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.