A Qualitative Chemostat Competition Model Analysis

This paper deals with the n-dimensional reaction-diffusion system with the Ivlev type response function as follows: with the boundary conditions forWe study the following problems of the steady-state system of the above system, including the existence and stability of the coexistence solutions, the long time behavior of the system and the effects of parameters on the number and existence region of positive solutions.The main content of this paper is as follows:In chapter1, we study the global bifurcation of the steady-state system corre-sponding to the above system and the asymptotic behavior of the system. First, a priori estimates of coexistence solutions are established by the maximum principle and eigenvalue theory. On this basis, the fact that the local solution branch can be extended to a global solution branch is proved by using fixed point index theory and bifurcation theory. Moreover, the global solution branch joins up with the two semi-trivial branches of the solutions of the model. Second, we analyze the stability of the coexistence solutions close to the bifurcation point. Finally, we discuss the long time behavior of the system.In chapter2, we study the existence region of positive solutions of the steady-state system. First, it is shown that if a≠λ1and b≠σ1, then the necessary and sufficient conditions are a> r1(a, b) and b> r2(a, b) when the system possesses positive solutions by using the fixed point theory and the upper and lower solution method. Then, combining with the monotone method and the fixed point theory, it is proved that A is a connected unbounded region in R2, whose boundary consists of two monotone nondecreasing curves T1:a=F1(b) and F2:b=F2(a). Finally, it is shown that the system has at least two positive solutions in certain subregion of A.