Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 92959, 25 pages
Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis
Department of Higher Mathematics, Donetsk State University of Management, 163a Chelyuskintsev Street, Donetsk 83015, Ukraine
Received 23 January 2007; Accepted 11 June 2007
Copyright © 2007 Nataliya Bradul and Leonid Shaikhet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.