Mean Square Stability of the Zero Equilibrium of the Nonlinear Delay Differential Equation: Nicholson's Blowflies Application

Abstract We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson's blowflies equation is one of the major problems in ecology, necessary conditions for the possible extinction of the Nicholson's blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler-Maruyama scheme which is mean square stable \cite{Maruyama1955,Cao2004}.