Lotka-volterra System Hopf Bifurcation And Chaos

Population ecology is an important branch of ecology, but also mathematics in ecology to date the most widely used and in-depth development of the most system-atic and mature branches.And population dynamics of interaction between predation, competition, mutualism and so on. The interaction between populations can change or is conditional, that can change the population conditional or depend on the interaction of different factors (such as size, environmental conditions, the individual stages of de-velopment, etc.). The branch theoretical plays a very important role in the research of the biological model, especially Hopf bifurcation theory, which for biological systems with time delay to determine the existence of periodic solutions has a very important role.ChapterⅡwill use the Hopf bifurcation theory to determine a stable limit cycle in a class of predator-prey system with delay and stocking items. Using time delay as a bifurcation parameter, through the branch point, the stability of equilibrium will occur significant change. While taking delay as a bifurcation parameter, we also derived the conditions for Hopf bifurcation, and then use Hassard method and center manifold theorem to determine the branch direction and the direction of bifurcation periodic solutions, at last we can have the Stability calculations and the determine formula.ChapterⅢis to consider a class of the competitive model with delay and stocking items, and let delay as a bifurcation parameter to study the conditions of existence of Hopf bifurcation and the change of balance between stability when pass through the fulcrum, then from use methods of Hassard and center manifold theorem to determine the branch direction and the direction of bifurcation periodic solutions, at last we can have the Stability calculations and the determine formula.