TY - JOUR
T1 - Using moment equations to understand stochastically driven spatial pattern formation in ecological systems
AU - Bolker, Benjamin
AU - Pacala, Stephen Wilson
N1 - Funding Information:
We express our great appreciation to Simon Levin for suggesting the topic of moment closure in the first place, and for his subsequent guidance and useful suggestions in approaching this problem. Thanks go to Juan Lin for helpful conversations and for the suggestion of back-calculating parameters from the equilibrium distribution. In addition, we thank many other people who have helped with the development of these ideas, including Jonathan Dushoff, George Hurtt, Takuya Kubo, Mark Lewis, Paul Moorcroft, several anonymous reviewers, and Rick Durrett, whose comments helped improve the manuscript. We are pleased to acknowledge the support of NASA Grant NAGW-4688 to Stephen Pacala and Simon Levin.
PY - 1997/12
Y1 - 1997/12
N2 - Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present s new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean end spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by 'moment closure,' neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.
AB - Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present s new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean end spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by 'moment closure,' neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.
UR - http://www.scopus.com/inward/record.url?scp=0031408804&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0031408804&partnerID=8YFLogxK
U2 - 10.1006/tpbi.1997.1331
DO - 10.1006/tpbi.1997.1331
M3 - Article
C2 - 9466960
AN - SCOPUS:0031408804
SN - 0040-5809
VL - 52
SP - 179
EP - 197
JO - Theoretical Population Biology
JF - Theoretical Population Biology
IS - 3
ER -