Coexistence in the three species predator–prey model with diffusion

The three species food chain model is discussed, in which the third species is the predator of the second one and the second species is the predator of the first one. We consider coexistence states of the associated weakly-coupled elliptic problem under the homogeneous Neumann boundary conditions. It is shown that there are no non-constant solutions if the diffusion rates of species are strong or if the intrinsic growth rate of a prey is slow and the intrinsic drop rates of predators are fast. It is also shown that the weakly-coupled parabolic system has a unique global solution for any non-negative initial data.