Dynamics Behavior In A Multi-Species Keller-Segel Chemotaxis Model With Lotka-Volterra Cooperative Source

In this dissertation,we investigate nonlinear dynamics behavior in a multi-species chemotaxis Keller-Segel model with Lotka-Volterra cooperative source.The contents of this dissertation are divided into two parts.In firt part,we mainly study the stability of a unique positive equilibrium point for the model under the condition of equal diffusion coefficients.First,we give a sufficient condition to guarantee the existence and uniqueness of positive equilibrium point in the model by applying the nonsingular M-matrix theory.Then,by using the linearization,the local stability of the unique positive equilibrium point is proved.Finally,we discuss the global asymptotical stability of the positive equilibrium point by the relationship between this model and a corresponding system of ordinary differential equations.In second part,we study nonlinear dynamics behavior near unstable constant equilibrium in the model under the condition of unequal diffusion coefficients.It is show that the linear unstable equilibrium point caused by chemotaxis also is full nonlinear unsta-ble.Our result provides a quantitative characterization for early pattern formation in the model.