A Class Of Economator Model With Congestion Effect

The chemostat models are the basic mathematical models, which describe the competition of species in the open ecosystem and laboratory bioreactors. This paper deals with the following unstirred chemostat model with crowding effects: with boundary conditions and initial conditionsS(x,0)=S0(x)≥0, u(x,0)=u0(x)≥0,≠0, v(x,0)=v0(x)≥0,≠0.The introduction of crowding effects makes the law of conservation of biomass invalid. We study the following problems of the steady-state system of the above system, including the existence and uniqueness of the semi-trivial solution, the exis-tence and stability of coexistence solutions, and the crowding effects in the unstirred chemostat, by means of the generalized maximum principle, the degree theory and the bifurcation theory. The main contents of this paper are as follows:In chapter 1, the biological background and recent work on this model are described in details, and some preliminaries which are very useful in the forthcoming chapters are given.In chapter 2, we consider the existence and uniqueness of the semi-trivial so-lution, and the existence of coexistence solutions by the degree theory. First of all, with the help of the generalized maximum principle and fixed point index theory, we obtain the existence and uniqueness of semi-trivial solution. At the same time, some elementary properties of semi-trivial solutions are given. Finally, we get the existence of positive solutions by the degree theory.In chapter 3, we investigate the bifurcation structure of coexistence solutions, and analyze the crowding effects on asymptotic behavior of coexistence solutions in the unstirred chemostat. First, we take b as the bifurcation parameter to construct a local and global branch that bifurcates from the semi-trivial branch. It turns out that the bifurcation curve connects one semi-trivial solution with another. Second, we carry out the existence and stability of bifurcation solutions that bifurcates from a double eigenvalue. It is shown that there exist stable positive solutions near the trivial solution. Finally, we discuss the crowding effects on asymptotic behavior of coexistence solutions in the unstirred chemostat.