Study On The Properties Of Several Randomized Biological Models With Logistic Growth

In this thesis, we investigate some stochastic epidemic models with Logistic growth and non-autonomous Logistic system with feed-back by applying the theory of stochastic differen-tial equation. By inducing random perturbations to some deterministic model, we get stochastic models and study their property of dynamics. The thesis is divided into five chapters, which is organized as follows:In chapter 1, we introduce the historical background and the recent development of the prob-lems considered in this thesis, as well as the main work done in this dissertation. Moreover, we present some preliminary lemmas and definitions.In chapter 2, we introduce stochasticity into a model of SIR with density dependent birth rate. We get three stochastic models and Whose contact coefficient, natural death rate and diseased death rate are subject to the environ-mental white noise respectively. We show that the models possesses non-negative solutions as desired in any population dynamics. We also carry out the globally asymptotical stability of the equilibrium through the stochastic Lyapunov functional method if R0≤1. Furthermore when Ro> 1, we give the the asymptotic behavior of the stochastic systems around the endemic equi-librium of the deterministic model and show that the solution will oscillate around the endemic equilibrium. Finally, numerical simulations are present to illustrate our results.[n chapter 3, we study a stochastic non-autonomous logistic system with feedback control Sufficient conditions for stochastic asymptotically bounded, extinction, non-persistence in the mean, weak persistence and persistence in the mean are established. The critical number between weak persistence and extinction is obtained. A very important fact is found in our results, that is the feedback control is harmless to the permanence of species even though under the randomized environment. Finally, the numerical simulation is done.In chapter 4, we propose a stochastic disease model with chronic stage and Logistic growthThe existence, uniqueness, and positivity of the solution are studied. There is no disease-free equilibrium for the stochastic models after adding stochastic perturbation in the corresponding deterministic model. Hence we discuss the behavior around disease-free equilibrium of the deter-ministic model. Finally, the numerical simulation is done.In chapter 5, we summarize the main results in this thesis and propose some problem which can be considered in the future.