Periodic solution of a bioeconomic fishery model by coincidence degree theory

. In this article we use coincidence degree theory to study the existence of a positive periodic solutions to the following bioeconomic model in fishery dynamics


Introduction
In [17], Moussaoui and Auger introduced following system of three ordinary differential equation describing the fishery dynamics with price depending on supply and demand n(t) is the fish stock biomass, E(t) is the fishing effort and (p(t) is the price per unit of the resource at time t). Authors assumed that the price varies at a fast time scale τ, while fish growth and investment in the fishery by boat owners occur at a slow time scale t = ετ, with τ ≪ 1 being a small dimensionless parameter.
The study of existence, uniqueness and asymptotic behavior of solutions of mathematical models can be found in all applied sciences in the recent years. Many of the mathematical models occur in terms of differential equations or a system of differential equations.The increasing expansion of branches of system of differential equations has attracted many researchers to study the dynamical nature of solutions, especially, on existence and uniqueness of solutions. One of the models that attracts the attention of researchers in applied science is the bioeconomic model, similar to classical bioeconomic models of fishery dynamics [1,3].
Using regression [17], we can transform model (1.1) into the following system of two differential equations.
  Since the variation of environment, in particular the periodic variation of the environment, play an important role in many biological and ecological system, especially, in fish stock biomass and fishing effort, it is natural to study the existence and asymptotic behavior of periodic solutions of the model (1.2). From the application point of view, only positive periodic solutions are important. Hence, it is realistic to assume the periodicity of the coefficient functions in (1.2). Thus, assuming r, q, A, c, and α to be positive T-periodic functions, we have the following nonautonomous model where r, q, A, c, and α are continuous positive T-periodic functions with ecological meaning as n the fish stock biomass, E the fishing effort, r fish growth rate, K carrying capacity, q catchability per fishing effort unit, D half saturation level, A carrying capacity of the market or maximum demand and α slope of the linear demand function decreasing with the price. Setting f (t, n, E) = r(t) K n 2 + q(t)En n + D and we can express (1.3) into the following systems of equations System of equations of the form (1.4) with general f and g have been studied by many authors [2,11,14,[20][21][22][23][24][25]28] using various types of fixed point theorems to study the existence of positive T-periodic of (1.4) when f and g are positive continuous functions. Further, they were applied to many mathematical models [11,14,[20][21][22][23][24][25]28] to study the existence of positive Tperiodic solutions. One may refer to [19] for applications of fixed point theorems [7,9,10,12] on the existence of positive periodic solutions of mathematical models. As far as our knowledge is concerned, there exist no results on the existence and uniqueness of positive Tperiodic solutions of (1.3). We have used Mawhin's coincidence degree theory to study the existence of T-periodic solution of (1.3). Although there exist hundreds of research articles in the literature on the use of Schauder's fixed point theorem and Krasnosel'skii's fixed point theorem, the use of Mawhin's coincidence degree theory to study the existence of positive T-periodic solutions of (1.3) is relatively scarce in the literature. Previous papers based on Mawhin's coincidence degree theory for different biological models are [4-6, 8, 15, 26, 27, 29].
In order to obtain our results, we assume r(t), q(t), A(t), c(t) and α(t) in (1.3) are all positive T-periodic functions. Further then we assume f and g are T-periodic functions with respect to the first variable.
This work has been divided into four sections. Section 1 is Introduction. Basic theory and Mawhin's coincidence degree theory is given in Section 2. Section 3 contains the main results of this paper. Examples are given to illustrate our results. Section 4 discusses the conclusion of this article.

Preliminaries
Before presenting our results on the existence of periodic solution of system (1.3), We provide the essentials of the coincidence degree theory. Let Z and W be the real Banach spaces, and Let L : dom(L) ⊂ Z → W be Fredholm operator of index zero, If P : Z → Z and Q : W → W are two continuous projectors such that Im(P) = Ker(L), Ker(Q) = Im(L), Z = Ker(L) ⊕ Ker(P) and W = Im(L) ⊕ Im(Q), then the inverse operator of L| dom(L)∩Ker(P) : dom(L) ∩ Ker(P) → Im(L) exists and is denoted by Theorem 2.1 ([16]). Let L be a Fredholm operator of index zero and let N be the L-compact on Ω. Assume the following conditions are satisfied: Then, the equation Lx = Nx has at least one solution in dom(L) ∩ Ω.

