Effect of small time delay in a predator-prey model within random environment

Abstract

We consider a non-Kolmogorov type predator-prey model with and without delay. The local stability and Hopf bifurcation results are stated for both the cases of the deterministic system. The effect of stochastic perturbation in the form of parametric white and colored noise are considered. The exponential mean square stability of the trivial solutions for the stochastic differential equations and stochastic delay differential equations are studied under the Ito interpretation. To study the exponential mean square stability of the interior equilibrium point for the delayed model system in the presence of parametric white and colored noise, an approximation procedure is followed for the system of stochastic delay differential equation model system under the assumption that the magnitude of time delay is small. We have obtained threshold magnitude of discrete time delay. Critical magnitude for noise intensities are obtained for stochastic stability aspects. Numerical simulation results are provided in support of our analytical findings.