Hua Nie

A SYSTEM OF REACTION–DIFFUSION EQUATIONS IN THE UNSTIRRED CHEMOSTAT WITH AN INHIBITOR

A system of reaction–diffusion equations is considered in the unstirred chemostat with an inhibitor. Global structure of the coexistence solutions and their local stability are established. The asymptotic behavior of the system is given as a function of the parameters, and it is determined when neither, one, or both competing populations survive. Finally, the results of some numerical simulations indicate that the global stability of the steady-state solutions is possible. The main tools for our investigations are the maximum principle, monotone method and global bifurcation theory.

Uniqueness and stability for coexistence solutions of the unstirred chemostat model

This article deals with the uniqueness and stability of coexistence solutions of a basic N-dimensional competition model in the unstirred chemostat by Lyapunov–Schmidt procedure and perturbation technique. It turns out that if the parameter G ≠ 0, which is given in Theorem 1.1, this model has a unique coexistence solution provided that the maximal growth rates a, b of u, v, respectively, lie in a certain range. Moreover, the unique coexistence solution is globally asymptotically stable if G > 0, while it is unstable if G < 0. In the later case, the semitrivial equilibria are both stable.

Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin

We investigate the effects of toxins on the multiple coexistence solutions of an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. It turns out that coexistence solutions to this model are governed by two limiting systems. Based on the analysis of uniqueness and stability of positive solutions to two limiting systems, the exact multiplicity and stability of coexistence solutions of this model are established by means of the combination of the fixed-point index theory, bifurcation theory and perturbation theory.

Algal competition in a water column with excessive dioxide in the atmosphere

This paper deals with a resource competition model of two algal species in a water column with excessive dioxide in the atmosphere. First, the uniqueness of positive steady state solutions to the single-species model with two resources is established by the application of the degree theory and the strong maximum principle for the cooperative system. Second, some asymptotic behavior of the single-species model is given by comparison principle and uniform persistence theory. Third, the coexistence solutions to the competition system of two species with two substitutable resources are obtained by global bifurcation theory, various estimates and the strong maximum principle for the cooperative system. Numerical simulations are used to illustrate the outcomes of coexistence and competitive exclusion.

Coexistence states of a three-species cooperating model with diffusion

In this article, a Lotka--Volterra three-species time-periodic mutualism model with diffusion is investigated. Some sufficient conditions for the existence and estimates of coexistence states are established. Meanwhile, with the assistance of functional analysis methods, some sufficient or necessary results for the existence of positive steady state of the model are presented. Our approach to the discussion is mainly based on the skill of sub- and super-solutions for a general reaction--diffusion system.

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.