Almost periodic solutions for Fox production

Abstract. By utilizing the continuation theorem of coincidence degree theory, we shall prove that a Foxproduction harvesting model with delay has at least one positive almost periodic solution. Some pre-liminary assertions are provided prior to proving our main theorem. We construct a numerical examplealong with graphical representations to illustrate feasibility of the theoretical result.Keywords. Almost periodic solution; Fox production harvesting model; Coincidence degree theory. AMS subject classiﬁcation: 34K14. 1 Introduction Consider the following equation of population dynamics [1, 2]x ′ (t) = −xF(t,x) +xG(t,x), x ′ (t) =dxdt, (1)where x = x(t) is the size of the population, F(t,x) is the per–capita harvesting rateand G(t,x) is the per–capita fecundity rate. Let G(t,x) and F(t,x) be deﬁned in theformF(t,x) = α(t) and G(t,x) = β(t)ln γ K(t)x(t), γ > 0then equation (1) becomesx ′ (t) = −α(t)x(t) +β(t)x(t)ln γ K(t)x(t), (2)where α(t) is a variable harvesting rate, β(t) is an intrinsic factor and K(t) is a varyingenvironmental carrying capacity. The positive parameter γ is referred to as an interac-tion parameter [1, 3, 4]. Indeed, if γ > 1 then intra–speciﬁc competition is high whereasif 0 < γ < 1 then the competition is low. For γ = 1, equation (2) reduces to a classical


Introduction
Consider the following equation of population dynamics [1,2] x ′ (t) = −xF (t, x) + xG(t, x), x ′ (t) = dx dt , where x = x(t) is the size of the population, F (t, x) is the per-capita harvesting rate and G(t, x) is the per-capita fecundity rate. Let G(t, x) and F (t, x) be defined in the form F (t, x) = α(t) and G(t, x) = β(t) ln γ K(t) x(t) , γ > 0 then equation (1) becomes where α(t) is a variable harvesting rate, β(t) is an intrinsic factor and K(t) is a varying environmental carrying capacity. The positive parameter γ is referred to as an interaction parameter [1,3,4]. Indeed, if γ > 1 then intra-specific competition is high whereas if 0 < γ < 1 then the competition is low. For γ = 1, equation (2) reduces to a classical Gompertzian model with harvesting [2,5]. Equation (2) is called a Fox surplus production model that has been used to build up certain prediction models such as microbial growth model, demographic model and fisheries model. This equation is considered to be an efficient alternative to the well known γ-logistic model. Specifically, Fox model is more appropriate upon describing lower population density; we refer the reader to the papers [1,3,4,6,7,8] for more informations.
One of the most important behaviors of solutions which has been a main object of investigations among authors is the periodic behavior of solutions [9,10,11,12,13,14,15,16]. To consider periodic environmental factors acting on a population model, it is natural to study the model subject to periodic coefficients. Indeed, the assumption of periodicity of the parameters in the model is a way of incorporating the time-dependent variability of the environment (e.g. seasonal effects of weather, food supplies, mating habits and harvesting). On the other hand, upon considering long-term dynamical behavior, it has been found that the periodic parameters often turn out to experience some perturbations that may lead to a changing in character. Thus, the investigation of almost periodic behavior is considered to be in more accordance with reality; see the remarkable monographs [17,18,19] for more details. In [20], the author has declared that population models involving delayed argument provide better description for real phenomena. For this reason, we shall consider system (2) in the form where τ (t) < t and the parameters are to satisfy certain almost periodic assumptions that will be specified later. System (3) has been rarely investigated in the literature; see for instance the papers [21,22] where the authors concentrated on studying the existence of periodic solutions, stability, oscillation and the global attractivity of the solutions. It is worth pointing out, however, that all the above mentioned papers are presented under periodic assumptions and to the best of authors' observation no paper has been published regarding the almost periodicity of system (3).
Although it has more widespread applications in real life, the notion of almost periodicity has been less considered among researchers. We mention here some recent works [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] in which several methods have been used to prove, in particular, the existence of almost periodic solutions. Following this trend, we shall utilize the continuation theorem of coincidence degree theory to prove that system (3) has at least one positive almost periodic solution.
The remaining part of this paper is organized as follows: In Section 2, some preliminary assertions are provided prior to proving our main theorem. In section 3, sufficient conditions are established to investigate the existence of almost periodic solutions of the said system. Section 4 contains a numerical example along with graphical representations to illustrate the feasibility of the theoretical result.

