| In this paper, several degenerated bifurcations of dynamical system which consist of differential equations with time delays are discussed. Mainly by generalizing and using cen-ter manifold theorem and normal form theory, the dynamic behavior of the corresponding prey predator system and neural network model are analyzed. Then, some normal forms of those dynamic systems are obtained. Furthermore, we get some interesting bifurcation phenomenons. Concretely, this paper has done the following work:In the first chapter, this paper introduces the background and status of mathematical ecology research and some related basic concepts.In the second chapter, the problem of BT bifurcation for a delayed predator prey system with stage structure are researched. A constant harvesting h which is related with predator is introduced to this model. By analyzing the corresponding characteristic equation, stability of boundary equilibrium and internal equilibrium to be researched. Through using center manifold theorem, we can get different bifurcation situations in the degenerated equilibrium.In the third chapter, the bifurcation problems of triple-zero bifurcation of a delayed ratio-dependent Holling-Tanner predator prey system are considered. We firstly give the existing conditions of triple-zero singular point. Then, by selecting appropriate bifurcation parameters and using center manifold theorem and normal form theory, the normal form of the system at the point of triple-zero singularity and the corresponding bifurcation results are given.In the final chapter, the BT and triple-zero bifurcation analysis in a recurrent neural network model with delays are mainly discussed. By discussing the distribution of charac-teristic roots of the characteristic equation, we give the existence conditions of the BT and triple-zero singularities. So, respectively, quadratic standard form of the system which has corresponding bifurcation is given by the center manifold theorem. |
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