Coexistence solutions and their stability of an unstirred chemostat model with toxins

We study a reaction–diffusion system of the unstirred chemostat model with toxins in the NN-dimensional case. Firstly, some sufficient conditions for the existence of positive steady-state solutions are obtained by means of the fixed point index theory. Secondly, the local structure of positive steady-state solutions and their stability are discussed by the bifurcation theory. Lastly, the effect of the toxins is investigated by the perturbation technique. The results show that when the parameter ββ, which measures the effect of toxins, is large enough, this model has no coexistence solution provided that the maximal growth rate bb of the species vv is greater than σˆ11−k, and all positive solutions of this model are controlled by a limit system provided that bb belongs to the interval (σ11−k,σˆ11−k).