Abstract
Known Nicholson's blowflies equation (which is one of the most important models in ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial) point of equilibrium of this equation and also a capability of its discrete analogue to preserve 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 to verify stability in probability of nonlinear origin equation. From known previous results connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense in the continuous case and necessary and sufficient conditions for the discrete case 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.
Highlights
IntroductionIn the case if the order of nonlinearity is more than 1, these conditions are sufficient ones (both for continuous and discrete time [15,16,17,18,19]) for stability in probability of the initial nonlinear equation by stochastic perturbations
Consider the nonlinear differential equation with exponential nonlinearity x (t) = ax(t − h)e−bx(t−h) − cx(t), (1.1)Equation (1.1) is enough popular with researches [1,2,3,4,5,6,7,8,9,10,11]
Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories
Summary
In the case if the order of nonlinearity is more than 1, these conditions are sufficient ones (both for continuous and discrete time [15,16,17,18,19]) for stability in probability of the initial nonlinear equation by stochastic perturbations. This method was used already for stability investigation of other biological systems with delays: SIR epidemic model [15] predator-prey model [19]. Conditions for asymptotic mean square stability that are used here were obtained via the general method of Lyapunov functionals construction for stability investigation of stochastic differential and difference equations [22,23,24,25,26,27,28]
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