Properties Of Positive Steady-state Solutions To Two Kinds Of Biological Model

A chemostat often called a continuous culture, is a important biomathematics model, a piece of laboratory apparatus used for the continuous culture of microorganisms. It is used an ecological model of a simple lake, as a model of a simple lake, as a model of the growth of unicelluar phytoplankton in lake and sea. Moreover, it has been widely applied to the commercial pruduction of microorganisms, waste treatment, biological pharmacy, and processing and the management and prediction of the ecology system, particulary the marine ecology, and the control of the environment pollution. The nutrient is pumped continuously at a constant rate into a culture vessel whose volume is kept constant by pumping the mixture of the nutrient, metabolites and microorganisms out at the same rate. The input and output of the nutrient in the chemostat simulate approximately the continuous metabolism in nature and outputs of microorganisms correspond to species migration or unnatural death which ofen happan in nature. So we can get our expected goals by controlling some microorganisms concentration or adjusting some parameters in the system.In chapter 1, an unstirred chemostat model with inhibitior is investigated. In this model, one species released inhibitor by withering oneself inhibits the other species. The chemostat model can be depicted mathematically as following:S"-af₁(S)u-bf₂(S)v=0, x∈(0, 1),u"+af₁(S)u-βpu=0, x∈(0, 1),v"+b(1-k)f₂(S)v=0, x∈(0, 1),p"+bkf₂(S)v=0, x∈(0, 1),with boundary conditionsS'(0)=-1, S'(1)+γS(1)=0,u'(0)=0, u'(1)+γu(1)=0,v'(0)=0, v'(1)+γv(1)=0,p'(0)=0, p'(1)+γp(1)=0.Here f_i(S)=S/(a_i+s)(i=1, 2)is Monod functional response. S(x) is the concentration of the nutrent, u(x) is the concentration of the inhibited spaces, v(x)is the concentration of the spces who rleases the inhibitor, p(x) is the concentration of the inhibitor, a and b are the maximum growth rates of u and v.β＞0, k∈[0, 1). The existence of the global bifurcation solutions can be determined by the bifurcation theory, respectly treated the growth rates of a and b as bifurcation parameters. The conditional stability of the bifurcation solution is determined by the stability theory.In chapter 2, the local bifurcation and stability of positive steady-state solutins for a class of predator-prey model are investigated. The steady-state system takes the following formâ–³u+au-u²-((a₁v)/(u+k₁))u=0, x∈Ω,â–³v+bv-((a₂)/(u+k₂))v²=0, x∈Ω,u=v=0,x∈(?)Ω.WhereΩis bounded domain in R^N wtih sufficiently soomth boundary (?)Ω. u and v are the density of the prey and predator inΩ. The parameters a, a₁, a₂, b, k₁, k₂ are the positive constants, a and b is the maximum rates of u and v. k₁, k₂ are the protecting extent of the prey u and predator v in the entironment. We have the bifurcation solutions and stabitily.