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Modified mean-field models and self-organisation in spatial systems for antagonistic interactions

Presented by: 
M Pascual [Michigan]
Friday 14th December 2001 - 10:00 to 11:00
INI Seminar Room 1
Session Title: 
Immunology, Ecology and Epidemiology (main themes)
A simple modification of the mean-field equations provides an accurate approximation of the macroscopic dynamics of predator and prey densities in an individual-based model that is both stochastic and nonlinear. This approach modifies the parameters but preserves the functional forms of the mean-field equations in spite of the elaborate spatial patterns present in the system. The spatial patterns reduce the per-capita rates of predation and prey growth but do so in a way that is both simple and specific. In particular, these rates become those one would expect in a well-mixed system but with a smaller `effective neighborhood' of interaction. A connection is described between the proposed modified mean-field equations and the geometry of the system. The cluster geometry is characterized by power-law scalings that appear robust to changes in the parameters and in the microscopic rules specifying growth and inhibition. Implications for models of macroscopic dynamics when microscopic behaviour is not well known are discussed.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons