Dynamic Analysis Of Two Classes Of Biological Mathematical Models

In this thesis,by using the comparison theorem、Lyapunov stability theory,iter-ative method and applying fundamental inequality,we study the dynamical behaviors of two classes of biological mathematical models(food chain model with stage struc-ture and time delays;Leslie-Gower model with nonlinear harvesting and prey refuge)with practical background,which mainly include the positivity and boundedness,the existence of equilibria,the stability of the equilibria.The dissertation consists of three chapters.In chapter one,the history and background of two classes of models are summa-rized.The research contents and some basic preliminaries of this thesis are briefly addressed.In chapter two,based on the existing food chain model,a new three species food chain model was constructed by incorporating the stage structure and maturi-ty time(delays)for predator and top predator species.The positivity,boundedness of solutions and the existence of the equilibrium are discussed.By using the approximate-ly linearized system,Lyapunov functional,comparison theorem and iterative technique,the stability of equilibria are studied.Numerical simulations are great well agreement with the theoretical results.In chapter three,based on the traditional Leslie-Gower model,a new Leslie-Gower model was constructed by incorporating the feeding rate and the nonlinear harvesting function for prey.The positivity,boundedness of solutions and the existence of the positive equilibrium are discussed.By using Routh-Hurwitz method,some sufficient conditions for the locally asymptotically stable of the positive equilibrium are derived.With the help of Lyapunov function,the globally asymptotically stable of the positive equilibrium is studied.Applying Pontryagin maximum principle,the optimal tax pa-rameters and the optimal equilibrium solution of the system axe analyzed.Numerical simulations are great well agreement with the theoretical results.