The Existence Of Positive Steady-state Solutions To Two Kinds Of Biological Models

Partial differential equations (PDE) are often used to describe, explain and foresee all kinds of biological phenomena. Many scholars and specialists have been paying more attention to Chemostat model and Volterra-Lotka model, and have gained a good many important and useful results. The whole thesis is made up of three chapters, the existence and stability of steady-state solutions to two kinds of biological models are investigated: one is an un-stirred Chemostat model with inhibitor, the other is a cooperative system with saturation.In chapter 1, the existence of positive steady-state solutions for an un-stirred Chemostat with inhibitor is investigated. On the assumption of un-stirred, by reducing the dimension of system, the steady-state system takes the following form.here, u v are two competitive microorganic species. The necessary condition and the prior bound for positive solutions of steady-state system are obtained by the maximum principle and monotone method. Further, a sufficient condition for the coexistence of steady states is determined by using degree theories, calculating the index of fixed points and combining with bifurcation theories and spectrum analysis of operators. Moreover, some corresponding numerical simulations are given.In chapter 2, we characterize a set A of points (a,b) in R₊² so that for every (a, b) inside A, the system (I) has at least one positive solution and for (a, b) outside A there are only trivial and semi-trivial solutions. It is shown that A is a connected unbounded region in R₊² whose boundary consists of two monotone nondecreasing curveshere, the functions H₁(b) and H₂(a) are constructed in terms of the limit of certain...