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]: Ṅ=N[L(t,N)-M(t,N)]-NF(t),
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 (1.1), let L(t,N) be a Hills' type function ([1, 2])L(t,N)=a1+(N/K)γ
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] Ṅ(t)=N(t)[a(t)1+(N(θ(t))/K(t))γ-b(t)],
where b(t)=M(t,N)+F(t).

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 N(t)=ey(t), (1.3) reduces toẏ(t)=a(t)1+(ey(θ(t))/K(t))γ-b(t)
for γ>0, with the initial function and the initial value y(t)=ϕ(t),y(0)=y0,t∈(-∞,0).
For (2.1) and (2.2), we assume the following conditions:

a(t),b(t)∈C([0,+∞),[0,+∞)) and K(t)∈C([0,+∞),(0,+∞));

θ(t) is a continuous function on [0,+∞) that satisfies θ(t)≤t;

ϕ(t):(-∞,0)→[0,∞) is a continuous bounded function, ϕ(t)≥0,y0>0.

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:DomL⊂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 dimKerL=codimImL<+∞ and ImL 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 ImP=KerL, KerQ=ImL=Im(I-Q), it follows that the mapping L|DomL∩KerP:(I-P)X→ImL is invertible. We denote the inverse of that mapping by KP. 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 KP(I-Q)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Let Ω⊂X be an open bounded set and let N:X→Y be a continuous operator which is L-compact on Ω¯. Assume that

Ly≠λNy for every y∈∂Ω∩DomL and λ∈(0,1);

QNy≠0 for every y∈∂Ω∩KerL;

the Brouwer degree deg{JQN,Ω∩KerL,0}≠0.

Then Ly=Ny has at least one solution in DomL∩Ω¯.3. 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 Λ(f)={λ̃∈R:limT→∞1T∫0Tf(s)e-iλ̃sds≠0},mod(f)={∑j=1mnjλ̃j:nj∈Z,m∈N,λ̃j∈Λ(f),j=1,2,…,m},
the set of Fourier exponents and the module of f, respectively. Let K(f,ɛ,S) denote the set of ɛ-almost periods for f with respect to S⊂C((-∞,0],ℝ), l(ɛ) denote the length of the inclusion interval, and m(f)=limT→∞(1/T)∫0Tf(s)ds denote the mean value of f.

Definition 3.1.

y(t)∈C(ℝ,ℝ) is said to be almost periodic on ℝ if for any ɛ>0 the set K(y,ɛ)={δ:|y(t+δ)-y(t)|<ɛ,∀t∈ℝ} is relatively dense; that is, for any ɛ>0 it is possible to find a real number l(ɛ)>0 for any interval with length l(ɛ); there exists a number δ=δ(ɛ) in this interval such that |y(t+δ)-y(t)|<ɛ for any t∈ℝ.

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

In our case, we set X=Y=V1⊕V2,
where V1={y∈AP(R):mod(y)⊆mod(F),∀μ∈Λ(y)satisfies|μ|>α},V2={y(t)≡k,k∈R},
where F=F(t,ϕ)=a(t)1+(eϕ(θ(0))/K(t))γ-b(t),ϕ∈C([-∞,0],R)
and α is a given constant; define the norm ‖y‖=supt∈R|y(t)|,y∈X(orY).

Remark 3.2.

If f is ɛ-almost periodic function, then ∫tf(s)ds is ɛ-almost periodic if and only if m(f)=0. Whereas f∈AP(ℝ) does not necessarily have an almost periodic primitive, m(f)=0. That is why we can not make V1={z∈AP(ℝ):m(z)=0} and let V1={y∈AP(ℝ):mod(y)⊂mod(F),∀μ∈Λ(y)satisfy|μ|>α}.

We start with the following lemmas.

Lemma 3.3.

X and Y are Banach spaces endowed with the norm ∥·∥.

Proof.

If yn∈V1 and yn converge to y0, then it is easy to show that y0∈AP(ℝ) with mod(y0)⊂mod(F). Indeed, for all |λ̃|≤α we have
limT→∞1T∫0tyn(s)e-iλ̃sds=0.
Thus
limT→∞1T∫0Ty0(s)e-iλ̃sds=0,
which implies that y0∈V1. One can easily see that V1 is a Banach space endowed with the norm ∥·∥. The same can be concluded for the spaces X and Y. The proof is complete.

Lemma 3.4.

