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International Journal of Differential Equations
Volume 2011, Article ID 354016, 12 pages
http://dx.doi.org/10.1155/2011/354016
Research Article

Positive Almost Periodic Solutions for a Time-Varying Fishing Model with Delay

Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China

Received 19 May 2011; Revised 8 August 2011; Accepted 12 August 2011

Academic Editor: Dexing Kong

Copyright © 2011 Xia Li 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 time-varying fishing model with delay. By means of the continuation theorem of coincidence degree theory, we prove that it has at least one positive almost periodic solution.

1. Introduction

Consider the following differential equation which is widely used in fisheries [14]: where is the population biomass, is the per capita fecundity rate, is the per capita mortality rate, and is the harvesting rate per capita.

In (1.1), let be a Hills' type function ([1, 2]) and take into account the delay and the varying environments; Berezansky and Idels [5] proposed the following time-lag model based on (1.1) [16] where .

The model (1.3) has recently attracted the attention of many mathematicians and biologists; see the differential equations which are widely used in fisheries [1, 2]. However, one can easily see that all equations considered in the above-mentioned papers are subject to periodic assumptions, and the authors, in particular, studied the existence of their periodic solutions. On the other hand, ecosystem effects and environmental variability are very important factors and mathematical models cannot ignore, for example, reproduction rates, resource regeneration, habitat destruction and exploitation, the expanding food surplus, and other factors that affect the population growth. Therefore it is reasonable to consider the various parameters of models to be changing almost periodically rather than periodically with a common period. Thus, the investigation of almost periodic behavior is considered to be more accordant with reality. Although it has widespread applications in real life, the generalization to the notion of almost periodicity is not as developed as that of periodic solutions; we refer the reader to [718].

Recently, the authors of [19] proved the persistence and almost periodic solutions for a discrete fishing model with feedback control. In [20, 21], the contraction mapping principle and the continuation theorem of coincidence degree have been employed to prove the existence of positive almost periodic exponential stable solutions for logarithmic population model, respectively. A primary purpose of this paper, nevertheless, is to utilize the continuation theorem of coincidence degree for this purpose. To the best of the authors’ observation, there exists no paper dealing with the proof of the existence of positive almost periodic solutions for (1.3) using the continuation theorem of coincidence degree. Therefore, our result is completely different and presents a new approach.

2. Preliminaries

Our first observation is that under the invariant transformation , (1.3) reduces to for , with the initial function and the initial value For (2.1) and (2.2), we assume the following conditions: (A1) and ;(A2) is a continuous function on that satisfies ; (A3) is a continuous bounded function, .

By a solution of (2.1) and (2.2) we mean an absolutely continuous function defined on satisfying (2.1) almost everywhere for and (2.2). As we are interested in solutions of biological significance, we restrict our attention to positive ones.

According to [22], the initial value problem (2.1) and (2.2) has a unique solution defined on .

Let be normed vector spaces, be a linear mapping, and be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , it follows that the mapping is invertible. We denote the inverse of that mapping by . If is an open bounded subset of , then the mapping will be called -compact on , if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .

Theorem 2.1 (see [19]). Let be an open bounded set and let be a continuous operator which is -compact on . Assume that (1) for every and ; (2) for every ; (3)the Brouwer degree . Then has at least one solution in .

3. Existence of Almost Periodic Solutions

Let denote the set of all real valued almost periodic functions on , for we denote by the set of Fourier exponents and the module of , respectively. Let denote the set of -almost periods for with respect to , denote the length of the inclusion interval, and denote the mean value of .

Definition 3.1. 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 for any interval with length ; there exists a number in this interval such that for any .

Throughout the rest of the paper we assume the following condition for (2.1): (H).

In our case, we set where where and is a given constant; define the norm

Remark 3.2. If is -almost periodic function, then is -almost periodic if and only if . Whereas does not necessarily have an almost periodic primitive, . That is why we can not make and let .

We start with the following lemmas.

Lemma 3.3. and are Banach spaces endowed with the norm .

Proof. If and converge to , then it is easy to show that with . Indeed, for all we have Thus which implies that . One can easily see that is a Banach space endowed with the norm . The same can be concluded for the spaces and . The proof is complete.

Lemma 3.4. Let and where . Then is a Fredholm mapping of index zero.

Proof. It is obvious that is a linear operator and . It remains to prove that . Suppose that . Then, there exist and such that From the definitions of and , one can deduce that and are almost periodic functions and thus , which implies that . This tells us that On the other hand, if then we have . Indeed, if then we obtain It follows that Thus Note that is the primitive of in ; therefore we have . Hence, we deduce that which completes the proof of our claim. Therefore, Furthermore, one can easily show that is closed in and Therefore, is a Fredholm mapping of index zero.

Lemma 3.5. Let , , and such that Then, is L-compact on ( is an open and bounded subset of ).

Proof. The projections and are continuous such that It is clear that Therefore In view of we can conclude that the generalized inverse (of ) exists and is given by Thus where is defined by
The integral form of the terms of both and implies that they are continuous. We claim that is also continuous. By our hypothesis, for any and any compact set , let be the inclusion interval of . Suppose that and uniformly converges to . Because , there exists such that . Let be the inclusion interval of and It is easy to see that is the inclusion interval of both and . Hence, for all , there exists such that . Therefore, by the definition of almost periodic functions we observe that By applying (3.26), we conclude that is continuous and consequently and are also continuous.
From (3.26), we also have that and are uniformly bounded in . In addition, it is not difficult to verify that is bounded and is equicontinuous in . Hence by the Arzelà-Ascoli theorem, we can immediately conclude that is compact. Thus is -compact on .

Theorem 3.6. Let condition (H) hold. Then (2.1) has at least one positive almost periodic solution.

Proof. It is easy to see that if (2.1) has one almost periodic solution , then is a positive almost periodic solution of (1.3). Therefore, to complete the proof it suffices to show that (2.1) has one almost periodic solution.
In order to use the continuation theorem of coincidence degree theory, we set the Banach spaces and the same as those in Lemma 3.3 and the mappings , , , the same as those defined in Lemmas 3.4 and 3.5, respectively. Thus, we can obtain that is a Fredholm mapping of index zero and is a continuous operator which is -compact on . It remains to search for an appropriate open and bounded subset . Corresponding to the operator equation we may write Assume that is a solution of (3.28) for a certain . Denote In view of (3.28), we obtain and consequently, which implies from that Similarly, we can get By inequalities (3.32) and (3.33), we can find that there exists such that where Then from (3.26), we have or Choose the point , where satisfies . Integrating (3.28) from to , we get However, from (3.28) and (3.38), we obtain Substituting back in (3.37) and for , we have where Let . Obviously, it is independent of . Take It is clear that satisfies assumption (1) of Theorem 2.1. If , then is a constant with . It follows that which implies that assumption (2) of Theorem 2.1 is satisfied. The isomorphism is defined by for . Thus, . In order to compute the Brouwer degree, we consider the homotopy For any , , we have . By the homotopic invariance of topological degree, we get Therefore, assumption (3) of Theorem 2.1 holds. Hence, has at least one solution in . In other words, (2.1) has at least one positive almost periodic solution. Therefore, (1.3) has at least one positive almost periodic solution. The proof is complete.

4. An Example

Let . Then (1.3) has the form One can easily realize that and ; thus condition holds. Therefore, by the consequence of Theorem 3.6, (4.1) has at least one positive almost periodic solution (Figure 1).

354016.fig.001
Figure 1: Transient response of state when .

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.

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