Existence of the periodic solution
For the sake of convenience and simplicity, we use the notations: where f is a continuous t-Periodic function. Set: Also, there exist positive numbers L i (i = 1, 2, . . . , 4) such that . . , 4) will be calculated as in the proof of following theorem.
Then, system (1.3) has at least one positive T-periodic solution Proof. Firstly, we make a change of variables. Consider Banach spaces with the norm ∥ · ∥ as follows: For any z = (z 1 , z 2 ) ∈ Z, the periodicity of system (3.1) implies are T-periodic functions. In fact Obviously, Γ 2 (z, t) is also periodic function by similar way. Define operators L, P, Q as follows, respectively It is easy to see that For any z ∈ W, let z 1 = z − Qz, we can obtain that Thus, L is a Fredholm operator of index zero, which implies that L has a unique generalized inverse operator.
Next we show that N is L-compact. Define the inverse of L as K P : Im(L) → Ker(P) ∩ dom(L) and is given by Therefore, for any z(t) ∈ Z, we have Clearly, QN and K P (I − Q)N are continuous. Due to Z is Banach space, using the Arzelà-Ascoli theorem, we have that N is L-compact on U for any open bounded set U ⊂ Z. Next, in order to apply the coincidence degree theory, we need to construct an appropriate open bounded subset U. Therefore, the operator equation is defined by Lz = λNz, λ ∈ (0, 1), that is, e 2z 1 (t) e z 2 (t) (e z 1 (t) +D) 2 − c(t) .
For the second condition of Theorem 2.1, we prove that QN(z 1 , z 2 ) T ̸ = (0, 0) T for each (z 1 , z 2 ) ∈ ∂Ω ∩ Ker(L). When (z 1 , z 2 ) T ∈ ∂Ω ∩ Ker(L) = ∂Ω ∩ R 2 , (z 1 , z 2 ) T is a constant vector in R 2 and |z 1 | + |z 2 | = Λ. If the system (3.12) has a solution, then Since, (3.12) does not have solution then, it is evident that QN(z 1 , z 2 ) T ̸ = 0, thus the second condition of Theorem 2.1 is satisfied. Finally, we prove that the last condition of Theorem 2.1 is satisfied, to do so, we define the following mapping By using the invariance property of homotopy in topological degree theory, we get,

Furthermore, the system of algebraic equation
has a unique solution (x * , y * ), where x * = r 1 − αK+αc−Aq αK+Drq > 0 and y * = Dr(αK+αc−Aq) Now, all the conditions in Theorem 2.1 have been verified. This implies that system (3.1) has at least one T-periodic solution. Consequently, system (1.3) has at least one positive T-periodic solution. The theorem is proved.

Conclusion
Using Mawhin's coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions of the model (1.3). By formulating the model as a system of differential equations and introducing appropriate transformations, we were able to apply the coincidence degree theory and obtain our main results. The conditions (A1), (A2), and (A3) played a crucial role in establishing the existence of periodic solutions. Set r(t) ≡ r, q(t) ≡ q, A(t) ≡ A, α(t) ≡ α and c(t) ≡ c be constants; then (1.  In a recent paper, Moussaoui and Auger [17], studied the equilibrium points of (4.1). They proved that if Aq < αc, (4.2) then the system (4.1) has no positive equilibrium point provided that holds. It is worth noting that our conditions (A1), (A2), and (A3) are different from the condition (4.2). By Theorem 3.1, the system (4.1) has positive T-periodic solution. We note that the condition (4.3) can be satisfied for large K. On the other hand, by Theorem 1 b) of [17], the system (4.1) has a unique positive equilibrium, which is a positive T-periodic solution of (4.1). Our Theorem 3.1 strengthens this observation. While this research paper has successfully addressed the existence of positive periodic solutions for the bioeconomic fishery model, there are several avenues for further exploration. It would be interesting to study global attractivity and uniqueness of the solution for the system investigated in this paper. Another promising direction is to examine problem (1.3) by introducing a delay in the system, such as incorporating a time lag in fish stock biomass. Conducting further investigations in these areas have potential implications for understanding and managing fisheries dynamics.