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Abstract This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.

Abstract We consider the second-order rational difference equation xn+1=γ+δxnxn−12, $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$ where γ, δ are positive real numbers and the initial conditions x−1 $x_{-1}$ and x0 $x_{0}$ 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.

Abstract By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form xn+1=f(xn)xn−1,n=0,1,… $x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots $ , f:(0,+∞)→(0,+∞) $f:(0,+\infty )\to (0,+\infty )$, f is sufficiently smooth and the initial conditions are x−1,x0∈(0,+∞) $x_{-1}, x _{0}\in (0,+\infty )$. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α1 $\alpha _{1}$. Then we apply the results to several difference equations.

We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

We investigate the local stability and the global asymptotic stability of the difference equation xn+1=αxn2+βxnxn-1+γxn-1/Axn2+Bxnxn-1+Cxn-1, n=0,1,… with nonnegative parameters and initial conditions such that Axn2+Bxnxn-1+Cxn-1>0, for all n≥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, where β=B=0, in which case we show that such equation exhibits a global period doubling bifurcation.