The Hopf Bifurcation Analysis In Competing Systems With Time Delay

In this paper, by using qualitative method and stability theory of ordinarydifferential equations, Hopf bifurcation for two classes with time delay in competitionmodels are studied. First of all, we can get the existence of the positive equilibriumsystem conditions. By analyzing the characteristic equation, we can get condition ofthe system occurring Hopf bifurcation. Finally by using the center manifold theoremand normal form method, we can get the related properties of the bifurcation. Thepaper is divided into five parts.The first part is the introduction, the predator prey model background and theresearch present situation, then introduces the main work of this paper.The second part is some theoretical tools, including qualitative and stabilitytheory of ordinary differential equations and Hopf bifurcation theory related lemmasand theorems.The third part is a discussion with two delays in competition Hopf bifurcationmodel, by using qualitative method and stability theory of ordinary differentialequations, selecting delay for bifurcation parameter, and discuss the existence of Hopfand the positive equilibrium of the stability. Finally, we give a numerical example.The fourth part discusses a distributed delay in competition Hopf bifurcationmodel, and get the balance of the stability conditions.The last part has an summarize of this paper, and we have a future prospects ofthe future research direction of mathematical biology.