Spatial Self-Organization in a Cyclic Resource-Species Model

Stephen M. Krone and Yongtao Guan


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.