Stochastic Analysis of a Hassell-Varley Type Predation Model

We investigate a Hassell-Varley type predator-prey model with stochastic perturbations. By perturbing the growth rate of prey population and death rate of predator population with white noise terms, we construct a stochastic differential equation model to discuss the effects of the environmental noise on the dynamical behaviors. Applying the comparison theorem of stochastic equations and Itô’s formula, the unique positive global solution to the model for any positive initial value is obtained. We find out some sufficient conditions for stochastically asymptotically boundedness, permanence, persistence in mean and extinction of the solution. Furthermore, a series of numerical simulations to illustrate our mathematical findings are presented. The results indicate that the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model when the noise intensity is small under some conditions, but while the noise intensity is sufficiently large, the species may die out, which does not happen in the deterministic model.