Let L:X→Y and
Ly=a(t)1+(ey(θ(t))/K(t))γ-b(t),
where Ly=y′=dy/dt. Then L is a Fredholm mapping of index zero.

Proof.

It is obvious that L is a linear operator and KerL=V2. It remains to prove that ImL=V1. Suppose that ϕ(t)∈ImL⊂Y. Then, there exist ϕ1∈V1 and ϕ2∈V2 such that
ϕ=ϕ1+ϕ2.
From the definitions of ϕ(t) and ϕ1(t), one can deduce that ∫tϕ(s)ds and ∫tϕ1(s)ds are almost periodic functions and thus ϕ2(t)≡0, which implies that ϕ(t)∈V1. This tells us that
ImL⊂V1.
On the other hand, if φ(t)∈V1∖{0} then we have ∫0tφ(s)ds∈AP(ℝ). Indeed, if λ̃≠0 then we obtain
limT→∞1T∫0T[∫0tφ(s)ds]e-iλ̃tdt=1iλ̃limT→∞1T∫0Tφ(t)e-iλ̃tdt.
It follows that
Λ[∫0tφ(s)ds-m(∫0tφ(s)ds)]=Λ(φ(t)).
Thus
∫0tφ(s)ds-m(∫0tφ(s)ds)∈V1⊂X.
Note that ∫0tφ(s)ds-m(∫0tφ(s)ds) is the primitive of φ in X; therefore we have φ∈ImL. Hence, we deduce that
V1⊂ImL,
which completes the proof of our claim. Therefore,
ImL=V1.
Furthermore, one can easily show that ImL is closed in Y and
dimKerL=1=codimImL.
Therefore, L is a Fredholm mapping of index zero.

Lemma 3.5.

Let N:X→Y, P:X→X, and Q:Y→Y such that
Ny=a(t)1+(ey(θ(t))/K(t))γ-b(t),y∈X,Py=m(y),y∈X,Qz=m(z),z∈Y.
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
ImP=KerL,ImL=KerQ.
It is clear that
(I-Q)V2={0},(I-Q)V1=V1.
Therefore
Im(I-Q)=V1=ImL.
In view of
ImP=KerL,ImL=KerQ=Im(I-Q),
we can conclude that the generalized inverse (of L) KP:ImL→KerP∩DomL exists and is given by
KP(z)=∫0tz(s)ds-m[∫0tz(s)ds].
Thus
QNy=m[a(t)1+(ey(θ(t))/K(t))γ-b(t)],KP(I-Q)Ny=f[y(t)]-Qf[y(t)],
where f[y(t)] is defined by
f[y(t)]=∫0t[Ny(s)-QNy(s)]ds.

The integral form of the terms of both QN and (I-Q)N implies that they are continuous. We claim that KP is also continuous. By our hypothesis, for any ɛ<1 and any compact set S⊂C((-∞,0],ℝ), let l(ɛ,S) be the inclusion interval of K(F,ɛ,S). Suppose that {zn(t)}⊂ImL=V1 and zn(t) uniformly converges to z0(t). Because ∫0tzn(s)ds∈Y(n=0,1,2,3,…), there exists ρ(0<ρ<ɛ) such that K(F,ρ,S)⊂K(∫otzn(s)ds,ɛ). Let l(ρ,S) be the inclusion interval of K(F,ρ,S) and
l=max{l(ρ,S),l(ɛ,S)}.
It is easy to see that l is the inclusion interval of both K(F,ɛ,S) and K(F,ρ,S). Hence, for all t∉[0,l], there exists μt∈K(F,ρ,S)⊂K(∫0tzn(s)ds,ɛ) such that t+μt∈[0,l]. Therefore, by the definition of almost periodic functions we observe that
‖∫0tzn(s)ds‖=supt∈R|∫0tzn(s)ds|≤supt∈[0,l]|∫0tzn(s)ds|+supt∉[0,l]|(∫0tzn(s)ds-∫0t+μtzn(s)ds)+∫0t+μtzn(s)ds|≤2supt∈[0,l]|∫0tzn(s)ds|+supt∉[0,l]|∫0tzn(s)ds-∫0t+μtzn(s)ds|≤2∫0l|zn(s)|ds+ɛ.
By applying (3.26), we conclude that ∫0tz(s)ds(z∈ImL) is continuous and consequently KP and KP(I-Q)Ny are also continuous.

