Coexistence And Numerical Simulation Of Two Kinds Of Biological Models

In recent decades, all kinds of reaction-diffusion equations have attracted con-siderable attention by many biologists and mathematicians, especially to the preda-tor-prey models with different functional responses and boundary conditions. From the biological significance of reality, a key issue for a predator-prey model study is whether the various species can coexist. Therefore, the steady-state system of predator-prey model becomes the main object being studied. The chemostat is a laboratory apparatus used for the continuous culture of microorganisms. In micro-biology research, chemostat has been widely applied to waste treatment, microor-ganisms culture, biology pharmacy, sewage treatment, food processing, and control of environmental pollution. Therefore, the chemostat model has an important sig-nificance in reality.Based on the current researches of the predator-prey model and chemostat model, we have systematically investigated two kinds of mathematical model for bi-ological:predator-prey models (a one-prey and two-competing-predators predator-prey model、a predator-prey model with Crowley-Martin functional response) and an unstirred chemostat model. Research contents include the asymptotic behavior of the positive solutions to the parabolic system, the existence, uniqueness, stabil-ity and multiplicity of positive steady-state solutions. The tools used here include comparison principle, supper and sub-solution methods, perturbation theory, fixed points index theory, implicit function theorem, regularity theorem, bifurcation the-ory, Lyapunov-Schmidt reduction and numerical simulation.The main structure and contents of this paper are as follows:In chapter1, the background and research situation of predator-prey model and chemostat model are introduced. Some preliminaries that will be used in the forthcoming chapter are presented.In chapter2, a one-prey and two-competing-predators predator-prey model is studied. Firstly, the stability of the trivial solution and semi-trivial solution and the asymptotic behavior of positive solutions of parabolic system are investigated by means of the comparison principle. Secondly, by using the fixed point index theory, the sufficient conditions for the existence and non-existence of coexistence solutions are determined. In addition, the bifurcation from a double eigenvalue is discussed by virtue of space decomposition and implicit function theorem and the appropriate expressions of bifurcating positive solutions are given. The results show that when parameters c1and c2are small enough and e1and e2are large enough, the three species can coexist. Finally, by means of Matlab programming and finite-difference scheme, we make some numerical simulations to verify and complement the above theoretical analysis.In chapter3, a predator-prey model with Crowley-Martin functional response is considered. Firstly, we discuss the existence of positive solutions by means of the fixed point index theory. Secondly, we discuss the stability and uniqueness of positive solutions when b1is small. Thirdly, the stability and multiplicity of positive solutions are investigated when c1is large. The methods used include the pertur-bation theory, bifurcation theory and degree theory. It turns out that c1have an effect on the stability and exact multiplicity of positive solutions. Then, we inves-tigate the existence of the bifurcation solutions from a double eigenvalue by virtue of the implicit function theorem and space decomposition. In addition, we obtain the stability of the bifurcating positive solutions by virtue of perturbation theorem and Lyapunov-Schmidt reduction. Finally, we make some numerical simulations to verify and complement the theoretical analysis in this chapter.In chapter4, a food chain model in an unstirred chemostat is analyzed. By regarding b as bifurcation parameter and using the bifurcation theory, the local and global bifurcation from the semi-trivial solution are investigated, and the necessary and sufficient conditions for coexistence of the steady-state are obtained. Secondly, the stability of bifurcation solutions under some conditions is investigated by the perturbation theorem for linear operators and the stability theorem for bifurcation solutions. Thirdly, the stability and uniqueness of coexistence solutions are inves- tigated when m2is large by virtue of the perturbation theory and degree theory. The sufficient condition of the stability and uniqueness of coexistence solutions is given. The results indicate that if m2is sufficiently large, the system has a unique asymptotically coexistence solution as long as maximum growth rate b of the species v lies in certain range.