Codimension Bifurcation Analysis For Several Predator-prey Systems With Multiple Delays

In this thesis, we analyze the bifurcation problems of two types of predator-prey systems: one is a predator-prey model with double Allee effect, and the other one is a modified Leslie-Gower predator-prey model with two time-delays. By generalizing and using the normal form theory and center manifold theorem for delay differential systems,we compute the unfolding normal forms at corresponding interior positive equilibrium of these models, and then obtain a series of interesting bifurcation phenomena.In the first chapter, we introduce the background of this research and the development status. Afterwards, some of the necessary theoretical knowledge is listed.In the second chapter, we mainly consider the bifurcation problem of a predator-prey model with double Allee effect. Firstly, the sufficient, conditions for the model to be Bogdanov-Takens (B-T) or triple-zero singular points at the positive equilibrium point are given. Then, by selecting the appropriate bifurcation parameters and using the normal form theory and center manifold theorem, we deduce the unfolding normal forms and the corresponding bifurcation results at B-T or triple-zero singularity.In the third chapter, the bifurcations of a delayed modified Leslie-Gower predator-prey system are investigated. By analyzing the distribution of the root of the corresponding characteristic equation at the singularity, we get the conditions for the existence of the B-T or triple-zero singularity. By applying center manifold theory and qualitative theo-ry of differential equations, some bifurcation results of the system at the singularity are presented.