A Class Chemostat Model Positive Equilibrium State Solutions Of Nature

This paper deals with an unstirred chemostat model with toxins: st=Δs-auf1(s)-bvf2(s), x∈Ω, t>0, ut=Δu+auf1(s)-βpu, x∈Ω, t>0, vt=Δv+b(1-k)vf2(s),x∈Ω, t>0, pt=Δp+bkvf2(s), x∈Ω, t>0with boundary conditions with initial conditions s(x,0)＝s0(x)≥0,(x,0)＝u0(x)≥0,(?)0,x∈Ω, s(x,0)=v0(x)≥0,(?)0,p(x,0)=p0(x)>0,(?)0,x∈Ω.By means of the fixed point index theory, the existence of positive steady-state solutions of this model is studied. Moreover, the bifurcation structure, stability of positive steady-state solutions and the effects of the toxins on the number and sta-bility of coexistence solutions are discussed by bifurcation theory and perturbation theory.The main contents of this paper are as follows:In chapter1, the biological background of the model is described in detail, and some preliminaries are given.In chapter2, the properties of positive steady-state solutions are considered. At first, some sufficient conditions and a necessary condition for the existence of positive steady-state solutions are obtained by means of the fixed point index theory. Moreover, the local structure of positive steady-state solutions and its stability when the effect of toxins is large enough are discussed by bifurcation theory. Moreover, the local branch can be expanded to the global branch. At last, the effects of the toxins on the number and stability of coexistence solutions are investigated by perturbation technique. The results show that if the effect of toxins is large enough, this model has no coexistence solution when b, which models the maximal growth rate of the species v, is larger than σ1/1-k, but when all positive solutions of this model are controlled by a limit system.