We consider a strongly coupled reaction-diffusion system describing three interacting species in a simple food chain structure. Based on the Leray-Schauder degree theory, the existence of non-constant positive steady states is investigated. The results indicate that, when the intrinsic growth rate of the middle species is small, cross-diffusions of the predators versus the preys are helpful to create global coexistence (stationary patterns).
"Non-constant positive steady states for a strongly coupled nonlinear reaction-diffusion system arising in population dynamics." Rocky Mountain J. Math. 45 (4) 1333 - 1355, 2015. https://doi.org/10.1216/RMJ-2015-45-4-1333