window is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that Open image in new window is compact for any open bounded set Open image in new window . Moreover, Open image in new window is bounded, thus, Open image in new window is Open image in new window -compact on Open image in new window for any open bounded set Open image in new window . Corresponding to the operator equation Open image in new window , we have
Suppose that Open image in new window is a solution of (3.5) for certain Open image in new window . Integrating on both sides of (3.5) from Open image in new window to Open image in new window with respect to Open image in new window , we have
It follows from (3.5) to (3.9) that
Multiplying (3.6) by Open image in new window and integrating over Open image in new window gives
which yields
By using the inequality Open image in new window , we have
By using the inequality Open image in new window , we derive from (3.17) that
Similarly, multiplying (3.7) by Open image in new window and integrating over Open image in new window , then synthesize the above, we obtain
It follows from (3.18) and (3.19) that
so, there exists a positive constant Open image in new window such that
which together with (3.19), there also exists a positive constant Open image in new window such that
This, together with (3.11), (3.12), and (3.21), leads to
Since Open image in new window , there exist some points Open image in new window , such that
It follows from (3.21) and (3.22) that
From (3.8) and (3.9), we obtain that
This, together with (3.12), (3.13), and (3.26), deduces
From (3.6) and (3.24), we have
From (3.7) and (3.24), it yields that
Noticing that Open image in new window , from (3.8) and (3.9), deduces
There exist two points Open image in new window such that
where Open image in new window . Then, this, together with (3.12), (3.13), (3.23), (3.28), (3.29), and (3.32), deduces
It follows from (3.27) to (3.33) that
From (3.34), we clearly know that Open image in new window are independent of Open image in new window , and from the representation of Open image in new window , it is easy to know that there exist points Open image in new window such that Open image in new window , where

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.

Finally, we will prove that the condition (c) of Lemma 2.11 is valid. Define the homotopy Open image in new window by
where Open image in new window is a parameter. From (3.37), it is easy to show that Open image in new window . Moreover, one can easily show that the algebraic equation
has a unique positive solution Open image in new window in Open image in new window . Note that Open image in new window (identical mapping), since Open image in new window , according to the invariance property of homotopy, direct calculation produces

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.

If we only consider the prey population in one-patch environment and ignore the dispersal process in the system (1.4), then the classical ratio-dependence two species predator-prey model in particular of (1.4) with Michaelis-Menten functional response and time delay on time scales

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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Copyright information

© Zhenjie Liu. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.School of Mathematics and ComputerHarbin UniversityHarbinChina

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