Existence of positive steady-state solutions for the competition model in the unstirred chemostat

Aim To discuss the existence of positive steady-state solutions for the competition model in an unstirred chemostat.Methods Using lower-upper solution method,principal eigenvalue theory,the maximum principle and the index of fixed point of compact map in cone.Results Some conditions of existence of positive steady-state solutions to the system are obtained.Conclusion Positive steady-state solutions exist when the maximal birth-rates of the two competing populations are large enough.