Abstract and Applied Analysis

Volume 2014 (2014), Article ID 980218, 11 pages

http://dx.doi.org/10.1155/2014/980218

Research Article

Global Exponential Stability of Positive Almost Periodic Solutions for a Fishing Model with a Time-Varying Delay

Hong Zhang,¹ Shuhua Gong,² and Jianying Shao²

¹College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, China

²College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

Received 18 October 2013; Revised 3 February 2014; Accepted 17 February 2014; Published 28 April 2014

Academic Editor: Shengqiang Liu

Copyright © 2014 Hong Zhang 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 concerned with a nonautonomous fishing model with a time-varying delay. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of positive almost periodic solutions of the model with almost periodic coefficients and delays. Moreover, an example and its numerical simulation are given to illustrate the main results.

1. Introduction

In the classic study of population dynamics, the differential equation

is widely used in fisheries [1–4], where denotes the population biomass, denotes the per-capita fecundity rate, denotes the per-capita mortality rate, and is the harvesting rate per-capita.

Let be a Hills type function [1, 3]

Denote . Taking account of the delay and the varying environments, Berezansky and Idels [5] proposed the following time-lag model based on (1):

where , , , are almost periodic functions and parameter . Consequently, some theorems on the stability and existence of periodic solutions for (3) were established in Berezansky and Idels [5] and Wang [6]. For more details, we refer to the article of Wang [6].

In the real-world phenomena, the two foundations for the theory of nature selection are periodically and almost periodically varying environment. And the almost periodic effects are more frequent than the periodic effects (see [7, 8]). Therefore, the effects of the almost periodic environment on the evolutionary theory have been extensively studied by a large number of researchers and some of these results can be found in [9–12]. For the reason of seasonal variation, it is not necessary to let (3) be exactly periodic but almost periodic instead. The problem of finding the global stability conditions for the positive almost periodic solution of (3) with almost periodic coefficients and delays becomes important. As pointed out in [7, 8], significant differences often appear in almost periodic problem by comparison with the periodic case. For example, contrary to periodic functions, there exists an almost periodic function such that for all and . Hence, it is difficult to establish the existence, uniqueness, and global exponential stability of positive almost periodic solutions of (3). Moreover, to the best of our knowledge, there is no literature considering the global exponential stability of positive almost periodic solutions problem for (3) and its generalized equations.

Motivated by the above discussions, in this paper, a new approach will be developed to obtain a condition for the global exponential stability of the positive almost periodic solutions of (3), and the exponential convergent rate can be unveiled.

For convenience, we introduce some notations. Given a bounded continuous function defined on , let and be defined as

It will be assumed that

Then,

Let denote nonnegative real number space, let be the continuous functions space equipped with the usual supremum norm , and let . If is defined on with , then we define where for all .

Due to the biological interpretation of model (3), only positive solutions are meaningful and therefore admissible. Thus, we just consider admissible initial conditions

We denote for an admissible solution of the admissible initial value problems (3) and (9). Also, let be the maximal right-interval of existence of .

2. Preliminary Results

In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

Definition 1 (see [7, 8]). Let be continuous in . is said to be almost periodic on if, for any , the set is relatively dense; that is, for any , it is possible to find a real number with the property that, for any interval with length , there exists a number in this interval such that for all .

From the theory of almost periodic functions in [7, 8], it follows that, for any , it is possible to find a real number , and,for any interval with length , there exists a number in this interval such that

for all .

From (5) and Lemma  1.1 in Berezansky and Idels [5], we obtain the following lemma.

Lemma 2. Every solution of (3) and (9) is persistent on , and ; that is, there exist two positive constants and such that

Lemma 3. For every solution of (3) and (9), there exists such that

Proof. Let . It follows from Lemma 2 that

We now give two cases to prove that there exists such that

Case  i. Suppose that

Then,

By the fluctuation lemma [13, Lemma  A.1], there exists a sequence such that

Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that

From (16), we get

By the boundedness of the coefficients and delays, there is a subsequence of , still denoted by , such that , , , and are convergent to , , , and , respectively. In view of the facts and , it follows from

that (taking limits)

which yields

This implies that (14) holds.

Case  ii. If there exists such that , then (3) yields

and we have

Integrating (24) from to , in view of (23), we obtain

For any , with the same approach as that in derivation of (25), we can show

On the other hand, if and , we can choose such that

which, together with (26), yields

Thus, there must exist such that (14) holds.

Again from the fluctuation lemma [13, Lemma  A.1], there exists a sequence such that

Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that

We next divide our proof in two steps to show that there exists such that

First, assume that

Then,

According to (30) and (33), we get

Without loss of generality, we assume that all , , , and are convergent to , , , and , respectively. This can be achieved because of almost periodicity. In view of (3) and (30), it follows from

that (taking limits)

which, together with (8), yields

This implies that (31) holds.

Second, there exists such that

This, together with (3), yields

and we have

for all . Integrating (40) from to , in view of (6), we obtain

For any , with the same approach as that in derivation of (41), we can show

On the other hand, if and , we can choose such that

which, together with (42), yields

Thus, there must exist such that (31) holds.

