Global stability in stochastic difference equations for predator-prey models.

There are many publications on theoretical analysis of deterministic difference equationsand stochastic differential equations. However, relatively few theoretical papers are pub-lished to consider the positivity of solutions of discrete-time stochastic difference equations(DSDEs), and no theoretical papers investigate the global stability of nontrivial solutionsof DSDEs with nonlinear terms. In this paper, we consider a DSDE model that is ageneralization of two-dimensional nonlinear models of stochastic predator-prey interac-tions, and show the positivity and global stability of the nontrivial solutions by using ournew discretized version of the Itô formula. In addition, our results are compared withthose of continuous-time stochastic differential equations and discrete-time deterministicdifference equations. Numerical simulations are introduced to support the results. [ABSTRACT FROM AUTHOR]