Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

In this paper, we explore the dynamics of a certain class of Beddington host-parasitoid models, where in each generation a constant portion of hosts is safe from attack by parasitoids, and the Ricker equation governs the host population. Using the intrinsic growth rate of the host population that is not safe from parasitoids as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability. Then, we apply the new theory to the following well-known cases: May’s model, [Formula: see text]-model, Hassel and Varley (HV)-model, parasitoid-parasitoid (PP) model and [Formula: see text] model. We use numerical simulations to confirm our theoretical results.

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.

In this paper, we consider the cooperative system [Formula: see text] where all parameters [Formula: see text] are positive numbers and the initial conditions [Formula: see text] are nonnegative numbers. We describe the global dynamics of this system in a number of cases. An interesting feature of this system is that it exhibits a coexistence of locally stable equilibrium and locally stable periodic solutions as well as the Allee effect.

In this paper, our aim is to study the dynamical behavior of third-order system of rational difference equations xn+1 = αxn−2 β + γxnxn−1xn−2 , yn+1 = α1yn−2 β1 + γ1ynyn−1yn−2 , n = 0, 1, · · · . where the parameters α, β, γ, α1, β1, γ1 and initial conditions x0, x−1, x−2, y0, y−1, y−2 are positive real numbers. Some numerical examples are given to verify our theoretical results.