A spatial model for selection and cooperation

We study the evolution of cooperation in an interacting particle system with two types. The model we investigate is an extension of a two-type biased voter model. One type (called defector) has a (positive) bias $\alpha$ with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate $\gamma$. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at $\alpha=\gamma$. For $\alpha>\gamma$ cooperators always die out, but if $\gamma>\alpha$, cooperation is the winning strategy.