The Asymptotic Properties Analysis Of Several Stochastic Differential Equations With Delay

The Gilpin-Ayala model is one of the most important mathematical models in ecology. The study of various forms of Gilpin-Ayala model has been favored by several authors. It is difficult to research the Gilpin-Ayala model which is a nonlinear system. By combining inequality skills with Lyapunov method,exponential stability method, M-matrix theory and other methods, the asymptotic behaviors of stochastic Gilpin-Ayala models have been investigated. In this paper,we considered better methods to solve the problems of asymptotic property of stochastic Gilpin-Ayala models.It is well-known that Lyapunov method is an important tool to investigate the asymptotic properties of the system. Instead of this method, we apply Feller’s test to study the asymptotic behaviors of three one-dimensional Gilpin-Ayala models with different stochastic perturbations in the second chapter of this article. Compared with the method about asymptotic properties of the current researches, our method has many advantages, such as simple structure, less constraints, independence of Lyapunov function and so on. Finally, examples with numerical simulations are given to illustrate our results.Population models in the real world are inevitably subject to the environmental noises. At the same time, time delays must be taken into account. Hence, it is significant to understand how these two factors influence stability of population ecology. The third chapter of this article shows how the known results of stability theory can be simply applied to stability of equilibrium points of two nonlinear differential equations with stochastic perturbations and time delay. Firstly, it is given a systematic method that solves asymptotic mean-square stability and stability in probability of nonlinear differential equations. Secondly, sufficient conditions for stability in probability of the nonlinear system with white noises perturbations are obtained based on the theory. The results show that the delays have an impact on stability of nonlinear stochastic differential equations. Finally, we give a numerical simulation to support the theoretical analysis in this paper.