In summary, (14) and (31) imply that there exists such that (12) holds. This ends the proof of Lemma 3.

Lemma 4. Let

where , , and . Moreover, assume that is a solution of (3) with initial condition (9) and is bounded continuous on . Then, for any , there exists , such that every interval contains at least one number for which there exists satisfying

Proof. Define a continuous function by setting

Then, we have

which implies that there exist two constants and such that

For , we add the definition of with . Set

By Lemma 3, the solution is bounded and

which implies that the right side of (3) is also bounded, and is a bounded function on . Thus, in view of the fact that for , we obtain that is uniformly continuous on . From (10), for any , there exists , such that every interval , , contains a for which

Let . For , denote . Then, for all , we get

where lies between   and .

Calculating the upper right derivative of yields

Let

It is obvious that and is nondecreasing.

Now, we distinguish two cases to finish the proof.

Case  One. Consider

We claim that

Assume, by way of contradiction, that (57) does not hold. Then, there exists such that . Since

There must exist such that

which contradicts (56). This contradiction implies that (57) holds. It follows that there exists such that

Case  Two. There is a such that . Then, in view of (49), (52), and (54), we get

which yields that

For any , with the same approach as that in derivation of (62), we can show

if .

On the other hand, if and , we can choose such that

which, together with (63), yields

With a similar argument as that in the proof of Case One, we can show that

which implies that

In summary, there must exist such that holds for all . The proof of Lemma 4 is now complete.

3. Main Results

In this section, we establish sufficient conditions on the existence and global exponential stability of almost periodic solutions of (3).

Theorem 5. Under the assumptions of Lemma 4, (3) has at least one positive almost periodic solution . Moreover, is globally exponentially stable; that is, there exist constants and such that

Proof. Let be a solution of (3) with initial conditions satisfying the assumptions in Lemma 4. We also add the definition of with for all . Set

where is any sequence of real numbers. By Lemma 3, the solution is bounded and

which implies that the right side of (3) is also bounded, and is a bounded function on . Thus, in view of the fact that for , we obtain that is uniformly continuous on . Then, from the almost periodicity of , , , and , we can select a sequence such that

Since is uniformly bounded and equiuniformly continuous, by Arzala-Ascoli Lemma and diagonal selection principle, we can choose a subsequence of , such that (for convenience, we still denote it by ) uniformly converges to a continuous function on any compact set of , and

Now, we prove that is a solution of (2). In fact, for any and , from (71), we have

where . Consequently, (73) implies that

Therefore, is a solution of (3).

Secondly, we prove that is an almost periodic solution of (3). With the help of Lemma 4, for any , there exists , such that every interval contains at least one number for which there exists satisfying

Then, for any fixed , we can find a sufficient large positive integer such that, for any ,

Let ; we obtain

which implies that is an almost periodic solution of (3).

Finally, we prove that is globally exponentially stable.

Let and , where . Then,

where and lies between and .

It follows from Lemma 3 that there exists such that

We consider the Lyapunov functional

Calculating the upper left derivative of along the solution of (78), we have

We claim that

Contrarily, there must exist such that

Together with (55), (81), and (83), we obtain

Thus,

which contradicts (49). Hence, (82) holds. It follows that

This completes the proof of Theorem 5.

4. An Example

In this section, we present an example to check the validity of the results we obtained in the previous sections.

Example 1. Consider the following fishing model with time-varying delay:

Obviously, , , , , , . By calculating, we obtain

which imply that (87) satisfies the assumptions of Theorem 5. Therefore, (87) has a unique positive almost periodic solution , which is globally exponentially stable with the exponential convergent rate . The numerical simulation in Figure 1 strongly supports the conclusion.

Figure 1: Numerical solution of system (87) for initial value .

Remark 6. Most recently, by using Mawhin continuation theorem, criteria ensuring the local existence of almost periodic solutions for the fishing model (3) are established in Li et al. [14], where the global exponential convergence for almost periodic solution is not touched. Unfortunately, as pointed out by Wang and Zhang [15] and Ortega [16], for the essential reason that the compact condition is not suitable for the almost periodic function family, the coincidence degree cannot be used to solve almost periodic problem. Hence, the mapping of Lemma 3.3 in Li et al. [14] is not -compact and the existence of almost periodic solutions for (3) cannot hold. Moreover, to the best of our knowledge, there is no research on the global exponential stability of positive almost periodic solutions to the fishing model (3). We also mention that all results in [5, 6, 9, 10, 12] cannot be applied to imply that all solutions of (87) with initial values (9) converge exponentially to the positive almost periodic solution. Here, we employ a novel proof to establish some criteria to guarantee the existence and global exponential stability of positive almost periodic solutions for fishing model with time-varying coefficients and delays. This implies that the results of this paper are contributed to complement the previous references in this topic.

Conflict of Interests

The authors declare no conflict of interests. They also declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this present paper.

Acknowledgments

The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors express the sincere gratitude to Professor Bingwen Liu for the helpful discussion when this revision work is carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).

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