Stochastic Spatial Models: A User's Guide to Ecological Applications

Spatial pattern, how it arises and how it is maintained, are central foci for ecological theory. In recent years, some attention has shifted from continuum models to spatially discrete analogues, which allow easy treatment of local stochastic effects and of non-local spatial influences. Many of these fall within the area of mathematics known as `interacting particle systems9, which provides a body of results that facilitate the interpretation of the suite of simulation models that have been considered, and point towards future analyses. In this paper we review the basic mathematical literature. Three influential examples from the ecological literature are considered and placed within the general framework, which is shown to be a powerful one for the study of spatial ecological interactions.

[1]  Richard C. Brower,et al.  Critical Exponents for the Reggeon Quantum Spin Model , 1978 .

[2]  Hal Caswell,et al.  Predator-Mediated Coexistence: A Nonequilibrium Model , 1978, The American Naturalist.

[3]  Rick Durrett,et al.  On the Growth of One Dimensional Contact Processes , 1980 .

[4]  Maury Bramson,et al.  Asymptotics for interacting particle systems onZd , 1980 .

[5]  Peter Donnelly,et al.  The transient behaviour of the Moran model in population genetics , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Maury Bramson,et al.  Consolidation rates for two interacting systems in the plane , 1986 .

[7]  J. T. Cox,et al.  Diffusive Clustering in the Two Dimensional Voter Model , 1986 .

[8]  R. May,et al.  Population dynamics and plant community structure: Competition between annuals and perrenials , 1987 .

[9]  David Griffeath,et al.  Recent Results for the Stepping Stone Model , 1987 .

[10]  Rick Durrett,et al.  Crabgrass, measles, and gypsy moths: An introduction to interacting particle systems , 1988 .

[11]  R. Durrett.  Lecture notes on particle systems and percolation , 1988 .

[12]  Maury Bramson,et al.  Statistical Mechanics of Crabgrass , 1989 .

[13]  J. T. Cox,et al.  Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$ , 1989 .

[14]  J. T. Cox,et al.  Mean field asymptotics for the planar stepping stone model , 1990 .

[15]  Kenneth S. Alexander,et al.  Spatial Stochastic Processes , 1991 .

[16]  Geoffrey Grimmett,et al.  Exponential decay for subcritical contact and percolation processes , 1991 .

[17]  Maury Bramson,et al.  A Useful Renormalization Argument , 1991 .

[18]  Hwachou Chen,et al.  On the Stability of a Population Growth Model with Sexual Reproduction on $Z^d, d \geq 2$ , 1992 .

[19]  T. Czárán,et al.  Spatiotemporal dynamic models of plant populations and communities. , 1992, Trends in ecology & evolution.

[20]  Lawrence Gray,et al.  Critical Attractive Spin Systems , 1994 .