Asymptotic Properties of a Lotka-Volterra Competition and Mutualism Model Under Stochastic Perturbations

Abstract

We consider and establish global asymptotic properties of a stochastically perturbed Lotka-Volterra model of competing or symbiotic populations. In this model, stochastic perturbations are assumed to be of the white noise type and are proportional to the current system state. While such a type of the stochastic perturbations is most realistic for real-life biosystems, models with such perturbations are also difficult to analyze because such model have no equilibrium states apart from the origin.

To analyze the dynamics of the stochastic model, in this paper, we firstly construct a Lyapunov function that is applicable to both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method enables us to find estimates for this region.

We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization (stabilization by noise). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (stochastic stabilization) are found as well applying the Khasminskii-type Lyapunov functions.

Keywords: Competition Model, Ito's Stochastic Differential Equation, Stochastic Extinction, Stabilization By Noise