Global Behavior and Bifurcation in a Class of Host-Parasitoid Models with a Constant Host Refuge

In this paper, by using the analytical approach, we investigate the global behavior and bifurcation in a class of host-parasitoid models when a constant number of the hosts are safe from parasitism. We find the conditions for the existence and stability of the equilibria. We detect the existence of the Neimark-Sacker bifurcation under certain conditions. We explicitly derived the approximation of the limit curve depending on the parameters that appear in the model. We show that a locally asymptotically stable equilibrium can never be transformed into unstable by increasing a constant number of hosts that are using a refuge. Specially, we consider the effect of constant host refuge in (S), (HV), and (PP) models.The obtained results show that the constant number of hosts in refuge affects the qualitative behavior of these models in comparison to the same models without refuge. The theory is confirmed and illustrated numerically.