Interplay between Turing Pattern Formation and Domain Coarsening in Spatially Extended Population Models

Hiroki Sayama^1,2*, Marcus A. M. de Aguiar^1,3, Yaneer Bar-Yam^1,4 and Michel Baranger^1,5

¹New England Complex Systems Institute, Cambridge, MA 02138, U.S.A.
²Department of Human Communication, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
³Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas,13081-970, Campinas, S‹o Paulo, Brazil
⁴Department of Molecular and Cellular Biology, Harvard University, Cambridge, MA 02138, U.S.A.
⁵Physics Department, University of Arizona, Tucson, AZ 85721, U.S.A.
*E-mail address: sayama@hc.uec.ac.jp

(Received February 26, 2003; Accepted March 5, 2003)

Keywords: Turing Pattern Formation, Domain Coarsening, Spatially Extended Population Models, Genetic Distribution, Evolution

Abstract. We introduce a model of spatially distributed populations of organisms that mate and compete with others in local neighborhoods. Competition for local finite resources causes Turing instability in population distribution, possibly leading to the formation of isolated groups. In the presence of disruptive selection against genetic intermediates, this model also shows dynamically coarsening domains in genetic distribution. We examine an interplay of these two distinct dynamics, both analytically and numerically, and show that the domain coarsening process is strongly affected by the spatial separation between groups created by the Turing pattern formation process. The ratio between mating and competition ranges is found to be one of the crucial parameters to determine the long-term evolution of genetic distribution in the population.