Properties And Numerical Simulations Of Stochastic Non-autonomous Gilpin-Ayala Competition Models

The theory of differential equations plays an important role in ecology, especially inmathematical ecology. In recent years, more and more attention is paid to how to applystochastic differential equations to investigate population dynamical systems. Populationin the real world is inevitably affected by environmental noises，so using the theory ofstochastic differential equations to investigate the population dynamics has a bettertheoretical and practical significance.The research of persistence and extinction for population dynamics is one of themost significant topics. And the classical model is Lotka-Volterra competition model.Cosidering that this model is too idealistic，Gilpin-Ayala competition model whichaccords with the realistic better was brought into being.This paper investigates two widely used stochastic non-autonomous Gilpin-Ayalacompetition models, which are derived by adding environmental noise on the coefficientsof the deterministic model. For the first system, we give some conditions to ensure themodel has a global positive solution, and discuss the following properties: extinction,non-persistence in the mean and stochastic permanence. For the second system, we givea formal solution, and discuss the following properties: extinction, non-persistence in themean, strong persistence in the mean and stochastic permanence. Finally, we use theMilstein method to verify the correctness of the results and analyze how the stochasticpermanence of the models changes by changing the parameter.