International Journal of Differential Equations

Volume 2011, Article ID 354016, 12 pages

http://dx.doi.org/10.1155/2011/354016

## 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 [1–4]: 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) [1–6] 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 [7–18].

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).

#### Acknowledgment

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

#### References

- M. Kot,
*Elements of Mathematical Ecology*, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Berezansky, E. Braverman, and L. Idels, “Delay differential equations with Hill's type growth rate and linear harvesting,”
*Computers & Mathematics with Applications*, vol. 49, no. 4, pp. 549–563, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. - Y. Kuang,
*Delay Differential EquationsWith Applications in Population Dynamics*, vol. 191, Academic Press, Boston, Mass, USA, 1993. - L. Berezansky and L. Idels, “Stability of a time-varying fishing model with delay,”
*Applied Mathematics Letters*, vol. 21, no. 5, pp. 447–452, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Wang, “Stability and existence of periodic solutions for a time-varying fishing model with delay,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 5, pp. 3309–3315, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ahmad and G. Tr. Stamov, “Almost periodic solutions of n-dimensional impulsive competitive systems,”
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 3, pp. 1846–1853, 2009. View at Publisher · View at Google Scholar - S. Ahmad and G. Tr. Stamov, “On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations,”
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 5, pp. 2857–2863, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Li and F. Chen, “Almost periodic solutions of a discrete almost periodic logistic equation,”
*Mathematical and Computer Modelling*, vol. 50, no. 1-2, pp. 254–259, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Lou and X. Chen, “Traveling waves of a curvature flow in almost periodic media,”
*Journal of Differential Equations*, vol. 247, no. 8, pp. 2189–2208, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. O. Alzabut, J. J. Nieto, and G. Tr. Stamov, “Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis,”
*Boundary Value Problems*, Article ID 127510, 10 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - R. Yuan, “On almost periodic solutions of logistic delay differential equations with almost periodic time dependence,”
*Journal of Mathematical Analysis and Applications*, vol. 330, no. 2, pp. 780–798, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Wu and Y. Ye, “Existence and stability of almost periodic solutions of nonautonomous competitive systems with weak Allee effect and delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 11, pp. 3993–4002, 2009. View at Publisher · View at Google Scholar - G. T. Stamov and N. Petrov, “Lyapunov-Razumikhin method for existence of almost periodic solutions of impulsive differential-difference equations,”
*Nonlinear Studies*, vol. 15, no. 2, pp. 151–161, 2008. View at Google Scholar · View at Zentralblatt MATH - G. T. Stamov and I. M. Stamova, “Almost periodic solutions for impulsive neutral networks with delay,”
*Applied Mathematical Modelling*, vol. 31, pp. 1263–1270, 2007. View at Google Scholar - Q. Wang, H. Zhang, and Y. Wang, “Existence and stability of positive almost periodic solutions and periodic solutions for a logarithmic population model,”
*Nonlinear Analysis*, vol. 72, no. 12, pp. 4384–4389, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. S. Besicovitch,
*Almost Periodic Functions*, Dover Publications, New York, NY, USA, 1954. - A. Fink,
*Almost Periodic Differential Equations: Lecture Notes in Mathematics*, vol. 377, Springer, Berlin, Germany, 1974. - T. Zhang, Y. Li, and Y. Ye, “Persistence and almost periodic solutions for a discrete fishing model with feedback control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1564–1576, 2011. View at Publisher · View at Google Scholar - J. O. Alzabut, G. Tr. Stamov, and E. Sermutlu, “On almost periodic solutions for an impulsive delay logarithmic population model,”
*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 625–631, 2010. View at Google Scholar · View at Zentralblatt MATH - J. O. Alzabut, G. T. Stamov, and E. Sermutlu, “Positive almost periodic solutions for a delay logarithmic population model,”
*Mathematical and Computer Modelling*, vol. 53, no. 1-2, pp. 161–167, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Kolmanovskii,
*Introduction to the Theory and Applications of Functional Differential Equations*, vol. 463, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.