Asymptotic behavior of an unstirred chemostat model with internal inhibitor

This paper deals with a chemostat model with an internal inhibitor. First, the elementary stability and asymptotic behavior of solutions of the system are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which measures the effect of the inhibitor, plays a very important role in deciding the stability and longtime behavior of solutions of the system. The results show that if μ is sufficiently large, this model has no coexistence solution and one of the semitrivial equilibria is a global attractor when the maximal growth rate a of the species u lies in certain range; but when a belongs to another range, all positive solutions of this model are governed by a limit problem, and two semitrivial equilibria are bistable. The main tools used here include monotone system theory, degree theory, bifurcation theory and perturbation technique.