Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Strong cooperation or tragedy of the commons in the chemostat

1 Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA
2 Department of Microbiology, Oregon State University, Corvallis, OR 97331, USA
3 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

Special Issues: Resource Explicit Population Models

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.
  Article Metrics


1. K.L. Asfahl, J. Walsh, K. Gilbert and M. Schuster, Non-social adaptation defers a tragedy of the commons in Pseudomonas aeruginosa quorum sensing, ISME J., 9 (2015), 1734–1746.

2. J. S. Chuang, O. Rivoire and S. Leibler, Simpson's Paradox in a synthetic microbial system, Science, 323 (2009), 272–275.

3. J. Cremer, A. Melbinger and E. Frey, Growth dynamics and the evolution of cooperation in microbial populations, Sci Rep, 2 (2012), 281.

4. A.A. Dandekar, S. Chugani and E.P. Greenberg, Bacterial quorum sensing and metabolic incentives to cooperate, Science, 338 (2012), 264–266.

5. F. Fiegna and G.J. Velicer, Competitive fates of bacterial social parasites: persistence and selfinduced extinction of Myxococcus xanthus cheaters, Proc. Biol. Sci., 270 (2003), 1527–1534.

6. G.R. Hardin, The tragedy of the commons, Science, 162 (1968), 1243–1248.

7. R. Kümmerli, A.S. Griffin, S. A.West, A. Buckling and F. Harrison, Viscous medium promotes cooperation in the pathogenic bacterium Pseudomonas aeruginosa, Proceedings of the Royal Society B, 276 (2009), 3531–3538.

8. W.F. Lloyd, Two lectures on the checks to population (Oxford Univ. Press, Oxford, England, 1833), reprinted (in part, in: Population, Evolution, and Birth Control, G. Hardin, Ed. (Freeman, San Francisco, 1964), 37.

9. T. Pfeiffer, S. Schuster and S. Bonhoeffer, Cooperation and competition in the evolution of ATPproducing pathways, Science, 292 (2001), 504–507.

10. P.B. Rainey and K. Rainey, Evolution of cooperation and conflict in experimental bacterial populations, Nature, 425 (2003), 72–74.

11. M. Schuster, E. Foxall, D. Finch, H. Smith and P. De Leenheer, Tragedy of the commons in the chemostat, PLOS ONE, Dec 2017. Available from: https://doi.org/10.1371/journal.pone.0186119.

12. Introduction to Evolutionary Game Theory, Proceedings of Symposia in Applied Mathematics, 69 (2011), 1–25.

13. H.L. Smith, Monotone Dynamical Systems, American Mathematical Society, 1995.

14. H.L. Smith and P. Waltman, The theory of the chemostat, Cambridge University Press, 1994.

15. A.J. Waite and W. Shou, Adaptation to a new environment allows cooperators to purge cheaters stochastically, PNAS, 109 (2012), 19079–19086.

16. S.A. West, A.S. Griffin, A. Gardner and S.P. Diggle SP, Social evolution theory for microorganisms, Nat. Rev. Microbiol., 4 (2006), 597–607.

17. S.A. West, A.S. Griffin and A. Gardner, Evolutionary explanations for cooperation, Curr. Biol. 17 (2007), R661–72.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved