Properties Of Solutions To A Competition Model Between Microorganisms

The chemostat is a important model in microbial ecology. It is used as an ecological model of a simple lake, as a model of waste-treatment, and as a model for commercial production of fermentation processes. The chemostat is also used as a model for the manufacture of products by using genetically altered organisms. The alteration is accomplished by the introduction of DNA into the cell in the form of a plasmid. Unfortunately, the plasmid can be lost in the reproductive process. Ryder Dibiasio once deduced a model of plasmid-bearing and plasmid-free competition in chemostat. Comparing with the common chemostat model, this model has a parameter q(0 < q < 1) which is the probability that a plasmid is lost in the production.In this paper, basing on the current biological model and combining with the theories of reaction diffusion equations , we discuss a model of plasmid-bearing and plasmid-free competition in unstirred chemostat. Sufficient conditions and necessary conditions for the positive solutions of steady-state system, the stability and the asymptotic behavior of the solutions are obtained by the maximum principle, the monotone method, the degree theory, perturbation theorem, means of the comparison principle, Lyapunov function, numerical simulation and so on. There are two competitive microorganisms and a growth-limiting nutrient whose concentrations are denoted by u, v and S in the system which takes the form:The paper is made of four sections to investigate above-mentioned model.In chapter 1, the existence of positive steady-state solutions for system (1) is investigated. The necessary condition and the prior estimate for positive solutions of steady-state system are obtained by the maximum principle and the monotonemethod, then sufficient conditions for the positive solutions of steady-state system are determined by using the degree theory, the index of fixed point and bifurcation theory. It is also shown that the system generates the bifurcation from the semi-trivial solution (0,6) at the parameter a = j^j.In chapter 2, basing on the chapter l,we know that the positive solutions in the neighborhood of (^,0,0) exist with the bifurcation parameter a . Then the stability of the trivial solution,the semi-trivial solutions and the above-mentioned positive solution is established by the perturbation theorem, the eigenvalue theorem for linear operators and the stability theorem for bifurcation solution.In chapter 3, we obtain sufficient conditions under which a single organism will be survival or extinct in the given environment . Moreover the asymptotic behavior of the solutions of (1) is proved by means of the comparison principle, regularity theorem and Lyapunov function.In chapter 4, we study the one-dimension equations of (1) and its steady-state equations . First, the way to judge the sign of a — yf-,a — j^f:,b — H\d is given. Then we take some numerical simulation to complement and illustrate the foregoing ideas.