It is well known that if reacting species experience unequal diffusion rates, then dynamics that lead to a constant steady state in a "well-mixed" environment can in a spatial setting lead to interesting patterns. In this paper, we focus on complementary pattern formation mechanisms that operate even when the diffusion rates are equal. In particular, we can say that when the mean-field ODE has an attracting periodic orbit then the stochastic spatial model will have large-scale spatial structures in equilibrium. We explore this mechanism in depth through the dynamics of the simulator WATOR.