Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 368176, 8 pages

http://dx.doi.org/10.1155/2013/368176

Research Article

Periodic Solutions for Gause-Type Ratio-Dependent Predator-Prey Systems with Delays on Time Scales

Xiaoquan Ding, Hongyuan Liu, and Fengye Wang

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China

Received 28 February 2013; Accepted 27 May 2013

Academic Editor: Qiru Wang

Copyright © 2013 Xiaoquan Ding et al. 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.

This paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numerical responses on time scales. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solutions. In particular, our results improve and generalize some known ones.

1. Introduction

In this paper, we consider the following periodic Gause-type ratio-dependent predator-prey system with time delays on a time scale :

Here is a periodic time scale which has the subspace topology inherited from the standard topology on . The symbol stands for the delta-derivative which gives the ordinary derivative if , and the forward difference operator if .

The theory of dynamic equations on time scales, which was introduced in order to unify differential and difference equations, has recently received a lot of attention, and many results on the issue have been documented in monographs [1, 2] and papers [3–12].

In system (1), set , . If , then system (1) reduces to the standard Gause-type ratio-dependent predator-prey system governed by the ordinary differential equations

where and stand for the population of the prey and the predator, respectively. The function is the growth rate of the prey in the absence of the predator. The function is the death rate of the predator. The function , called functional response of predator to prey, describes the change in the rate of exploitation of prey by a predator as a result of a change in the prey density. The function , called numerical response of predator to prey, describes the change in reproduction rate with changing prey density. If , then system (1) is reformulated as

which is the discrete time ratio-dependent predator-prey system and is a discrete analogue of (2).

We note that Ding [13], Ding and Jiang [14], Fan and Li [15], Fan et al. [16], Fan and Wang [17], Fan et al. [18], Hsu et al. [19], Kuang [20], Wang and Li [21], and Xia and Han [22] considered some special cases of system (2). Fan and Wang [23], and Xia et al. [24] discussed some special cases of system (3). Bohner et al. [4], and Shao [10] investigated some special cases of system (1). So far as we know, there is no published paper concerned system (1).

The main purpose of this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [25], to derive sufficient conditions for the existence of periodic solutions of system (1). Furthermore, we will see that our results for the above system can be easily extended to the one with distributed or state-dependent delays. Our results improve or generalize some theorems in [10, 13–18, 21, 23].

2. Preliminaries

In this section, we briefly give some elements of the time scale calculus, recall the continuation theorem from coincidence degree theory, and state an auxiliary result that will be used in this paper.

First, let us present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, we refer the reader to [1, 2].

A time scale is an arbitrary nonempty closed subset of the real numbers , which inherits the standard topology of . Thus, the real numbers , the integers , and the natural numbers are examples of time scales, while the rational numbers and the open interval are no time scales.

Let . Throughout this paper, the time scale is assumed to be -periodic; that is, implies . In particular, the time scale under consideration is unbounded above and below.

For , the forward and backward jump operators are defined by

respectively.

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

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

For and we define , the delta-derivative of at , to be the number (provided it exists) with the property that, given any , there is a neighborhood of (i.e., for some ) in such that

is said to be delta-differentiable if its delta-derivative exists for all . The set of functions that are delta-differentiable and whose delta-derivative are rd-continuous functions is denoted by .

A function is called a delta-antiderivative of provided , for all . Then, we define the delta integral by

If is continuous, then is rd-continuous, and if is delta-differentiable at , then is continuous at . Moreover, every rd-continuous function on has a delta-antiderivative.

Next, let us recall the continuation theorem in coincidence degree theory. To do so, we need to introduce the following notation.

Let , be real Banach spaces, be a linear mapping, and be a continuous mapping.

The mapping is said to be a Fredholm mapping of index zero, if and is closed in .

If is a Fredholm mapping of index zero, then there exist continuous projectors and , such that , . It follows that the restriction of to is invertible. Denote the inverse of by .

