Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis

Bradul, Nataliya; Shaikhet, Leonid

doi:https://doi.org/10.1155/2007/92959

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2007 | Article ID 092959 | https://doi.org/10.1155/2007/92959

Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis

Nataliya Bradul¹and Leonid Shaikhet¹

Received23 Jan 2007

Accepted11 Jun 2007

Published14 Aug 2007

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.

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Copyright

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.

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