This paper deals with the unstirred chemostat model in the presence of an internal inhibitor. The main purpose of the paper is to determine the exact range Λ of the maximal growth (a,b) of two species where the system possesses positive solutions. It turns out that Λ is a connected unbounded region in R+2, whose boundary consists of two monotone nondecreasing curves Γ1:a=H1(b) and Γ2:b=H2(a). For every (a,b) inside Λ the system has positive solutions and for (a,b) outside Λ there exists no positive solution. The functions H1(b) and H2(a) are constructed in terms of the limit of the corresponding time-dependent solution with a specific initial function. In particular, it is also shown that the system has at least two positive solutions in certain subregion of Λ.