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 α with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate γ. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at α = γ. A special choice of interaction kernels yield that for α > γ cooperators always die out, but if γ > α, cooperation is the winning strategy.