Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting

Abstract

<p style='text-indent:20px;'>The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue <inline-formula><tex-math id="M1">$ v $</tex-math></inline-formula>. We show that a positive equilibrium point is locally asymptotically stable when the profit <inline-formula><tex-math id="M2">$ v $</tex-math></inline-formula> is less than a certain critical value <inline-formula><tex-math id="M3">$ v^{*}_1 $</tex-math></inline-formula>, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.</p>