Preliminary assertions
Through the change of variables x(t) = e y(t) , we observe that (3) reduces to the equation We consider equation (4) together with the initial condition Equations (4) and (5) are considered under the following assumptions: By a solution of (4) and (5) we mean an absolutely continuous function y(t) defined on R satisfying (4) almost everywhere for t ≥ 0 and (5). Due to certain biological purposes, we focus our discussion on the positive solutions of (4).
To prove the main results of this paper we shall utilize the continuation theorem of coincidence degree theory [39,40]. However, before proceeding to the main results we set forth some basic concepts in the framework of this theory.
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 dimKer L = codimIm L < +∞ and Im L is closed in Y . If L is a Fredholm mapping of index zero and there exists 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∩KerP : (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 K P (I − Q)N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
We mention the statement of the continuation theorem.  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) and l(ε) denote the length of the inclusion interval.
is 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 In our case, we set where V 1 = y ∈ AP (R) : (mod y(t)) ⊂ (mod F ), ∀ µ ∈ Λ(y(t)) satisfying |µ| > c , R) and c is a given positive constant. Proof. If {y n } ⊂ V 1 and y n converges 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 which implies that y 0 ∈ V 1 . One can easily see that V 1 is a Banach space endowed with the norm · . The same can be concluded for the spaces X and Y. The proof is complete.
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 = V 2 . It remains to prove that Im 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 that Im L ⊂ V 1 .
On the other hand, if ϕ(t) ∈ V 1 \{0} then we have We note that t 0 ϕ(s) ds − m( t 0 ϕ(s) ds) is the primitive of ϕ(t) in X, therefore we have ϕ(t) ∈ Im L. Hence, we deduce that V 1 ⊂ Im L which completes the proof of our claim. Therefore, Im L = V 1 .
Furthermore, one can easily show that Im L is closed in Y and dimKerL = 1 = codimImL.
Therefore, L is a Fredholm mapping of index zero.
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 Im P = Ker L and Im L = Ker Q.
It is clear that In view of Im P = Ker L and Im L = Ker Q = Im (I − Q) we can conclude that the generalized inverse (of L) K p : Im L → Ker P ∩ DomL exists and is given by Thus QN y = m − α(t) + β(t) ln γ K(t) e y(τ (t)) and The integral form of the terms of both QN and (I −Q)N imply 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([−σ, 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 converge 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 0 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)}.
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 µ t ∈ K(F, ρ, S) ⊂ K( t 0 z n (s) ds, ε) such that t + µ t ∈ [0, l]. Therefore, by the definition of almost periodic functions we observe that By applying (6), we conclude that t 0 z(s) ds (z ∈ Im L) is continuous and consequently K p and K p (I − Q)N y are also continuous.
From (6), we also have t 0 z(s) ds and K p (I − Q)N y are uniformly bounded in Ω. In addition, it is not difficult to verify that QN (Ω) is bounded and K p (I − Q)N y is equicontinuous in Ω. Hence by the Arzela-Ascoli theorem, we can immediately conclude that K p (I − Q)N (Ω) is compact. Thus N is L−compact on Ω.

The main result
One immediate observation is that if equation (4) has one almost periodic solution y, then x = e y is a positive almost periodic solution of (3). Therefore, in the proof of Theorem 2 it suffices to show that (4) has one almost periodic solution. Proof. 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 1 and the mappings L, N, P, Q the same as those defined in Lemma 2 and Lemma 3, respectively. Thus, we can deduce 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.
It follows that ) .
Therefore, we may write Similarly, we can get By inequalities (8) and (9), we can find that there exists t 0 ∈ R such that By virtue of (6), we get |y ′ (s)|ds + 1.

A Numerical example
Here we give an example that illustrates the almost periodic behavior of Fox production harvesting model with delay.