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Journal of Statistical Physics, Vol. 114, Nos. 5/6, March 2004 (© 2004)
Invasion and Extinction in the Mean Field
Approximation for a Spatial
Host-Pathogen Model
M. A. M. de Aguiar, 1, 2 E. M. Rauch, 1, 3 and Y. Bar-Yam 1, 4
Received February 4, 2003; October 2, 2003
We derive the mean field equations of a simple spatial host-pathogen, or preda-
tor-prey, model that has been shown to display interesting evolutionary proper-
ties. We compare these equations, and the equations including pair-correlations,
with the low-density approximations derived by other authors. We study the
process of invasion by a mutant pathogen, both in the mean field and in the pair
approximation, and discuss our results with respect to the spatial model. Both
the mean field and pair correlation approximations do not capture the key
spatial behaviors—the moderation of exploitation due to local extinctions, pre-
venting the pathogen from causing its own extinction. However, the results
provide important hints about the mechanism by which the local extinctions
occur.
KEY WORDS: Evolution; predator-prey; host-pathogen; mean-field; spatial.
1. INTRODUCTION
The importance of space in ecology, evolution and epidemiology has
become increasingly recognized in recent research. It has become apparent
that inhomogeneities in spatially distributed populations can fundamentally
change the dynamics of these systems.(1–5) In particular, spatial symmetry
breaking and the resulting patterns of inhomogeneity lead to dynamic
1 New England Complex Systems Institute, Cambridge, Massachusetts 02138.
2 Instituto de Física Gleb Wataghin, Universidade Estadual de Campin
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