Stability and bifurcation of a reaction-diffusion-advection model with nonlinear boundary condition

Abstract

The dynamics of the reaction-diffusion-advection population models with linear boundary condition has been widely studied. This paper is devoted to the dynamics of a reaction-diffusion-advection population model with nonlinear boundary condition. Firstly, the stability of the trivial steady state is investigated by studying the corresponding eigenvalue problem. Secondly, the existence and stability of nontrivial steady states are proved by applying the Crandall-Rabinowitz bifurcation Theorem, the Lyapunov-Schmidt reduction method and perturbation method, in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible. The general results are applied to a parabolic equation with monostable nonlinear boundary condition, and to a parabolic equation with sublinear growth and superlinear boundary condition. Our theoretical results show that the nonlinear boundary condition can lead to the occurrence of various steady state bifurcations. Meanwhile, compared with the linear boundary condition, the nonlinear boundary condition can induce the multiplicity and growing-up property of positive steady-state solutions for the model with logistic interior growth. Finally, the numerical results show that the advection can change the bifurcation direction of some bifurcation, and affect the density distribution of the species.