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.

Recommended by Panayiotis SiafarikasWe investigate global dynamics of the following systems of difference equations x n 1 α 1 β 1 x n /y n , y n 1 α 2 γ 2 y n / A 2 x n , n 0, 1, 2, . . ., where the parameters α 1 , β 1 , α 2 , γ 2 , and A 2 are positive numbers and initial conditions x 0 and y 0 are arbitrary nonnegative numbers such that y 0 > 0. We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.

We consider two systems of rational difference equations in the plane:; n ¼ 0; 1; 2; . . . ;andwhere the parameters a 1 ; a 2 ; b 2 ; g 1 are positive numbers and initial conditions x 0 and y 0 are positive numbers. For the first system we obtain the global dynamics, whereas for the second system we obtain substantial global results for all values of parameters. Global dynamics of two systems is substantially different to the contrary of their similarities.

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 investigate global dynamics of the following systems of difference equationsxn+1=β1xn/(B1xn+yn),yn+1=(α2+γ2yn)/(A2+xn),n=0,1,2,…, where the parametersβ1,B1,β2,α2,γ2,A2are positive numbers, and initial conditionsx0andy0are arbitrary nonnegative numbers such thatx0+y0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.

We investigate global dynamics of the following systems of difference equationswhere the parameters a 1 , b 1 , A 1 , g 2 , A 2 , B 2 are positive numbers, and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold.

We investigate the local stability and the global asymptotic stability of the difference equationxn+1=αxn2+βxnxn-1+γxn-1/Axn2+Bxnxn-1+Cxn-1,n=0,1,…with nonnegative parameters and initial conditions such thatAxn2+Bxnxn-1+Cxn-1>0, for alln≥0. We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, whereβ=B=0, in which case we show that such equation exhibits a global period doubling bifurcation.

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.