Evolutionary Games in Space

Issue		Math. Model. Nat. Phenom. Volume 4, Number 6, 2009 Ecology (Part 1)


Page(s)		54 - 90
DOI		10.1051/mmnp/20094602
Published online		27 November 2009

Math. Model. Nat. Phenom. Vol. 4, No. 6, 2009, pp. 54-90
DOI: 10.1051/mmnp/20094602

N. Kronik¹ and Y. Cohen²

¹ Department of Applied Mathematics, Holon Institute of Technology, Holon 58102, Israel
² Department of Fisheries, Wildlife, and Conservation Biology, University of Minnesota, St. Paul, MN 55118

natalieka@hit.ac.il

Published online: 27 November 2009

Abstract
The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.

Mathematics Subject Classification. 35B30, 35B20, 91A22

Key words: mathematical modeling -- game theory -- reaction-diffusion equation -- G-function -- evolutionary ecology

© EDP Sciences 2009

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