Non-constant positive steady states for the HP food chain system with cross-diffusions

We consider a strongly coupled reaction-diffusion system describing three interacting species in the HP food chain structure. Based on the Leray-Schauder degree theory, the non-existence and existence of non-constant positive steady states are investigated. The stability of the constant positive steady state is also discussed. The results indicate that strong cross-diffusions are helpful for the appearance of the non-constant positive steady state (stationary patterns).