Properties Of Solutions To Two Kinds Of Unstirred Chemostat Model

Since partial differential equations (PDE) were used to describe biological regulations and phenomena, many scholars and specialists have been paying more attention to PDE and many new subjects have been established which have more reality backgrounds, one of which is chemostat model.In this paper, two kinds of unstirred chemostat models are discussed. One is an unstirred chemostat model with two nutrients. There are two growth-limiting nutrients and a microorganism depending nutrients whose concentrations are denoted by S, R and u. The system takes the form:S_t = d₀S_xx-muf(S,R)R_t = d₁R_xx - nug{S, R), 0 < x < l,t > 0 (1)u_t = d₂u_xx + u(mf(S, R) + cng(S, R) - k)Heref (S, R) = S/{1 + aS + bR), g(S, R) = R/(l + aS + bR)are the response functions in (1). They are similar to the well-known Michaeli-Menten response function and a generalization of the Michaelis-Menten function. The parameters a, b are positive constants.The other is a competitive model with a single nutrient in the chemostat. There are two competitive microorganisms and a growth-limiting nutrient whose concentrations are denoted by u₁, u₂ and S. The system takes the form:S_t = d₀ΔS-u₁f₁(S)-u₂f₂(S), x∈Ω,t>0u_1t = d₁Δu₁ + u₁(f₁(S) - k₁), x∈Ω,t>0 (2)u_2t = d₂Δ u₂ + u₂(f₂(S) -k₂), x∈Ω,t>0where d₀ is the diffusive coefficient for the nutrient 5, d₁, d₂ are the random motility coefficient of microbial populations u₁,u₂ with death rate k₁ and k₂, respectively, f₁, f₂ are the growth rates of u₁, u₂, respectively.The thesis is made up of three chapters and the properties of solutions to two kinds of unstirred chemostat models are investigated.In chapter 1, the existence of positive periodic solutions for (1) with same diffusive coefficient is investigated. By using the maximum principle, the monotonemethod and theory of periodic parabolic operators, a sufficient condition for coexistence of positive periodic solutions is obtained.In chapter 2, we focus on the steady-state of (1). The prior bound for positive solutions of steady state system is obtained by the maximum principle and the monotone method. Further, we discuss the global structure of the coexistence solutions to the system by using bifurcation theory. The sufficient conditions for the coexistence solutions of the chemostat model are established. The local stability for the coexistence solutions is obtained by the perturbation theorem for linear operators and the stability theorem for bifurcation solutions.In chapter 3, we discuss the coexistence of positive periodic solutions for (2). By means of comparison theorems for parabolic equation and stability theory, the existence and stability of semi-trivial periodic solutions to the system are discussed. Meanwhile, a sufficient condition for the coexistence of positive periodic solutions for (2) is obtained by the theory of Leray-Schauder degree.