From (3.26), we also have that ∫0tz(s)ds and KP(I-Q)Ny are uniformly bounded in Ω¯. In addition, it is not difficult to verify that QN(Ω¯) is bounded and KP(I-Q)Ny is equicontinuous in Ω¯. Hence by the Arzelà-Ascoli theorem, we can immediately conclude that KP(I-Q)N(Ω¯) is compact. Thus N is L-compact on Ω¯.

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 y¯, then N¯=ey¯ 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¯. It remains to search for an appropriate open and bounded subset Ω. Corresponding to the operator equation
Ly=λNy,λ∈(0,1),
we may write
ẏ(t)=λ[a(t)1+(ey(θ(t))/K(t))γ-b(t)].
Assume that y=y(t)∈X is a solution of (3.28) for a certain λ∈(0,1). Denote
y*=supt∈Ry(t),y*=inft∈Ry(t).
In view of (3.28), we obtain
m(a(t)-b(t))=m(b(t)Kγ(t)(ey(θ(t)))γ)
and consequently,
m(a(t)-b(t))≥m(b(t)Kγ(t))eγy*,
which implies from (H) that
y*≤γ-1ln(m[a(t)-b(t)]m(b(t)/Kγ(t))).
Similarly, we can get
y*≥γ-1ln(m(a(t)-b(t))m(b(t)/Kγ(t))).
By inequalities (3.32) and (3.33), we can find that there exists t1∈ℝ such that
|y(t1)|≤M,
where
M=|γ-1ln(m(a(t)-b(t))m(b(t)/Kγ(t)))|+1.
Then from (3.26), we have
‖y(t)‖≤|y(t1)|+supt∈R|∫t2ty′(s)ds|≤M+2supt∈[t2,t2+l]∫t2t|y′(s)|ds+ɛ
or
‖y(t)‖≤M+2∫t2t2+l|y′(s)|ds+1.
Choose the point ν-t2∈[l,2l]∩K(F,ρ,S), where ρ(0<ρ<ɛ) satisfies K(F,ρ)⊂K(z,ɛ). Integrating (3.28) from t2 to ν, we get
λ∫t2νa(t)1+[K(s)]-γeγy(θ(s))ds=λ∫t2ν|b(s)|ds+∫t2νy′(s)ds≤λ∫t2ν|b(s)|ds+ɛ.
However, from (3.28) and (3.38), we obtain
∫t2ν|y′(s)|ds≤λ∫t2ν|b(s)|ds+λ∫t2νa(t)1+[K(s)]-γeγy(θ(s))ds,≤2∫t2ν|b(s)|ds+ɛ≤2∫t2ν|b(s)|ds+1.
Substituting back in (3.37) and for ν≥t2+l, we have
‖y(t)‖≤M′,
where
M′=M+4∫0ν|b(s)|ds+3.
Let M̃=M+M′. Obviously, it is independent of λ. Take
Ω={y∈X:‖y‖<M̃}.
It is clear that Ω satisfies assumption (1) of Theorem 2.1. If y∈∂Ω∩KerL, then y is a constant with ∥y∥=M̃. It follows that
QNy=m(a(t)1+(ey(θ(t))/K(t))γ-b(t))≠0,
which implies that assumption (2) of Theorem 2.1 is satisfied. The isomorphism J:ImQ→KerL is defined by J(z)=z for z∈ℝ. Thus, JQNy≠0. In order to compute the Brouwer degree, we consider the homotopy
H(y,s)=-sy+(1-s)JQNy,0≤s≤1.
For any y∈∂Ω∩KerL, s∈[0,1], we have H(y,s)≠0. By the homotopic invariance of topological degree, we get
deg{JQN,Ω∩KerL,0}=deg{-y,Ω∩KerL,0}≠0.
Therefore, assumption (3) of Theorem 2.1 holds. Hence, Ly=Ny has at least one solution in DomL∩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.

4. An Example

Let a(t)=eπ(3+cos2t),b(t)=(1/2)eπ(3+cos2t),K(t)=4+sint,γ>0,θ(t)=t-2-sin3t. Then (1.3) has the formṄ(t)=N(t)[eπ(3+cos2t)1+(N(t-2-sin3t)/(4+sint))γ-12eπ(3+cos2t)].
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 (Figure 1).

Transient response of state N(t) when γ=2.

Acknowledgment

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

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