On the stochastic global dynamics of the delayed Nicholson's blowflies model

The well-known class of Nicholson's blowflies equations is considered under stochastic perturbations of the white noise type. We are concerned about the stability of the zero solution $x_0$ which means the extinction of the species of Nicholson's blowflies, and the positive equilibrium $x^{\ast }$ which means their persistence. Using appropriate Lyapunov functionals, sufficient conditions of stochastic stability, uniform stability and stochastic global exponential mean-square stability are derived. Moreover, we develop a new way of constructing a delayed-deterministic system by Lyapunov functional that leads to the extinction in the sense of the mean-square. Areas of stability with some numerical simulations are given to illustrate our results.