Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T04:41:41.726Z Has data issue: false hasContentIssue false
  • English
  • Français

A spatial model for selection and cooperation

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes

Published online by Cambridge University Press:  22 June 2017

Peter Czuppon  and
Peter Pfaffelhuber
Peter Pfaffelhuber*
Affiliation:
University of Freiburg
*
** Postal address: Department of Mathematical Stochastics, University of Freiburg, 79104 Freiburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the evolution of cooperation in an interacting particle system with two types. The model we investigate is an extension of a two-type biased voter model. One type (called defector) has a (positive) bias α with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate γ. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at α = γ. A special choice of interaction kernels yield that for α > γ cooperators always die out, but if γ > α, cooperation is the winning strategy.

Keywords

Interacting particle systemvoter modelcooperationphase transitionextinctionclustering

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82C22: Interacting particle systems 92D15: Problems related to evolution
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 522 - 539
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Archetti, M. and Scheuring, I. (2012). Review: game theory of public goods in one-shot social dilemmas without assortment. J. Theoret. Biol. 299, 9–20. CrossRefGoogle ScholarPubMed
[2] Blath, J. and Kurt, N. (2011). Survival and extinction of caring double-branching annihilating random walk. Electron. Commun. Prob. 16, 271–282. CrossRefGoogle Scholar
[3] Bramson, M. and Griffeath, D. (1981). On the Williams-Bjerknes tumour growth model I. Ann. Prob. 9, 173–185. Google Scholar
[4] Brockhurst, M. A., Buckling, A., Racey, D. and Gardner, A. (2008). Resource supply and the evolution of public-goods cooperation in bacteria. BMC Biology 6, 6pp. Google Scholar
[5] Clutton-Brock, T. (2009). Cooperation between non-kin in animal societies. Nature 462, 51–57. Google Scholar
[6] Crespi, B. J. (2001). The evolution of social behavior in microorganisms. Trends Ecol. Evol. 16, 178–183. Google Scholar
[7] Czuppon, P. (2016). Phenotypic heterogeneity in bacterial populations – a mathematical study. Doctoral thesis. University of Freiburg. Google Scholar
[8] Drescher, K. et al. (2014). Solutions to the public good dilemma in bacterial biofilms. Current Biol. 24, 50–55. CrossRefGoogle Scholar
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York. Google Scholar
[10] Evilsizor, S. and Lanchier, N. (2016). Evolutionary games on the lattice: death-birth updating process. Electron. J. Prob. 21, 29pp. Google Scholar
[11] Griffin, A. S. and West, S. A. (2003). Kin discrimination and the benefit of helping in cooperatively breeding vertebrates. Science 302, 634–636. CrossRefGoogle ScholarPubMed
[12] Hutzenthaler, M., Jordan, F. and Metzler, D. (2015). Altruistic defense traits in structured populations. Preprint. Available at https://arxiv.org/abs/1505.02154v1. Google Scholar
[13] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. CrossRefGoogle Scholar
[14] Louidor, O., Tessler, R. and Vandenberg-Rodes, A. (2014). The Williams–Bjerknes model on regular trees. Ann. Appl. Prob. 24, 1889–1917. Google Scholar
[15] Nowak, M. A. (2006). Five rules for the evolution of cooperation. Science 314, 1560–1563. CrossRefGoogle ScholarPubMed
[16] Penn, D. J. and Frommen, J. G. (2010). Kin recognition: an overview of conceptual issues, mechanisms and evolutionary theory. In Animal Behaviour: Evolution and Mechanisms, Springer, Heidelberg, pp. 55–85. Google Scholar
[17] Sturm, A. and Swart, J. M. (2015). A particle system with cooperative branching and coalescence. Ann. Appl. Prob. 25, 1616–1649. Google Scholar
[18] Williams, T. and Bjerknes, R. (1972). Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19–21. Google Scholar
[19] Wingreen, N. S. and Levin, S. A. (2006). Cooperation among microorganisms. Plos Biol. 4, 3pp. Google Scholar