Analysis Of Coexistence For Two Models

It has been a hot discussion using reaction-diffusion equation solves problems in our daily life, such as microorganisms culture. Resently, many scholars and specialists have been paying more attention to Lengyel-Epstein model, chemo- stat model and flow reactor model, and many important and useful results have been gained. Depending on these methods and theories, the author studied the n- dimensional Lengyel-Epstein model and the flow reactor in time vary environment.The paper contained three chapters to investigate the coexistence for the two models.In chapter 1, the author introduced some concepts, such as sub-and-super so- lution,ωlimit set and etc, and stated several theories, which would be used in the proof of propositions, for example, local bifurcation theorem and global bifurcation theorem.In chapter 2, the coexistence of n-dimensional Lengyel-Epstein model was re- searched. This model takes the form as following whereâ–³is the Laplace operator, u = u(x, t), v = v(x, t) denote the chemical con- centrations of iodide and chlorite, respectively, a and b are the parameters related to the feed concentrations, c the ratio of the diffusion coefficients,σ＞1, a rescaling parameter depending on the concentration of the starch,Ωis a bounded domain in Rⁿ andΩis open with smooth boudry. In this chapter, the author shall assume accordingly the all constants a, b, c, and a are positive. The backgroud and the research situation about Lengyel-Epstein were introduced in the first section. In the second section, the author studied the local bifurcation of the constant steady states in n-dimensional Lengyel-Epstein model, by the means of local bifurcation theorem and introducing a basic hypothesis. Consequently, the sufficient conditons for the exitence of non-constant steady-state solutions were obtained. Using the version of the initial system and global bifurcation theorem, the existence of the global bifurcation was proved in the last section.In chapter 3, the author invesgated the coexistence of the flow reactor model in time varying environment. And this model takes the form as belowS_t= dS_xx-r(t)S_x-f₁(S)_u-f₂(S)_v, u_t=du_xx-r(t)u_x+f₁(S)_u, v_t=dv_xx-r(t)v_x+f₂(S)_v.Where S(t, x) denotes the concentration of the nutrient in the flow reactor, and u(t, x), v(t, x) denote the concentrations of the competitors, f_i(S), i=1, 2 satisfy the follow conditions: (a) f_i(S)∈C¹, i=1, 2, (b) f_i^′(S)＞0, (c) f_i(0)=0, and (?) f_i(S) exists. In the first section, the backgroud and the research situation about the flow reactor model were introduced. By the means of compare theorem and introducing a basic hypothesis, the initial system was reduced to a simple periodic-parabolic system about u, v, or the limiting system of the flow reactor in time varying environment was obtained, in the second section. At last, the author proved the existence of periodic solutions of the flow reactor in time varying environment, depending on compatitive ordering, periodic map and momotone dynamical system.