Stability of positive equilibrium of a Nicholson blowflies model with stochastic perturbations

Abstract

This paper is concerned with the stability problem of the positive equilibrium of a Nicholson's blowflies model with nonlinear density-dependent mortality rate subject to stochastic perturbations. More specifically, the existence of a unique positive equilibrium of a Nicholson's blowflies model described by the delay differential equation \begin{equation*} N'(t)=-\left(a-be^{-N(t)}\right)+\beta N(t-\tau)e^{-\gamma N(t-\tau)} \end{equation*} is first quoted. It is assumed that the underlying model in noisy environments is exposed to stochastic perturbations, which are proportional to the derivation of the state from the equilibrium point. Then, by utilizing a stability criterion formulated for linear stochastic differential delay equations, explicit stability conditions are obtained. An extension to models with multiple delays is also presented.

Keywords

• nicholson's blowflies model
• nonlinear mortality rate
• stochastic perturbations
• asymptotic stability