Stability and bifurcation analyses of a Beverton–Holt host–parasitoid model with constant refuge and a generalized escape response
In this paper, we study a discrete-time host–parasitoid model in which the host survival probability is governed by a general escape function that satisfies natural biological constraints. The host population follows a Beverton–Holt growth model, while a constant number of individuals remains in a refuge each generation. Both host availability and current parasitoid density influence parasitoid population dynamics. We show that the system admits three possible equilibrium outcomes: no equilibrium, a parasitoid-free equilibrium, or a unique interior equilibrium where both species coexist. Conditions are established under which the parasitoid-free equilibrium is globally asymptotically stable. In certain parameter ranges, we prove the occurrence of transcritical bifurcation at the boundary equilibrium and Neimark–Sacker bifurcation at the interior equilibrium. Despite the possibility of complex behavior, we prove that, within specific parameter ranges, the system is uniformly persistent and permanent, ensuring the long-term survival of both populations. Numerical simulations are included to support and illustrate the theoretical results.