A time-periodic oscillatory hexagonal solution in a 2-dimensional integro-differential reaction-diffusion system

An oscillatory hexagonal solution in a two component reaction-di¤usion system with a non-local term is studied. By applying the center manifold theory, we obtain a four-dimensional dynamical system that informs us about the bifurcation structure around the trivial solution. Our results suggest that the oscillatory hexagonal solution can bifurcate from a stationary hexagonal solution via the Hopf bifurcation. This provides a reasonable explanation for the existence of the oscillatory hexagon.