  The classical idea of evolutionarily stable strategy (ESS) modeling animal
behavior does not involve any spatial dependence. We considered a spatial
Hawk-Dove game played by animals in a patchy environment with wrap around
boundaries. We posit that each site contains the same number of individuals. An
evolution equation for analyzing the stability of the ESS is found as the mean
dynamics of the classical frequency dependent Moran process coupled via
migration and nonlocal payoff calculation in 1D and 2D habitats. The linear
stability analysis of the model is performed and conditions to observe spatial
patterns are investigated. For the nearest neighbor interactions (including von
Neumann and Moore neighborhoods in 2D) we concluded that it is possible to
destabilize the ESS of the game and observe pattern formation when the
dispersal rate is small enough. We numerically investigate the spatial patterns
arising from the replicator equations coupled via nearest neighbor payoff
calculation and dispersal.
