Strong cooperation or tragedy of the commons in the chemostat

In [11], a proof of principle was established for the phenomenon of the tragedy of the commons, a center piece for many theories on the evolution of cooperation. A general chemostat model with two species, the cooperator and the cheater, was formulated where the cooperator allocates a portion of the nutrient uptake towards the production of a public good which is needed to digest an externally supplied resource. The cheater does not produce the public good, and instead allocates all nutrient uptake towards its own growth. It was proved that if the cheater is present, both the cooperator and the cheater will go extinct. A key assumption was that the cheater and cooperator share a common nutrient uptake rate and yield constant. Here, we relax that assumption and find that although the extinction of both types holds in many cases, it is possible for the cooperator to survive and exclude the cheater if it can evolve so as to have a lower break-even concentration for growth than the cheater. Coexistence of cooperator and cheater is generically impossible.