In this paper, we consider the dynamics of a certain class of host-parasitoid models, where some hosts are completely free from parasitism either with or without a spatial refuge and the host population is governed by the Beverton–Holt equation. We assume that, in each generation, a constant portion of the host population may find a refuge and be safe from the attack by parasitoids. We derive some criteria for the Neimark–Sacker bifurcation. Then, we apply the developed theory to the three well-known cases: [Formula: see text] model, Hassel and Varley model, and parasitoid–parasitoid model. Intensive numerical calculations suggest that the last two models undergo a supercritical Neimark–Sacker bifurcation.

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. ARTICLE HISTORY

We consider the second-order rational difference equation x n+1 = γ + δ x n x 2 n-1 , where γ , δ are positive real numbers and the initial conditions x-1 and x 0 are positive real numbers. Boundedness along with global attractivity and Neimark-Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form x n+1 = f (x n) x n-1 , n = 0, 1,. .. , f : (0, +∞) → (0, +∞), f is sufficiently smooth and the initial conditions are x-1 , x 0 ∈ (0, +∞). We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α 1. Then we apply the results to several difference equations.

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