Motivated by the article [Getz and Mills. Host–parasitoid coexistence and egg-limited encounter rates. Am Nat. 1996;148:301–315], in this paper, we explore a discrete model involving a host and a parasitoid with search and egg limitations and arbitrary host-escape function. We added proportional refuge for the hosts to the model. We focus on the system's behavior at the equilibrium points and the nearby regions. In addition to the topological classification of these points, we examined local behavior. In the case of the extinction equilibrium, we obtain the global result. We describe the dynamical behavior scenarios in the neighborhood of the non-isolated exclusion equilibrium point (1:1 resonant). For the unique coexisting equilibrium, we prove the emergence of the Neimark–Sacker bifurcation and calculate the first Lyapunov exponent. This bifurcation can be either super or sub-critical. We have also established the occurrence of the Chenciner bifurcation. Our findings indicate that proportional refuge may or may not stabilize the system, and the choice of host-escape function plays a crucial role in shaping the system's dynamics. We also provide numerical examples to support our theoretical results.

This research delves into the generalized Beddington host–parasitoid model, which includes an arbitrary parasitism escape function. Our analysis reveals three types of equilibria: extinction, boundary, and interior. Upon examining the parameters, we discover that the first two equilibria can be globally asymptotically stable. The boundary equilibrium undergoes period-doubling bifurcation with a stable two-cycle and a transcritical bifurcation, creating a threshold for parasitoids to invade. Furthermore, we determine the interior equilibrium’s local stability and analytically demonstrate the period-doubling and Neimark–Sacker bifurcations. We also prove the permanence of the system within a specific parameter space. The numerical simulations we conduct reveal a diverse range of dynamics for the system. Our research extends the results in [Kapçak et al., 2013] and applies to a broad class of the generalized Beddington host–parasitoid model.

This research paper delves into the two-dimensional discrete plant-herbivore model. In this model, herbivores are food-limited and affect the plants' density in their environment. Our analysis reveals that this system has equilibrium points of extinction, exclusion, and coexistence. We analyze the behavior of solutions near these points and prove that the extinction and exclusion equilibrium points are globally asymptotically stable in certain parameter regions. At the boundary equilibrium, we prove the existence of transcritical and period-doubling bifurcations with stable two-cycle. Transcritical bifurcation occurs when the plant's maximum growth rate or food-limited parameter reaches a specific boundary. This boundary serves as an invasion boundary for populations of plants or herbivores. At the interior equilibrium, we prove the occurrence of transcritical, Neimark-Sacker, and period-doubling bifurcations with an unstable two-cycle. Our research also establishes that the system is persistent in certain regions of the first quadrant. We demonstrate that the local asymptotic stability of the interior equilibrium does not guarantee the system's persistence. Bistability exists between boundary attractors (logistic dynamics) and interior equilibrium for specific parameters' regions. We conclude that changes to the food-limitation parameter can significantly alter the system's dynamic behavior. To validate our theoretical findings, we conduct numerical simulations.