Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis

Abstract

Known Nicholson's blowflies equation (which is one of the most important models inecology) with stochastic perturbations is considered. Stability of the positive (nontrivial)point of equilibrium of this equation and also a capability of its discrete analogue topreserve stability properties of the original differential equation are studied. For this purpose, the considered equation is centered around the positive equilibrium and linearized. Asymptotic mean square stability of the linear part of the considered equation is used toverify stability in probability of nonlinear origin equation. From known previous resultsconnected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense inthe continuous case and necessary and sufficient conditions for the discretecase are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically in the deterministic and the stochastic cases for different values of the parameters and of the initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted.