Analysis On Stability And Existence Of Bifurcation Solutions Of A Biological Model

Various mathematical models that imitate some kinds of life process have been established by applying certain mathematical tools. So it is significant to study these models. In this thesis, two biological systems are investigated. We fulfil the purpose of investigating the biological phenomenon by reasoning, seeking solutions and calculating.In the first chapter, we discuss an unstirring chemostat model of competition with internal inhibitor. There are two competitive microorganic species u, v, and a nutrients in the system, u produces inhibitor-----p, and v is inhibited by p. The parameter krepresents the fraction of the consumption devoted to producing the inhibitor. We suppose the same diffusion coefficient d. The variables are scaled to non-dimensional ones and the equation takes the formwith boundary conditionsand initial conditionsHere the parameters d, o, 6, ai (i = 1. 2), u have real biological meanings respectively. According the biological meanings of this model, we are concerned with whether microorganisms can survive, and whether several species can coexist or not in the reactor whencertain parameters change. We take the maximum growth rate of species v as bifurcation parameter. By applying maximum principle, the monotone method and bifurcation from a simple eigenvalue we discuss the system of equilibrium. It is proved that the coexistent solution can exist and be stable under certain conditions. The effect of inhibitior on the model has been shown in the analytic process.In chapter 2, we analyze the conditions under which a reaction-diffuse equation with n species n (n > 3) is diffusion-driven instability. Theoretical studies in reaction-diffusion theory mainly focus on the analysis of systems composed of only two species, and good results have been achieved. However, in fact, chemical and biochemical reactions usually involve more than two dynamically independent species. In this thesis, we analyze the zeroth order term of eigenpolynomial of linearized matrix, and a necessary condition for diffusion-driven instability of n-species of a reaction-diffusion system is attained by using the Routh-Hurwitz criterion.