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The classical idea of evolutionarily stable strategy (ESS) modeling animalbehavior does not involve any spatial dependence. We considered a spatialHawk-Dove game played by animals in a patchy environment with wrap aroundboundaries. We posit that each site contains the same number of individuals. Anevolution equation for analyzing the stability of the ESS is found as the meandynamics of the classical frequency dependent Moran process coupled viamigration and nonlocal payoff calculation in 1D and 2D habitats. The linearstability analysis of the model is performed and conditions to observe spatialpatterns are investigated. For the nearest neighbor interactions (including vonNeumann and Moore neighborhoods in 2D) we concluded that it is possible todestabilize the ESS of the game and observe pattern formation when thedispersal rate is small enough. We numerically investigate the spatial patternsarising from the replicator equations coupled via nearest neighbor payoffcalculation and dispersal.

Discovering the effect of nonlocal payoff calculation on the stabilty of ESS: Spatial patterns of Hawk-Dove game in metapopulations