  We present a proof of principle for the phenomenon of the tragedy of the
commons that is at the center of many theories on the evolution of cooperation.
We establish the tragedy in the context of a general chemostat model with two
species, the cooperator and the cheater. Both species have the same growth rate
function and yield constant, but the cooperator allocates a portion of the
nutrient uptake towards the production of a public good -the "Commons" in the
Tragedy- which is needed to digest the externally supplied nutrient. The
cheater on the other hand does not produce this enzyme, and allocates all
nutrient uptake towards its own growth. We prove that when the cheater is
present initially, both the cooperator and the cheater will eventually go
extinct, hereby confirming the occurrence of the tragedy. We also show that
without the cheater, the cooperator can survive indefinitely, provided that at
least a low level of public good or processed nutrient is available initially.
Our results provide a predictive framework for the analysis of
cooperator-cheater dynamics in a powerful model system of experimental
evolution.