The mapping is said to be -compact on , if is an open bounded subset of , is bounded and is compact.

Since is isomorphic to , there exists an isomorphism .

Here we state the Gaines-Mawhin theorem, which is a main tool in the proof of our main result.

Lemma 1 (Continuation Theorem [25, page 40]). Let be an open bounded set, a Fredholm mapping of index zero, and   -compact on . Assume

(a)for each ;(b)for each ;(c).

Then has at least one solution in .

For convenience and simplicity in the following discussion, we always use the following notation:

where is an -periodic function, is rd-continuous and -periodic in its first variable.

In order to achieve the prior estimation in the case of dynamic equations on a time scale , we also require the following inequality which is proved in [11, Theorem 2.1].

Lemma 2. Let , , and . If is an -periodic real function, then

3. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of system (1). For the sake of generality, we make the following fundamental assumptions for system (1):

 (H₁) are rd-continuous and -periodic such that for all .  (H₂) is rd-continuous and -periodic such that .  (H₃) is rd-continuous and -periodic in the first variable and is continuously differentiable in the second variable such that , , and for all , .  (H₄) is rd-continuous and -periodic in the first variable and is continuously differentiable in the second variable.  (H₅) is rd-continuous and -periodic in the first variable and is continuously differentiable in the second variable such that for all .  (H₆) and , where for all .

From (H₃), we have

Thus is strictly decreasing on . From this, (H₃), and (H₆), one can easily see that each of the equations

has a unique positive solution. Let , be their solutions, respectively, then .

3.1. Monotone Case

In this section, we study the existence of -periodic solution of system (1) when the numerical response function satisfies the following monotone condition:

 (H₇) for , .

From (H₅) and (H₇), we have

Then is strictly increasing on . From this, (H₅), and (H₆), one can easily see that the equation has a unique positive solution. Let be its solution.

Theorem 3. Suppose that (H₁)–(H₇) hold. Then system (1) has at least one -periodic solution.

Proof. Take

then and are Banach spaces with the norm . Set

where and

With these notations system (1) can be written in the form

Obviously, , is closed in , and . Therefore is a Fredholm mapping of index zero. Now define two projectors and as

then and are continuous projectors such that

Furthermore, through an easy computation we find that the generalized inverse of has the form

Then and read as

Clearly, and are continuous. By using the Arzela-Ascoli theorem, it is not difficult to prove that is compact for any open bounded set . Moreover, is bounded. Therefore is -compact on with any open bounded set .

In order to apply Lemma 1, we need to find an appropriate open, bounded subset in . Corresponding to the operator equation , we have

Suppose that is a solution of (20) for a certain . Integrating (20) on both sides from to leads to

That is

From (22), we have

It follows from (20), (22), (23), (24), and (H₃)–(H₅) that

Since , there exist such that

Then from (22), (27), (H₃), (H₄), and the monotonicity of function , we have

which, together with the monotonicity of function , leads to

By Lemma 2, we obtain from (25) and (29) that for all

From (22), (27), (H₃), and the monotonicity of function , we have

which, together with the monotonicity of function , leads to

By Lemma 2, we get from (25) and (32) that for all

From (23), (27), (H₅), and the monotonicity of function , we have

which, together with the monotonicity of function , lead to

That is

By Lemma 2, we get from (26), (30), (33), and (36) that for all

In view of (30), (33), (37), we obtain

Clearly, and are independent of .

From (), (), and the monotonicity of functions and , it is easy to show that the algebraic system

has a unique solution .

We now take , where is taken sufficiently large such that . With the help of (38) and (39), it is easy to see that satisfies condition (a) in Lemma 1. When , is a constant vector in with . Thus, we have

This proves that condition (b) in Lemma 1 is satisfied.

Taking , , a direct calculation shows that

By now we have proved that satisfies all the requirements in Lemma 1. Hence, (1) has at least one -periodic solution. This completes the proof.

