Study On The Dynamics Of The Nonlinear Population Systems

Throughout the history of theoretical ecology, reaction-difusionequations have been intensively used to describe spatiotemporal dy-namics. In this thesis, the main results are as follows:In chapter2, we investigate the dynamics of a difusive predator–prey model with Holling–II functional response and the additive Alleeefect in prey. We show the stability of the positive equilibrium,and give the conditions of the existence of the Hopf bifurcation. Bycarrying out global qualitative and bifurcation analysis, it is shownthat the weak and strong Allee efects in prey can induce diferentdynamical behavior in the predator–prey model.In chapter3, we present the complex dynamics of a Leslie–Gowerpredation model with the additive Allee efect on prey. We show theconditions of the existence and the stability of the positive equilibriawith weak and strong Allee efect, respectively. We carry out theanalytical study for two–dimensional system in details and fnd out acondition for Turing instability of a locally stable equilibrium. Fur-thermore, we perform numerical simulations and fnd that the corre-sponding reaction-difusion model has rich dynamics, such as stripes,spot-stripe mixtures and spots patterns.In chapter4, we investigate the complex dynamics of a reaction-difusion epidemic model with nonlinear incidence rate of saturatedmass action. We give the analysis of the boundedness, dissipationand the stability of the positive equilibria. And we show the con-ditions of Turing instability and determine the Turing space in theparameters space. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction ofspatially distributed infectious with local difusion, and fnd that themodel dynamics exhibits a difusion–controlled formation growth tospots, stripes-spots, stripes, stripes-holes and holes pattern replica-tion. Furthermore, we indicate that the speed of disease spreadingare getting bigger with the parameter or the difusion of infectiousincreasing.In chapter5, we investigate the dynamics of a difusive epidemicmodel with strong Allee efect in the susceptible population. We showsome properties of solutions of the model, the asymptotic stability ofthe disease–free and the endemic equilibria. Furthermore, we givethe conditions of Turing instability and determine the Turing spacein the parameters space. Based on these results, we perform a seriesof numerical simulations and fnd that the model exhibits complexpattern replication: spots, spot-stripe mixtures and stripes patterns.And in the last chapter, we give some discussions and remarks.