Analysis Of A Class Of Non - Uniform Chemostat Model With Toxic

By means of the generalized maximum principle> the fixed point index theory and bifurcation theory, we deal with an unstirred chemostat model with an external toxin: St=dSxx-auf1(S)-bvf2(S), x∈(0,1), t>0, ut=duxx+auf1(S)-γpu, x∈(0,1), t>0, vt =dvxx +bvf2(S), x∈(0,1), t>0, Pt=dpxx-ch(p)v, x∈(0,1), t>0, with boundary conditions Sr(0,t)=-1,Sx(1,t)+vS(1,t)=0,t>0, ux(0,t)=0,ux(1,t)+vu(1,t)=0,t>0, vx(0,t)=0,vx(1,t)+vv(1,t)=0,t>0, px(0,1)=-1,px(1,t)+up(1,t)=0,t>0, and initial conditions S(x,0)=S0(x)≥0, u(x,0)=u0(x)≥0(0≠0), v(x,0)=v0(x)≥0(0≠0), p(x,0)=p0(x)≥0,We studied the existence and uniqueness of semi-trivial steady state solutions and the existence of positive solutions. The main contents of this paper are as follows:In chapter 1, the biological background and the recent work on this model are described in detail, and some preliminaries which are very useful in the forthcoming chapters are given, including the generalized maximum principle、the fixed point index theory and the bifurcation theory. Moreover, priori estimates for positive solutions is proved.In chapter 2, the properties of positive steady-state solutions of this model are considered. First of all, some elementary properties of semi-trivial solutions are given. Moreover, with the help of the the generalized maximum principle and fixed point index theory, we obtained the existence and uniqueness of semi-trivial solutions and the existence of positive solutions. Thirdly, we obtained the existence and the global structure of positive steady-state solutions by using bifurcation theory, and the fact that the local solution branch can be extended to a global solution branch is proved. On this basis, the global solution branch connects with the branch {(α,S*,u*,0,z):a∈R+}. Finally, we present some numerical simulations to verify and complement our theoretical analysis.