Characters Of Solutions To A Kind Of Unstirred Chemostat Model

Today, the so called Mathematical Biology becomes an activebranch of mordern science. A lot of mathematical models are established success-fully, and significant achievements are obtained.Chemostat model is one of the most significant models in Mathematical biology,using differential equations to describe the continuous culture of micro-organisms.It is an important device used for culturing micro-organisms in a continuous en-vironment, and a great intermedium between principles and applications. It hasbeen widely applied to the study of the increase in different populations of micro-organisms and their interactive law. In addition, it has been applied to the predic-tion and management of the marine ecosystem, and the control of the environmentalpollution.Base on the theories on nonlinear PDEs, especially those of reaction-diffusionequations and corresponding elliptic equations, we will study the following systemin this thesis. S_t=S_xx-auf(S)-bvg(S), x∈(0, 1), t＞0, u_t=u_xx+auf(S), x∈(0,1), t＞0, (1) v_t=v_xx+bvg(S), x∈(0, 1), t＞0,with boundary conditions S_x(0,t)=-S⁽⁰⁾, S_x(1, t)+γ^S(1, t)=0, t＞0, u_x(0,t)=0, u_x(1,t)+γ^u(1,t)=0, t＞0, (2) u_x(0,t)=0, u_x(1,t)+γ^v(1,t)=0, t＞0,and initial conditions S(x,0)= S₀(X)≥0,x∈(0,1), u(x,0)=u₀(x)≥0,(?)0, x∈(0,1), (3) v(x,0)=v₀(x)≥0,(?)0, x∈(0,1),where f(S)=S^l/(k₁+S^l); g(S)=S^r/(k₂+S^r).Here S(x,t), u(x,t), v(x,t) are the concentrations of the nutrient and two compet-ing populations respectively, a＞0, b＞0 are the maximal growth rates of thecompeting populations respectively, r, l, k₁, k₂,γare positive parameters. The mean results obtained in this thesis are summarized as following:In chapter one, by means of the maximum principle, methods of the super-subsolution and global bifurcation theory, starting with the semi-trivial solution whichhas only one zero-component, the significant conditions of existence for coexistencestate and the correspongding parameter regions for this system are established.Meanwhile, local stability for the coexistence states are obtained by the perturbationtheorem for linear operators and the stability theorem for bifurcation solutions.In chapter two, the asymptotic behavior of solutions to the systems (1)-(3) isdiscussed. Techniques include the comparison priciple, theory of uniform persis-tence in infinite-dimensional dynamical systems, and it is determined when bothcompeting populations survives.