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.

Abstract A certain class of a host–parasitoid models, where some host are completely free from parasitism within a spatial refuge is studied. In this paper, we assume that a constant portion of host population may find a refuge and be safe from attack by parasitoids. We investigate the effect of the presence of refuge on the local stability and bifurcation of models. We give the reduction to the normal form and computation of the coefficients of the Neimark–Sacker bifurcation and the asymptotic approximation of the invariant curve. Then we apply theory to the three well-known host–parasitoid models, but now with refuge effect. In one of these models Chenciner bifurcation occurs. By using package Mathematica, we plot bifurcation diagrams, trajectories and the regions of stability and instability for each of these models.