Stability and Bifurcations in a Logistic Host–Parasitoid Model with Strong Allee Effects and Host Refuge

In this study, we analyze a discrete two-dimensional host–parasitoid model in which the host population follows logistic growth and is additionally subject to a strong Allee effect on the proportion of hosts that avoid parasitism. The parasitoid population dynamics are driven by host availability, attack success rate, and the number of parasitoids produced per successful attack. We classify the equilibrium points and explore the system’s local and global dynamics. Our analysis shows that, in certain parameter regions, an extinction equilibrium can be globally stable. For the boundary equilibrium, we prove the existence of transcritical and period-doubling bifurcations. Regarding interior equilibria, when multiple equilibria exist, their stabilities alternate. We prove the occurrence of codimension-1 period-doubling and Neimark–Sacker bifurcations, indicating the emergence of complex dynamics, including quasi-periodic and even chaotic behavior. Despite the possibility of complex dynamics, we prove that the system can exhibit uniform persistence and permanence under specific conditions, thereby ensuring the long-term coexistence of the host and parasitoid populations.