Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

In this paper, an n -species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multi-species Lotka–Volterra type competition model and prey–predator model.