Global Existence And Stability Of Solutions To The Three-Species Model With Cross-Diffusion

This paper divides into two chapters. In chapter 1, we consider the uniform bound-edness and global existence of solutions to the three-species Lotka-Volterra competition model with self and cross-diffusionby using the energy estimate method and Gagliardo-Nirenberg-type inequalities. Mean-white, global asymptotic stability of the positive equilibrium point for the model will be proved by Lyapunov function.In chapter 2, using the energy estimate, shauder theory and bootstrap arguments, the global existence of classical solutions for the competitor-competitor-mutualist diffusion modelis proved in n-dimensional (n > 1) domains, when diffusion coefficients, self-diffusion coefficients and cross-diffusion coefficients satisfies some conditions. Under certain conditions of the coefficients of the reaction functions, convergence of solutions for the model is given by Lyapunov functions.