Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson’s blowflies application

We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from white Gaussian noise, which is time derivative of standard Brownian motion. main objective this paper to introduce a new technique using Lyapunov functional for study stability zero solution system. Constructing appropriate deterministic system in neighborhood origin an effective way investigate necessary and sufficient conditions sense mean square. Nicholson’s blowflies equation one major problems ecology; possible extinction population investigated. support our theoretical results by providing areas some numerical simulations Euler–Maruyama scheme, square stable Maruyama (Rendiconti del Circolo Matematico di Palermo 4(1):48, 1955), Cao et al. (Appl Math Comput 159(1):127–135, 2004).