Abstract: We will discuss some examples of interacting particle systems
with cyclic local dynamics and explore conditions that lead to self-organized
spatial patterns such as spiral waves. The cyclic dynamics in these stochastic
spatial models occur when a typical site progresses through a succession of
states in a prescribed order and a feedback mechanism completes the cycle.
Much of the talk will focus on a model that captures aspects of bacterial
cross-feeding (cooperative degradation of a resource) and feedback through
catalytic support for the primary degrader. We will show that, when the number
of states is large enough, coexistence requires a local synchronization of the
dynamics that leads to large-scale spatio-temporal patterns.