We study the evolution of cooperation as a birth-death process in spatially extended populations. The benefit from the altruistic behavior of a cooperator is implemented by decreasing the death rate of its direct neighbors. The cost of cooperation is the increase of a cooperators death rate proportional to the number of its neighbors. When cooperation has higher cost than benefit, cooperators disappear. Then the dynamics reduces to the contact process with defectors as the single particle type. Increasing the benefit-cost ratio above 1, the extinction transition of the contact process splits into a set of nonequilibrium transitions between four regimes when increasing the baseline death rate $p$ as a control parameter: (i) defection only, (ii) coexistence, (iii) cooperation only, (iv) extinction. We investigate the transitions between these regimes. As the main result, we find that full cooperation is established at the extinction transition as long as benefit is strictly larger than cost. Qualitatively identical phase diagrams are obtained for populations on square lattices and in pair approximation. Spatial correlations with nearest neighbors only are thus sufficient for sustained cooperation.
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