Jianhua Wu

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.

Coexistence states for cooperative model with diffusion

Abstract In this paper, we are mainly concerned with a cooperative system with a saturating interaction term for one species. Existence of coexistence states is investigated by global bifurcation theory, and exact results on regions in parameter space which have nontrivial nonnegative steady state solutions are given. The stability of coexistence states is also studied.

Modeling and analysis of a periodic delayed two-species model of facultative mutualism

In this paper, we propose a periodic delayed two-species system modeling facultative mutualism. By using the method of coincidence degree and Lyapunov functional, easily verifiable sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions of the above system.

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.

Uniqueness and stability of a predator-prey model with C-M functional response

In this paper, we are concerned with positive solutions of a predator-prey model with Crowley-Martin functional response under Dirichlet boundary conditions. First, we establish the existence of positive solutions when ( a , c ) is close to ( λ 1 , λ 1 ) . Next, the stability properties of nonnegative solutions are discussed by spectral analysis. In addition, we obtain a complete understanding of the uniqueness and non-uniqueness of positive solutions by the Lyapunov-Schmidt procedure and the perturbation technique. At last, some numerical simulations are presented to supplement the analytic results in one dimension.

Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model

We consider a diffusive Leslie-Gower predator-prey model subject to the homogeneous Neumann boundary condition. Treating the diffusion coefficient d as a parameter, the Hopf bifurcation and steady-state bifurcation from the positive constant solution branch are investigated. Moreover, the global structure of the steady-state bifurcations from simple eigenvalues is established by bifurcation theory. In particular, the local structure of the steady-state bifurcations from double eigenvalues is also obtained by the techniques of space decomposition and implicit function theorem.

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.

Positive solutions of a Lotka–Volterra competition model with cross-diffusion☆

Abstract This paper concerns a Lotka–Volterra competition reaction–diffusion system with nonlinear diffusion effects. We first briefly discuss the stability of trivial solutions by spectrum analysis. Based on the boundedness of the solutions, the existence of the positive steady state solution is also investigated by the monotone iteration method.

Bifurcation from a double eigenvalue in the unstirred chemostat

This article concerns a competition model in the unstirred chemostat. The bifurcation solution from a double eigenvalue is obtained. We see that this bifurcation solution connects the positive solution from the semitrivial solution (θ a , 0) with that from the other semitrivial solution (0, θ b ). Moreover, the asymptotic stability of the positive solution corresponding to this bifurcation is derived under certain conditions. The method we used here is based on spectral analysis, comparison principle, bifurcation theory and Lyapunov–Schmidt procedure.

Delay dynamic equations on time scales

This article is concerned with delay dynamic equations on time scales. Linear and nonlinear delay dynamic equations are discussed. By using a different theorem to that used S. Hilger, Analysis on measure chains. A unified approach to continous and discrete calculus. Results Math. 18 (1990), pp. 18–56], some criteria for the existence, uniqueness and continuous dependence of the solution for the nonlinear delay dynamic equations are established.