Abstract: In this paper, we consider a diffusion predator-prey model with nonlinear boundary conditions, which has more extensive application areas than the corresponding model with linear boundary conditions. We first show that all eigenvalues of a class of eigenvalue problems are positive by using Green's identity. The existence of positive solutions is also established by using the bifurcation theory. Moreover, we discuss the asymptotic stability of bifurcation solutions with the help of the perturbation theory. Finally, we give some examples of numerical simulations by Matlab to support the analytical results.

Key words: predator-prey, nonlinear boundary conditions, bifurcation, stability