Rigorous results in selection of steady needle crystals

This paper is concerned with a cross-diffusion system arising in a prey–predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steadystate solution branch under some conditions.