Positive solutions to the unstirred chemostat model with Crowley-Martin functional response

Abstract

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steady-state. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results.