Noticing that both systems (2) and (3) are special cases of system (1), by Theorem 3, we can obtain the following results.

Theorem 4. Suppose that (H₁)–(H₇) hold. Then system (2) has at least one positive -periodic solution.

Theorem 5. Suppose that (H₁)–(H₇) hold. Then system (3) has at least one positive -periodic solution.

3.2. Nonmonotone Case

In this section, we study the existence of -periodic solution of system (1) when the numerical response function satisfies the following nonmonotone condition:

 (H₈) for , and there exists a positive constant such that

By (H₅) and (H₈), we have

then is strictly increasing on and strictly decreasing on . By these, (H₅) and (H₆), one can easily see that the equation

has two distinct positive solutions, namely, , . Without loss of generality, we suppose that , then .

Theorem 6. In addition to (H₁)–(H₆) and (H₈), suppose further that the following condition holds:

(H₉).

Then system (1) has at least two -periodic solutions.

Proof. The proof is similar to that of Theorem 3. To complete the proof, we need to find two disjoint open bounded subsets in . Under condition (H₉), (34) is no longer valid. So, we need to make a corresponding change.

Designating , there also exist , such that

From (23), (27), and the monotonicity of functions and , we will show that and can not simultaneously lie in or . In fact, if , then

This is a contradiction. If , then

This is also a contradiction. If , then

This is also a contradiction. Consequently, the distribution of and only have the following two cases.

Case  1. . Noticing that

from (27), (30) and (33), we obtain

By Lemma 2, we find from (26) and (51) that for all

Case  2. . From (27), (30) and (33), we also obtain

By Lemma 2, we find from (26) and (53) that for all

By (H₉), we know that

Clearly, , , , and are independent of .

By the monotonicity of functions and , (H₃), (H₅), and (H₆), it is easy to show that algebraic system (40) has two distinct solutions:

where is the unique solution of equation . Obviously,

We now take

where is taken sufficiently large such that

Then both and are bounded open subsets of . It follows from (55), (57), and (59) that , , and . With the help of (38), (52), (54), and (59), it is easy to see that and satisfy condition (a) in Lemma 1. When , is a constant vector in . Thus, we have

This proves that condition (b) in Lemma 1 is satisfied. A direct calculation shows that

By now we have proved that and satisfy all the requirements in Lemma 1. Hence, system (1) has at least two -periodic solutions and in and , respectively. This completes the proof.

Noticing that both systems (2) and (3) are special cases of system (1), by Theorem 6, we can obtain the following results.

Theorem 7. Suppose that ()–() and ()-() hold. Then system (2) has at least two positive -periodic solutions.

Theorem 8. Suppose that ()–() and ()-() hold. Then system (3) has at least two positive -periodic solutions.

Remark 9. The proof of Theorems 3 and 6 shows that all the above results also remain valid if some delay terms are replaced by terms with distributed or state-dependent delays.

Remark 10. In their Theorem 2.2, Ding and Jiang [14] proved that (2) has at least two positive -periodic solutions if (H₁)–(H₆), (H₈), and

(),

hold. Obviously, the condition () implies (H₉). Hence, our Theorem 7 improves Theorem 2.2 of [14].

Remark 11. Shao [10] studied a special case of system (1) for . Therefore, our Theorem 3 generalizes Theorem 1 of [10].

Remark 12. Fan and Wang [17], Wang and Li [21], and Fan et al. [16, 18] studied some special cases of system (2) for . Therefore, our Theorem 4 generalizes Theorem 2.1 of [17], Theorem 3.1 of [16], Theorem 3.5 of [18], and Theorem 2.1 of [21].

Remark 13. Fan and Wang [23] and Xia et al. [24] studied some special cases of system (3) for . Therefore, our Theorem 5 generalizes Theorem 2.1 of [23], and Theorem 3.1 of [24].

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grant 12A110007, and the Scientific Research Start-up Funds of Henan University of Science and Technology.

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