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

Xia Li, Yongkun Li, and Chunyan He Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn Received 19 May 2011; Revised 8 August 2011; Accepted 12 August 2011 Academic Editor: Dexing Kong Copyright q 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. 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.


Introduction
Consider the following differential equation which is widely used in fisheries 1-4 : where N N t is the population biomass, L t, N is the per capita fecundity rate, M t, N is the per capita mortality rate, and F t is the harvesting rate per capita.In 1.

Preliminaries
Our first observation is that under the invariant transformation N t e y t , 1.3 reduces to ẏ t a t 1 e y θ t /K t γ − b t 2.1 for γ > 0, with the initial function and the initial value y t φ t , y 0 y 0 , t ∈ −∞, 0 .

2.2
For 2.1 and 2.2 , we assume the following conditions: A2 θ t is a continuous function on 0, ∞ that satisfies θ t ≤ t; By a solution of 2.1 and 2.2 we mean an absolutely continuous function y t defined on −∞, ∞ satisfying 2.1 almost everywhere for t ≥ 0 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 X, Y be normed vector spaces, L : Dom L ⊂ X → Y be a linear mapping, and N : X → Y be a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dim Ker L codim Im L < ∞ and Im L is closed in Y .If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that Im P Ker L, Ker Q Im L Im I − Q , it follows that the mapping L| Dom L∩Ker P : I − P X → Im L is invertible.We denote the inverse of that mapping by K P .If Ω is an open bounded subset of X, then the mapping N will be called L-compact on Ω, if QN Ω is bounded and Then Ly Ny has at least one solution in Dom L ∩ Ω.

Existence of Almost Periodic Solutions
Let AP R denote the set of all real valued almost periodic functions on R, for f ∈ AP R we denote by the set of Fourier exponents and the module of f, respectively.Let K f, ε, S denote the set of ε-almost periods for f with respect to S ⊂ C −∞, 0 , R , l ε denote the length of the inclusion interval, and m f lim relatively dense; that is, for any ε > 0 it is possible to find a real number l ε > 0 for any interval with length l ε ; there exists a number δ δ ε in this interval such that |y t δ − y t | < ε for any t ∈ R.
Throughout the rest of the paper we assume the following condition for 2.1 : In our case, we set where and α is a given constant; define the norm We start with the following lemmas.Proof.If y n ∈ V 1 and y n converge to y 0 , then it is easy to show that y 0 ∈ AP R with mod y 0 ⊂ mod F .Indeed, for all | λ| ≤ α we have lim 3.9 From the definitions of φ t and φ 1 t , one can deduce that t φ s ds and t φ 1 s ds are almost periodic functions and thus φ 2 t ≡ 0, which implies that φ t ∈ V 1 .This tells us that

3.10
On the other hand, if ϕ t e −i λt dt.

3.11
It follows that

3.13
Note that t 0 ϕ s ds − m t 0 ϕ s ds is the primitive of ϕ in X; therefore we have ϕ ∈ Im L. Hence, we deduce that which completes the proof of our claim.Therefore,

3.17
Then, N is L-compact on Ω (Ω is an open and bounded subset of X).
Proof.The projections P and Q are continuous such that It is clear that

3.19
Therefore In view of Im P Ker L, we can conclude that the generalized inverse of L K P : Im L → Ker P ∩ Dom L exists and is given by The integral form of the terms of both QN and I − Q N implies that they are continuous.We claim that K P is also continuous.By our hypothesis, for any ε < 1 and any compact set S ⊂ C −∞, 0 , R , let l ε, S be the inclusion interval of K F, ε, S .Suppose that {z n t } ⊂ Im L V 1 and z n t uniformly converges to z 0 t .Because t 0 z n s ds ∈ Y n 0, 1, 2, 3, . . ., there exists ρ 0 < ρ < ε such that K F, ρ, S ⊂ K t o z n s ds, ε .Let l ρ, S be the inclusion interval of K F, ρ, S and l max l ρ, S , l ε, S .

3.25
It is easy to see that l is the inclusion interval of both K F, ε, S and K F, ρ, S .Hence, for all t / ∈ 0, l , there exists |z n s |ds ε.

3.26
By applying 3.26 , we conclude that t 0 z s ds z ∈ Im L is continuous and consequently K P and K P I − Q Ny are also continuous.From 3.26 , we also have that t 0 z s ds and K P I − Q Ny are uniformly bounded in Ω.In addition, it is not difficult to verify that QN Ω is bounded and K P I − Q Ny is equicontinuous in Ω. Hence by the Arzelà-Ascoli theorem, we can immediately conclude that Proof.It is easy to see that if 2.1 has one almost periodic solution y, then N e y 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 X and Y the same as those in Lemma 3.3 and the mappings L, N, P , Q the same as those defined in Lemmas 3.4 and 3.5, respectively.Thus, we can obtain that L is a Fredholm mapping of index zero and N is a continuous operator which is L-compact on Ω.

Lemma 3 . 3 .
X and Y are Banach spaces endowed with the norm • .

Theorem 3 . 6 .
Let condition (H) hold.Then 2.1 has at least one positive almost periodic solution.
a Banach space endowed with the norm • .The same can be concluded for the spaces X and Y .The proof is complete.Proof.It is obvious that L is a linear operator and Ker L V 2 .It remains to prove that Im L where Ly y dy/dt.Then L is a Fredholmmapping of index zero.
ds, ε such that t μ t ∈ 0, l .Therefore, by the definition of almost periodic functions we observe that International Journal of Differential Equations It remains to search for an appropriate open and bounded subset Ω. Corresponding to the operator equation Transient response of state N t when γ 2. Thus, JQNy / 0. In order to compute the Brouwer degree, we consider the homotopy H y, s −sy 1 − s JQNy, 0 ≤ s ≤ 1. 3.44 For any y ∈ ∂Ω ∩ Ker L, s ∈ 0, 1 , we have H y, s / 0. By the homotopic invariance of topological degree, we get deg{JQN, Ω ∩ Ker L, 0} deg −y, Ω ∩ Ker L, 0 / 0. 3.45 Therefore, assumption 3 of Theorem 2.1 holds.Hence, Ly Ny has at least one solution in Dom L ∩ Ω.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.
It is clear that Ω satisfies assumption 1 of Theorem 2.1.If y ∈ ∂Ω ∩ Ker L, then y is a constant with y M. It follows that which implies that assumption 2 of Theorem 2.1 is satisfied.The isomorphism J : Im Q → Ker L is defined by J z z for z ∈ R.

4. An Example
One can easily realize that m b t / K t γ > 0 and m a t > m b t ; thus condition H holds. Therefore, by the consequence of Theorem 3.6, 4.1 has at least one positive almost periodic solution Figure1.