Stability and bifurcations of a host–parasitoid model with general host escape function and general stocking upon parasitoids

Abstract

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings.