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

TL;DR

This paper explores how nonlocal payoff calculations influence the stability of evolutionarily stable strategies in spatial Hawk-Dove games, revealing conditions under which spatial patterns emerge in metapopulations.

Contribution

It introduces a spatial Hawk-Dove game model with nonlocal payoff calculation and analyzes how local interactions and dispersal rates affect ESS stability and pattern formation.

Findings

Small dispersal rates can destabilize ESS and lead to pattern formation.

Nearest neighbor interactions can induce spatial patterns in 1D and 2D habitats.

Numerical simulations confirm the emergence of spatial patterns under certain conditions.

Abstract

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.