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

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 is chosen as or , 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.

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 [1–4] 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:

(1.1)

which incorporates mutual interference by predators, where is a Michaelis-Menten type functional response function. Equation (1.1) has been studied by many authors and seen great progress (e.g., see [6–11]).

Xu and Chen [11] studied a delayed two-predator-one-prey model in two patches which is described by the following differential equations:

(1.2)

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

(1.3)

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 :

(1.4)

with the initial conditions

(1.5)

where . In (1.4), represents the prey population in the th patch , and represents the predator population. is the prey for , and is the prey for so that they form a food chain. denotes the dispersal rate of the prey in the th patch . For the sake of generality and convenience, we always make the following fundamental assumptions for system (1.4):

• (H) are all rd-continuous positive periodic functions with period ; are nonnegative constants.

In (1.4), set . If , then (1.4) reduces to the ratio-dependent predator-prey diffusive system of three species with time delays governed by the ordinary differential equations

(1.6)

If , then (1.4) is reformulated as

(1.7)

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

Definition 2.1.

The forward jump operator , the backward jump operator , and the graininess are defined, respectively, by

(2.1)

If , then is called right-dense (otherwise: right-scattered), and if , then is called left-dense (otherwise: left-scattered).

If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

Definition 2.2.

Assume is a function and let . Then one defines to be the number (provided it exists) with the property that given any , there is a neighborhood of such that

(2.2)

In this case, is called the delta (or Hilger) derivative of at . Moreover, is said to be delta or Hilger differentiable on if exists for all . A function is called an antiderivative of provided for all . Then one defines

(2.3)

Definition 2.3.

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

Definition 2.4.

If , , and is rd-continuous on , then one defines the improper integral by

(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 is periodic if there exists such that if , then . For , the smallest positive is called the period of the time scale.

Definition 2.6 (see [14]).

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

If , one says that is periodic with period if is the smallest positive number such that for all .

Lemma 2.7.

Every rd-continuous function has an antiderivative.

Lemma 2.8.

Every continuous function is rd-continuous.

Lemma 2.9.

If and , then

• (a) ;

• (b) if for all , then ;

• (c) if on , then .

Lemma 2.10.

If , then is nondecreasing.

Notation.

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

(2.5)

where is an -periodic function, that is, for all , .

Notation.

Let be two Banach spaces, let be a linear mapping, and let be a continuous mapping. If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , then the restriction is invertible. 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 exists an isomorphism .

Lemma 2.11 (Continuation theorem [15]).

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

• (a) for each ;

• (b) for each ;

• (c) .

Then the operator equation has at least one solution in .

Lemma 2.12 (see [16]).

Let . If is -periodic, then

(2.6)

3. Existence of Periodic Solutions

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

Theorem 3.1.

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

• (i)

• (ii) ,

• (iii) ,

• (iv) ,

then the system (1.4) has at least one -periodic solution.

Proof.

Consider vector equation

(3.1)

Define

(3.2)

where is the Euclidean norm. Then and are both Banach spaces with the above norm . Let . Then

(3.3)

and . Since is closed in , then is a Fredholm mapping of index zero. It is easy to show that are continuous projectors such that . Furthermore, the generalized inverse (to ) exists and is given by , thus

(3.4)

Obviously, are continuous. Since is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that is compact for any open bounded set . Moreover, is bounded, thus, is -compact on for any open bounded set . Corresponding to the operator equation , we have

(3.5)

Suppose that is a solution of (3.5) for certain . Integrating on both sides of (3.5) from to with respect to , we have

(3.6)

(3.7)

(3.8)

(3.9)

It follows from (3.5) to (3.9) that

(3.10)

(3.11)

(3.12)

(3.13)

Multiplying (3.6) by and integrating over gives

(3.14)

which yields

(3.15)

By using the inequality , we have

(3.16)

Then

(3.17)

By using the inequality , we derive from (3.17) that

(3.18)

Similarly, multiplying (3.7) by and integrating over , then synthesize the above, we obtain

(3.19)

It follows from (3.18) and (3.19) that

(3.20)

so, there exists a positive constant such that

(3.21)

which together with (3.19), there also exists a positive constant such that

(3.22)

This, together with (3.11), (3.12), and (3.21), leads to

(3.23)

Since , there exist some points , such that

(3.24)

It follows from (3.21) and (3.22) that

(3.25)

From (3.8) and (3.9), we obtain that

(3.26)

This, together with (3.12), (3.13), and (3.26), deduces

(3.27)

From (3.6) and (3.24), we have

(3.28)

From (3.7) and (3.24), it yields that

(3.29)

Noticing that , from (3.8) and (3.9), deduces

(3.30)

There exist two points such that

(3.31)

Hence,

(3.32)

where . Then, this, together with (3.12), (3.13), (3.23), (3.28), (3.29), and (3.32), deduces

(3.33)

It follows from (3.27) to (3.33) that

(3.34)

From (3.34), we clearly know that are independent of , and from the representation of , it is easy to know that there exist points such that , where

(3.35)

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

When , is a constant vector in with , from the definition of , we can naturally derive 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 by

(3.36)

where

(3.37)

where is a parameter. From (3.37), it is easy to show that . Moreover, one can easily show that the algebraic equation

(3.)

has a unique positive solution in . Note that (identical mapping), since , according to the invariance property of homotopy, direct calculation produces

(3.39)

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

Remark 3.3.

If and 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

(3.40)

where are positive -periodic functions, 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) , (ii) hold, then (3.40) has at least one -periodic solution.

Remark 3.6.

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