Characterization of positive solutions of the unstirred chemostat with an inhibitor

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 ΛΛ of the maximal growth (a,b)(a,b) of two species where the system possesses positive solutions. It turns out that ΛΛ is a connected unbounded region in R+2, whose boundary consists of two monotone nondecreasing curves Γ1:a=H1(b)Γ1:a=H1(b) and Γ2:b=H2(a)Γ2:b=H2(a). For every (a,b)(a,b) inside ΛΛ the system has positive solutions and for (a,b)(a,b) outside ΛΛ there exists no positive solution. The functions H1(b)H1(b) and H2(a)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 ΛΛ.