Stability Of Positive Steady-state Solution Research Articles

Dynamics of a Reaction–Diffusion–Advection System with Nonlinear Boundary Conditions

In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.

Stability and bifurcation in a single species with nonlinear boundary conditions

A single-species model subject to nonlinear boundary conditions is investigated. Existence, nonexistence, and local/global asymptotical stability of positive steady-state solutions are shown to identify the ranges of parameters, the growth rate functions, and boundary reaction functions. In particular, we show that the model can have multiple equilibria and in some parameter ranges can support Allee effects.

Hopf bifurcation in a delayed reaction\u2013diffusion\u2013advection equation with ideal free dispersal

In this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results.

Existence and stability of positive steady-state solutions for a Lotka-Volterrasystem with intraspecific competition

In this paper, we investigate the existence and stability of the positivesteady-state solutions for a Lotka-Volterra system with intraspecific competition byusing the Lyapunov-Schmidt reduction technique. To do this, we must firstly obtainthe semi-trivial steady states as their base, which extend the method used in theprevious studies. Our results show that the two competition species withintraspecific competition can coexist for bigger regions of the diffusion rateμ and also complete the existing works. MSC: 35K57.

Multiplicity and stability of a predator–prey model with non-monotonic conversion rate

This paper is concerned with a predator–prey model possessing a non-monotonic conversion rate. The main purpose is to determine the multiple existence and stability of positive steady-state solutions to this system. The results show that if the parameter d is suitably large, then the system contains an S -shaped global bifurcation curve with respect to a bifurcation parameter. That is, the system has two or three positive solutions for a suitable range of parameters. Moreover, the stability of positive solutions on this curve is also given. If d is properly small, both uniqueness and non-uniqueness results can occur. The main tools used here include the bifurcation theory, the Lyapunov–Schmidt procedure, and the perturbation technique.

Bounds for the solution of a non-linear parabolic initial-boundary value problem

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form