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In this paper, a two-species cooperating model with free diffusion and self-diffusion is investigated. The existence of the global solution is first proved by using lower and upper solution method. Then the sufficient conditions are given for the solution to blow up in a finite time. Our results show that the solution is global if the intra-specific competition is strong, while if the intra-specific competition is weak and the self-diffusion rate is small, blow-up occurs provided that the initial value is large enough or the free diffusion rate is small. Numerical simulations are also given to illustrate the blow-up results.

Global existence and blow-up analysis to a cooperating model with self-diffusion