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.