Bifurcation and pattern formation in homogeneous diffusive predator-prey
AbstractIn this paper, we shall consider the existence of nonconstant solutions
of a preadtor-prey model under homogeneous Neumann boundary conditions.
In particular, the existence of the global bifurcations, the Hopf
bifurcations and the steady state bifurcations of system are
established, and the bifurcation direction of the bifurcating periodic
solutions and the steady state bifurcations of system are investigated.
Moreover, the stability of the Hopf bifurcation is also studied, and we
show that the spatially homogeneous periodic solutions are
asymptotically stable, while the spatially nonhomogeneous periodic
solutions are unstable. Furthermore, the effect of cross-diffusion for
existence of nonconstant solutions of system is studied, and we also
show that cross-diffusion is not always helpful to create Turing
instability. Our mathematical approach is based on the Leray-Schauder
degree theorem, the bifurcation theorem, the technique of space
decomposition and the implicit function theorem.