Paulo Amorim, Raimund Bürger, Rafael Ordoñez, Luis M. Villada:
Global existence in a food chain model consisting of two competitive preys, one predator and chemotaxis
We consider a mathematical model for the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition. Besides diffusing, the predator species moves toward higher concentrations of a chemical substance which is produced by the prey, which move away from a substance produced by the predators. The resulting reaction-diffusion system consists of three parabolic equations along with three elliptic equation describing the chemical. First the local existence of a nonnegative solutions is proved, then we provide uniform estimates in Lebesgue spaces which lead to boundedness and the global well-posedness for the system. Finally we report and discuss some numerical simulations.