Division of labor in bacterial populations

Cooperating behaviors abound across all domains of life, but are vulnerable to invasion by cheaters. An important evolutionary question is to determine mechanisms that stabilize and maintain cooperation levels and prevent population collapse. Policing is one strategy populations may employ to achieve this goal, and it has been observed in many natural populations including microbes. Here we present and analyze a division of labor model to investigate if, when and how policing can be a cooperation-stabilizing mediator. The model represents a chemostat where cooperators produce a public good that benefits all individuals, and where toxin-producers produce a toxin that harms both cooperators and cheaters. We show that in many cases, the mere presence of toxin-producers is not enough to avoid a Tragedy of the Commons in which all individuals go extinct. The main focus of our work is to identify conditions on various model parameters which ensure that a mixed population of cooperators and toxin-producers can stably coexist and can avoid invasion by a cheater population. This happens when all of the following conditions hold: (i) The cost of policing must exceed the cost of cooperation. (ii) There is enough "collateral damage" caused by policing, i.e. the toxicity rate experienced by cooperators is sufficiently high, and (iii) The toxin affects cheaters even more than cooperators, and we provide a precise mathematical condition of how much stronger this effect should be.