In this paper we present a local dynamics and investigate the global behavior of the following system of difference equations$x_{n+1}=ax_{n}^{3}+by_{n}^{3}$ $y_{n+1}=Ax_{n}^{3}+By_{n}^{3}$ $n\in\mathbb{N}_0$ with non-negative parameters and initial conditions $x_{0}$ and $y_{0}$ that are real numbers. We establish the relations for local stability of equilibriums and necessary and sufficient conditions for the existence of period-two solution(s). We then use this result to give global behavior results for special ranges of parameters and determine the basins of attraction of all equilibrium points.

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.

We investigate the period-doubling and Naimark–Sacker bifurcations of the equilibrium of the difference equation xn+1 = γxn−1 + δxn Cxn−1 + xn where the parameters γ, δ, C are positive numbers and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that x−1 + x0 > 0. AMS Subject Classifications: 39A10, 39A20, 39A60, 37B25.

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.

Here we examine the behavior of a rational Lotka - Volterra model which is a modification of the ordinary polynomial case. We find nonnegative equilibrium points and define conditions in the parametric space for the stable positive equilibrium point. We also prove existence of the stable limit cycle in the case of the unstable positive equilibrium point.