Global existence of classical solutions for a class of diffusive ecological models with two free boundaries and cross-diffusion

In this paper we consider a class of diffusive ecological models with two free boundaries and with cross-diffusion and self-diffusion in one space dimension. The systems under consideration are strongly coupled, and the position of each free boundary is determined by the Stefan condition. We first show local existence of the solutions for the ecological models under some assumptions, and then prove the global existence of the solutions under extra assumptions. Our approach to the problem is by suitable changes, fixed point theorems and various estimates. Applications of these results are given to a two-species diffusive predator–prey model and a two-species diffusive competition model.