Global Solutions To The Three-Species Food-Chain Model And Cooperative Model With Self-and Cross-Diffusion

Divided into three parts, this paper investigate the following strongly coupled three-species food-chain model and cooperative model with Lotka-Volterra type reaction functionsFirstly, the existence of the positive equilibrium state solutions to equations (M_?) with homogeneous Dirichlet boundary value condition is established by applying the method of upper and lower solutions for any spatial dimension. Secondly, using the method of energy estimates and Gagliardo-Nirenberg type inequalities, we proved the existence and uniform boundedness of nonnegative global solutions for equations (M_?) with homogeneous Neumann boundary value condition when the spatial dimension is one. Moreover, some criteria on the global asymptotic stability of the positive equilibrium points for (M_?) are also given by constructing Lyapunov function. Finally, by applying the energy methods, Sobolev embedding theorems and bootstrap arguments, the global existence of nonnegative classical solutions to equations (M_-) with homogeneous Neumann boundary value condition is proved when the space dimension is at most 5. Under certain conditions for the coefficients of the reaction functions, the convergence of the solutions is established for the system with large diffusion coefficients by constructing Lyapunov function.