  This paper studies a variant of the multi-type contact process as a model for
the competition between cooperators and defectors on integer lattices.
Regardless of their type, individuals die at rate one. Defectors give birth at
a fixed rate whereas cooperators give birth at a rate that increases linearly
with the number of nearby cooperators. In particular, it is assumed that only
cooperators benefit from cooperators, which is referred to as kin-recognition
in the ecological literature. To understand how the inclusion of space in the
form of local interactions affects the dynamics, the results for the
interacting particle system are compared with their counterpart for the
non-spatial mean-field model. Due to some monotonicity with respect to the
parameters, both the spatial and non-spatial models exhibit a unique phase
transition. Our analysis shows however a major difference: In the spatial
model, when cooperation is strong enough, the cooperators out-compete the
defectors even when starting at arbitrarily low density. In contrast,
regardless of the strength of cooperation, when the initial density of
cooperators is too low, the defectors out-compete the cooperators in the
non-spatial model. In particular, when cooperation is sufficiently strong, the
cooperators can invade the defectors in their equilibrium in the spatial model
but not in the non-spatial model, showing that space in the form of local
interactions promotes cooperation.
