The behavior of a host–parasitoid model with host logistic growth and proportional refuge

Abstract

This paper deals with the host–parasitoid model, where the logistic equation governs the host population growth, and a proportion of the host population can find refuge. The equilibrium points’ existence, number, and local character are discussed. Taking the parameter regulating the parasitoid’s growth as a bifurcation parameter, we prove that Neimark–Sacker and period-doubling bifurcations occur. Despite the complex behavior, it can be proved that the system is permanent, ensuring the long-term survival of both populations. Furthermore, it was observed that the presence of the proportional refuge does not significantly influence the system’s behavior compared to the system without a proportional refuge.