The G-function formalism has beenwidely used in the context of evolutionary games for identifyingevolutionarily stable strategies (ESS). This formalism wasdeveloped for and applied to point processes. Here, we examine the G-functionformalism in the settings of spatial evolutionarygames and strategy dynamics, based on reaction-diffusion models. We startbyextending the point process maximum principle to reaction-diffusion modelswith homogeneous, locally stable surfaces. We then develop the strategy dynamics forsuch surfaces. When the surfaces are locally stable, but nothomogenous, the standard definitions of ESS and the maximumprinciple fall apart. Yet, we show by examples that strategy dynamics leads toconvergent stable inhomogeneous strategies that are possibly ESS, in the sensethat for many scenarios which we simulated,invaders could not coexist with the exisiting strategies.
