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 paper examines the relationship between herbivores and plants with a strong Allee effect. When the plant reaches a particular size, the herbivore attacks it. We use the logistic equation to model plant growth and analyze its behavior without herbivores before investigating their interactions. Our study investigates the equilibrium points and their stability, discovering that different fixed points can become unstable due to various bifurcations such as transcritical, saddle-node, period-doubling, and Neimark–Sacker bifurcations. We have identified the Allee threshold, which, if exceeded, can cause both populations to become extinct below that level. However, we have discovered a coexistence equilibrium that is locally asymptotically stable for a range of parameter values above that threshold. Our additional numerical simulations suggest that this area of stability can be expanded. Our results indicate that this system is highly responsive to its parameters. We compare our findings to those of a system without strong Allee effects and conduct numerical simulations to verify our results. By including the Allee effect in the plant population, we enrich the local and global dynamics of the system.

This paper studies the dynamics of a class of host-parasitoid models with host refuge and the strong Allee effect upon the host population. Without the parasitoid population, the Beverton–Holt equation governs the host population. The general probability function describes the portion of the hosts that are safe from parasitism. The existence and local behavior of solutions around the equilibrium points are discussed. We conclude that the extinction equilibrium will always have its basin of attraction which implies that the addition of the host refuge will not save populations from extinction. By taking the host intrinsic growth rate as the bifurcation parameter, the existence of the Neimark–Sacker bifurcation can be shown. Finally, we present numerical simulations to support our theoretical findings.