## Spatial Self-Organization in a Cyclic Resource-Species Model

*Stephen M. Krone and Yongtao Guan*
**Abstract**

We introduce a lattice-based stochastic spatial model (interacting
particle system) with cyclic local dynamics. In this model we seek
to capture the essential features of a spatially extended
biological system for which a given site alternates its state
between resources and species in a prescribed order. Furthermore,
this succession of states (at a given site) is assumed to form a
cyclic pattern due to a natural feedback mechanism. As a
motivating example, we describe a situation in which a number of
microbial species are involved in the successive degradation of a
resource in such a way that the last species in the sequence
provides catalytic support for the primary degrader.
Mathematically, a key feature of this class of models is that the
transitions between different states at a given site alternate
between contact and spontaneous updating. We explore conditions
under which all the species are able to coexist and the extent to
which this coexistence requires the development of spatio-temporal
patterns, including spiral waves. This self-organization, if it
occurs, results when synchronization of the dynamics at the
microscopic level lead to macroscopic patterns. These patterns
result in consumer-driven resource fluctuations that generate a
form of spatio-temporal niche partitioning. As with most models of
this complexity, we employ a mixture of mathematical analysis and
simulations to develop an understanding of the resulting dynamics.