Stability of equilibrium states for a stochastically perturbed exponential type system of difference equations

In the paper it is shown how the known results of stability theory can be simply applied to stability investigation of equilibrium points of some systems of nonlinear difference equations with stochastic perturbations. A system of two difference equations with exponential nonlinearity is considered and it is shown that instead of the zero equilibrium this system can have also a positive equilibrium. Sufficient conditions for stability in probability of the both equilibriums of the initial nonlinear system with stochastic perturbations are obtained. Numerical simulations and figures illustrate a convergence of the positive solutions of the considered system to one of two (zero or positive) equilibriums in deterministic and stochastic cases. The proposed investigation procedure can be applied for arbitrary nonlinear equations with an order of nonlinearity higher than one.