The Existence Of Positive Steady-state Solutions To Two Kinds Of Biological Models With Inhibitor

We are interested in soluting biological problems by using partial differential equations. Many scholars and specialists have been paying more attention to chemostat model and have gained many important and useful results. The whole thesis is made up of two chapters, the existence of steady-state solutions to two kinds of biological models is investigated: one is an un-stirred chemostat with an external inhibitor, the other is plasmid-bearing and plasmid-free organisms in a chemostat with an internal inhibitor.In chapter 1, the existence of positive steady-state solutions for an un-stirred chemostat with an external inhibitor is investigated. There are two competitive species u, v in the system. We assume that the two organisms competing for a nutrient in the presence of an inhibitor. The inhibitor p is from an external source but is lethal to v, and not to u which can take it up without harm. We suppose the same diffusion coefficient d. The variables are scaled to non-dimensional ones and the equation takes the form:Here the parameters m₁,m₂ have real biological meanings respectively. In the case where the inhibitor p acts on the growth rate of v with a degree of inhibiton f(p), sometimes f(p) = e^－μp, and f₃ = δp/(K+p), δ being uptaken by u and K being a Michaels-Menten parameter. The necessary condition and a priori estimate for positive solutions of steady-state system are obtained by the maximum principle and monotone method. Further, a sufficient condition for the coexistence of steady state is determined by using the degree theory and calculating the index of fixed points.In chapter 2, we analyze the model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an internal inhibitor, the system...