Non-constant positive steady state of one resource and two consumers model with diffusion

In this paper, a predator–prey reaction–diffusion system with one resource and two consumers is considered. Assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington–DeAngelis functional response, and they compete for the common resource. First, it is proved that the unique positive constant steady state is stable for the ODE system and the reaction–diffusion system. Second, a prior estimates of positive steady state is given. Finally, the non-existence of non-constant positive steady state, the existence and bifurcation of non-constant positive steady state are studied.