The Dynamical Behavior Analysis Of The Autonomous Discrete Two-species Competition And Predator-prey Population Model

The autonomous discrete-time two-species population dynamical models describedby difference equations are widely investigated, which are more appropriate than thecontinuous-time models. This paper study a discrete-time autonomous two-species com-petition model and a discrete two-species ratio-dependent predator-prey model. A newsuffcient condition on the global asymptotic stability of the positive equilibrium is estab-lished by using an iteration scheme and the comparison principle of di?erence equations.By using center manifold theorem and bifurcation theory, we show that the system un-dergoes Flip bifurcation and Hopf bifurcation. The paper is organized as follows:The first section is introduction, in which we present research background, purposeand significance of discrete-time epidemic models, and then the research status quo andresults of discrete-time epidemic models are given. Finally the organization of this paperis also presented.In section2, as preliminaries,we introduce some lemmas to obtain our main result.The autonomous discrete-time competition model is investigated in Section 3. Westudy the local stability and global stability of the equilibria, flip bifurcation and chaos ofthe discrete-time competition model. Finally, some numerical simulations are presentedto verify the obtained results.The autonomous discrete two-species ratio-dependent predator-prey model is inves-tigated in Section 4. We study the local stability and global stability of the equilibria, ?ipbifurcation and chaos of the discrete-time competition model. Finally, some numericalsimulations are presented to verify the obtained results.In section 5, we show some discuss and summaries.