# Tragedy of the Commons in the Chemostat

## TL;DR

This paper demonstrates the tragedy of the commons in a chemostat model with cooperators and cheaters, showing how cheaters lead to extinction of both species, and provides a framework for analyzing cooperator-cheater dynamics.

## Contribution

It introduces a chemostat model that proves the tragedy of the commons occurs with cheaters, and explores conditions for cooperator survival without cheaters.

## Key findings

- Cheaters cause extinction of both species when present initially.
- Cooperators can survive indefinitely without cheaters if initial conditions include some public good.
- The model offers a predictive framework for cooperator-cheater dynamics in experimental evolution.

## Abstract

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.