Qualitative Study On Competitive Model Of Nonuniform Chemostat With C - M Functional Response Function

This article mainly focuses on a competition model in the unstirred chemostat with the C-M functional response. First of all, the significance of this issue is shown through introducing the concept of Chemostat and its application in the scientific research and industrial manufacturing as well as the categories of the mathemat-ic model of chemostat including three different functional response——Holling Ⅱ, Beddington-DeAngelis and Crowley-Martin——with their corresponding modeling research status. Later, we theoretically study this model in the chapter one to four, which is numerically improved in the fifth chapter.In the first chapter, the mathematic model concerned in this paper in the N-dimensional case is given out, so does its steady-state equation with some simplified work and a redefined functional response. Meanwhile, the positive property of the solution of a elliptic equation is proved by the strong maximal principle and Hopf lemma. And a eigenvalue problem and its properties are considered.In the second chapter,we consider the single species modeling. Firstly, a priori estimate of the nonnegative-nontrivial steady-state solution of this modeling is worked out by the positive property of the solution of a elliptic equation, which we get from the first chapter. Secondly, the existence and uniqueness of this solution are proved by the upper and lower solutions method and Guass-Green formula. Thirdly, its monotonous property is analyzed by using a series of theory such as the existence theorem of implicit function, generalized maximal principle, Sobolev embedding theorem, Lp estimate, the uniqueness theorem of linear ellipse equations.In the third chapter, we firstly give out a priori estimate of the nonnegative-nontrivial steady-state solution for the competition model. And then, we discuss the local existence of this solution by the theorem of local bifurcation. Finally, we get its stability from the perturbation theorem of linear operator.In the forth chapter, the global structure of this system is studied by using the theorem of global bifurcation.the existence and uniqueness of the global bifurca-tion solution, which bifurcates from one of the semi-trivial solution to another, is approved. This result indicates that the two species of microorganism can coexist when suitable parameters are selected.In the last chapter, first, we give out the competition model in the unstirred chemostat with the C-M functional response in one-dimensional case. And secondly, we deduce a difference scheme for one nonlinear parabolic system in one-dimensional case. Thirdly, applying the difference scheme, we get the numerical solution for the single species modeling by Matlab, meanwhile, the relationship between the parameters of the C-M functional response and the critical value of the maximal growth rates of microorganism when it can survive, which clearly illustrated in the following variation diagram by using Bisection Algorithm, is investigated. Similarly, we finally calculate the two species modeling to study the effect on the nonnegative-nontrivial steady-state solution came from the two maximal growth rates, further more, we draw a figure to show the coexistence region.