Periodic solution of a bioeconomic fishery model by coincidence degree theory

Abstract

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 \begin{equation*}\label{eq1.3} \begin{cases} \frac{dn}{dt} = n \left(r(t) \left(1-\frac{n}{K}\right)-\frac{q(t)E}{n+D}\right),\\ \frac{dE}{dt} = E\left(\frac{A(t)q(t)}{\alpha(t)} \frac{n}{n+D}-\frac{q^2(t)}{\alpha(t)} \frac{n^2E}{(n+D)^2}-c(t)\right), \end{cases} \end{equation*} where the functions $r,q,A,c$ and $\alpha$ are continuous positive $T$-periodic functions. This is the model of a coastal fishery represented as a single site with $n(t)$ is the fish stock biomass, and $E(t)$ is the fishing effort. Examples are given to strengthen our results.

Keywords

• periodic solution
• coincidence degree
• existence of solution