Coexistence And Stability Of Several Biochemistry Models

A Chiness proverb is "People goes to high place, water flows to low place", which try to discover the laws of convergence. In a assemblage of particles, for example, the individuals can be very small such as basic particles in physics, bacteria, molecules, cell and so on, and very large objects such that as animals, plants, epidemics, rumors and so on, which have a common law:In micro presentation, the particles spread out as a result of this irregular individual particle's motion; In macro presentation, it is a nature phenomenon that a substance goes from high density regions to low density region, we can think of it as a diffusion process. On the other hand, the number of particles may change because of other reasons like birth, death, hunting, or chemical reactions, e.ct. To unravel the underlying mechanisms involved in the biological or chemical process, A large class of reaction diffusion equation and system are arose. The idea of modelling the dynamics of interactions with a system of nonlinear differential equations dates bake at least to the pioneering work of Lotka and Volterra in the 1920s. The idea of using diffusion to model the spatial dispersal of population genetics was introduced by Fisher the 1930s and applied to population dynamics by Skellam and others in the early 1950s. Such a mechanism was proposed as a model for chemical basis of morphogenesis by Turing in 1952, and such systems have been widely studied since about 1970.In the light of the recent work in these several kinds of biological models and chemical models, mainly using the theories of nonlinear analysis and nonlinear par-tial differential equations, especially those of reaction-diffusion equations and corre-sponding elliptic equations, we have systematically studied the dynamical behavior of the autocatalytic reaction diffusion model and Lotka-Volterra model with non-monotonic conversion rate, such as coexistence, multiplicity, stability of positive steady states and the longtime behavior of species. The tools used here include super-sub solutions method, comparison principle, monotone system theory, global bifurcation theory, fixed-point theory of topology, Lyapunov function technique. The main contents and results in this dissertation are as follows:In chapter 1, the general autocatalytic reaction models are deduced, both the foreign and domestic status of research on the autocatalytic reaction diffusion model and Lotka-Volterra model are introduced and analyzed in detail, we list some ba- sic theory and classic results of reaction diffusion systems, such as the maximum principle, the fixed points index theory, bifurcation theory and so on.In chapter 2, an autocatalytic reaction-diffusion system is investigated. We consider the coexistence states of the system by the degree and bifurcation methods, under the boundary conditions of Dirichlet type, it turns out that if the parameter c (the reaction rate) is properly small, then the system has no coexistence state, if the parameter c is suitably large, then the system has at least two coexistence states, and if the parameter c is sufficiently large, then the system has at least one coexistence state. Moreover, the bifurcation direction and global bifurcation are determined.In chapter 3, a tri-molecular autocatalytic reaction-diffusion system with differ-ent boundary conditions is investigated. Under the boundary conditions of Dirichlet type or Robin type, the positive steady-state solutions is established by the fixed points index theory and local bifurcation theory, the stability of bifurcation solution is determined by the linearlization thoery, the bifurcation direction and global bifur-cation are studied, the uniqueness of the positive steady-state solutions in interval and parameter large enough, under the boundary conditions of Neumann type, the global stability of the positive constant equilibrium is obtained by means of Lya-punov function. The major difficulties of this section are the proof of uniqueness.In chapter 4, a reaction diffusion system arising from Schnakenberg chemical reaction is studied. In the one dimensional and Neumann boundary conditions case, Hopf and Turing bifurcation analyses are carried out in details, examples of numerical simulation are also shown to support our approach, which implies that the reaction is a rich dynamical system. The trait of the technique is that we obtained the algorithm to determine the bifurcation director and bifurcating from a double eigenvalue.In chapter 5, a reaction diffusion system arising from a ratio-dependent predator-prey model with stage structure is investigated. The local and global asymptotic stability of the constant equilibrium is obtained under suitable conditions by by the method of linearization and Lyapunov functions, base on priori estimate and lin-earization at positive equilibrium, existence and non-existence results of non-trivial solution are derived by means of energy method and the fixed points index theory. Emphasis is the boundedness of positive solution and the topological theory. In chapter 6, a two-competitor/one-prey reaction diffusion system with gen-cral functional response is investigated. As an example, we consider a model with a Holling II functional response. The local and global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions by the method of linearization and Lyapunov functions, non-existence results of non-trivial solution are derived by the energy method and implicit function theory, and exis-tence positive steady state is derived by the fixed points index theory. The major difficulties are the application of the implicit function theory and the construction of the homogeneous function.