Two Types Of Qualitative Analysis And Numerical Simulation Of The Biological Model

Researching biological phenomena by establishing biological models and using rich mathematics theories and methods now becomes an important aspect of modern science and technology development. A large number of biological models can be summarized as reaction diffusion equations, so it is a crucial research aspect in the region of partial differential equations for us to investigate these models by reaction diffusion equations.In the light of the current researches and applications of the theoretics of re-action diffusion systems, based on sonic pioneering works, using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of reaction diffusion equations and corresponding elliptic equations, we have system-atically studied two biological models:chemostat model and depletion model. By studying the coexistence, uniqueness, stability of positive steady states and the longtime behavior of species for the chemostat model, stability of positive constant steady states and existence of non-constant steady states for the depletion model, some valuable results are obtained. The tools used here include super-sub solutions method, comparison principle, local and global bifurcation theories, stability the-ory, fixed point index theory. regularity theorem, Lyapunov function, perturbation technique and numerical simulation.The structure and contains of this paper are as follows.In chapter 1, we introduce the background of models and some classical results of reaction diffusion systems, such as eigenvalue problems, fixed point index theory, bifurcation theories and so on. These are the basic parts that will be very useful in the forthcoming chapters.In chapter 2. an unstirred chemostat model with B-D functional response is studied. Firstly, we obtain the global structure of this system by global bifurcation theory. It turns out that the bifurcation curve bifurcating from one of the semi-trivial equilibria can finally meet the other at some point in certain condition. Secondly, by the means of comparison principle, regularity theorem and Lyapunov function. the asymptotic behavior of solutions of the system is investigated, and we obtain a sufficient condition of a global attractor for the limit of the system. Finally, the effect of the parameterβ1 in the B-D functional response which models mutual interference between speciesμis considered carefully by making use of the fixed point index theory and perturbation technique. The result shows that ifβ1 is sufficiently large, the solution of this model is determined by a limiting equation when the growth rate of the specialμlies in certain range. Especially, when the growth rates ofμ,τare suitable large, this model has a unique positive steady solution which is non-degenerate and linear stable.In chapter 3, an activator-substrate system with homogeneous Neumann bound-ary condition—a biological depletion model is discussed. For the case that no activator is supplied to the system, we mainly analyze the steady-state problem qualitatively and numerically. Firstly, we establish the fine apriori estimate for positive solutions and the non-existence of non-constant positive solution by the maximum principle and energy integral method. The result shows that the system has no any non-constant positive solution when the diffusion rate d of activator is large. Secondly, the stability of positive constant solutions is discussed in detail by means of stability theory. It turns out that there is Turing instable occurring when d is small. Thirdly, in the one dimensional case, we regard d as the bifurcation parameter to make a detailed description for the global bifurcation structure of the set of the non-constant solutions using bifurcation theory. The results indicate that if d is properly small, the bifurcation curve of the system from positive constant solutions finally reach infinity with respect toμ, and the system has at least one non-constant positive solution, which say that diffusion can create pattern forma-tion. Some results on extensive numerical studies are reported in the last confirming and complementing the previous results.In chapter 4, we continue to consider the depletion model, in which the activator is supplied at a constant rate. In this case, the constant steady states are complicated which leads to the difficulty of theoretical analysis. So we first establish the relation between constant steady states and parameters of the system in detail. Then, by a general discussion for the stability of constant steady states and their bifurcation, we get the stability and the bifurcation structure for the concrete constant steady state. Finally, the predictions from linear theory are confirmed through extensive numerical simulations.