Shanbing Li

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.

Qualitative Analysis of a Predator–Prey Model with Crowley–Martin Functional Response

In this paper, we are concerned with positive solutions of a predator–prey model with Crowley–Martin functional response under homogeneous Dirichlet boundary conditions. First, we prove the existence and reveal the structure of the positive solutions by using bifurcation theory. Then, we investigate the uniqueness and stability of the positive solutions for a large key parameter. In addition, we derive some sufficient conditions for the uniqueness of the positive solutions by using some specific inequalities. Moreover, we discuss the extinction and persistence results of time-dependent positive solutions to the system. Finally, we present some numerical simulations to supplement the analytic results in one dimension.

Invariant sets and solutions to a class of wave equations

In this paper, we investigate the invariant sets and exact solutions of the (1+2)-dimensional wave equations. It is proven that there exists a class of solutions to the equations, which belong to the invariant set E 0 = { u : u x = v x F ( u ) , u y = v y F ( u ) } .