Stability of Equilibrium Points of Nicholson’s Blowflies Equation with Stochastic Perturbations

Abstract

In Chap. 9 the stability of the trivial and positive equilibrium points of the well-known Nicholson blowflies equation with stochastic perturbations is investigated. Basic stages of the proposed research are the following. It is assumed that the considered nonlinear differential equation has an equilibrium point and exposed to stochastic perturbations of white noise type that are proportional to the deviation of the system current state from the considered equilibrium point. In this case the equilibrium point is a solution of a stochastic differential equation too. The constructed stochastic differential equation is centered around the considered equilibrium point and linearized in the neighborhood of this equilibrium point. Necessary and sufficient conditions for the asymptotic mean-square stability of the linear part of the considered equation are obtained that at the same time are sufficient conditions for the stability in probability of the equilibrium point of the initial nonlinear equation under stochastic perturbations. The results of investigation are illustrated by four figures with stability regions and trajectories of solutions in the cases of stability and instability.