The Asymptotic Stability And Asymptotic Periodicity For Two Kinds Of Fractional-order Mathematical Models

In this paper,by using Mittag-Leffler function and the comparison theorem of fractional-order differential equation,we research the permanence,asymptotic stability and asymptotic periodicity for two kinds of fractional-order mathematical models,and some of our results generalize and improve the results of the known literature.It is divided into five chapters.In the first chapter,we mainly present the background and significance of the topic,and introduce the development of fractional-order differential equation,the application in various fields and the relevance works had been done by forefathers.Then,we introduce two kinds of mathematical models:predator-prey system and Mackey-Glass respiratory system,and state the main work of this article briefly.In the second chapter,we give some definitions and lemmas of fractional-order differential equation and a number of relevant properties of Mittag-Leffler function.Based on these properties,we deduce some important lemmas.In order to obtain the permanence of these models,we prove the comparison theorem of fractional-order d-ifferential equation.In order to research the asymptotic stability and asymptotic pe-riodicity of these systems,we introduce the characteristic equation of fractional-order differential equation.In the third and fourth chapters,by applying properties of Mittag-Leffler function,the comparison theorem of fractional-order differential equation and the relationship between characteristic equation of Laplace transform and stability,we get the perma-nence,asymptotic stability and asymptotic periodicity of fractional-order predator-prey model and Mackey-Glass respiratory model.Then,examples illustrate the feasibility of our results.In the fifth chapter,we make a summary and review of the full text,and discuss the directions for future research.