Stability In Distribution Of Gilpin-ayala Equation With Markovian Switching

Using a deterministic mathematical biology model in the research of ecologyproblems has abundant achievement, but still a lot of time can not accurately describereality which is the reason why the stochastic differential equations become anecessary way in the study of ecology problems. The Logistic equation is one of themost important and basic mathematical model in the research of single populationgrowth. Gilpin-Ayala model, as a promotion of Logistic model, can better fit theactual data by adjusting the indices appropriately. So in this paper, we analyzerandomized Gilpin-Ayala equation.In this paper, properties and stability of stochastic Gilpin-Ayala model arestudied by analyzing the more general coefficient with the Markov switching.The first chapter introduces the research significance，background and researchstatus. It presents an overview of studies in this field as well as the research in thisthesis.The second chapter presents the theoretical foundation for the research, includingknowledge of probability theory, stochastic processes, the basic theory of stochasticdifferential equations with markov switching and stochastic inequalities. What’smore, the definition of random integral, Ito formula and asymptotic stability indistribution is proposed.The third chapter studies an SIS epidemic model with the logistic birth rate, wefind the existence of equilibrium and stability conditions. Besides, we prove thenonexistence of closed trajectory and singular closed trajectory by using Dulacfunctions.The fourth chapter finds the explicit solution of Gilpin-Ayala equation withMarkov switching, and proves the explicit solution is with the sense of the globalattraction solution and other useful properties. Based on the properties of the explicit solution in chapter Four, the fifth chapterproves that equation (4.2) is of asymptotic stability in distribution.