Abstract: A system of reaction-diffusion equations in the unstirred chemostat with internal storage and external inhibitor is considered in this paper. Firstly, in order to overcome the ratio singularity, the sharp priori estimates for positive solutions of the system are established by the maximum principle, and then a special cone smaller than the usual positive cone is constructed. Secondly, the sufficient conditions for the existence of the model coexistence solution are studied by the monotone method and the topological degree theory on the special cone. It turns out that there exists at least one positive solution when the principal eigenvalues of the corresponding nonlinear eigenvalue problems are both positive or negative.

Key words: chemostat, internal storage, external inhibitor, monotone method, fixed point theory