IDEAS home Printed from https://ideas.repec.org/a/eee/thpobi/v84y2013icp46-55.html
   My bibliography  Save this article

A mathematical description of the inclusive fitness theory

Author

Listed:
  • Wakano, Joe Yuichiro
  • Ohtsuki, Hisashi
  • Kobayashi, Yutaka

Abstract

Recent developments in the inclusive fitness theory have revealed that the direction of evolution can be analytically predicted in a wider class of models than previously thought, such as those models dealing with network structure. This paper aims to provide a mathematical description of the inclusive fitness theory. Specifically, we provide a general framework based on a Markov chain that can implement basic models of inclusive fitness. Our framework is based on the probability distribution of “offspring-to-parent map†, from which the key concepts of the theory, such as fitness function, relatedness and inclusive fitness, are derived in a straightforward manner. We prove theorems showing that inclusive fitness always provides a correct prediction on which of two competing genes more frequently appears in the long run in the Markov chain. As an application of the theorems, we prove a general formula of the optimal dispersal rate in the Wright’s island model with recurrent mutations. We also show the existence of the critical mutation rate, which does not depend on the number of islands and below which a positive dispersal rate evolves. Our framework can also be applied to lattice or network structured populations.

Suggested Citation

  • Wakano, Joe Yuichiro & Ohtsuki, Hisashi & Kobayashi, Yutaka, 2013. "A mathematical description of the inclusive fitness theory," Theoretical Population Biology, Elsevier, vol. 84(C), pages 46-55.
    • Handle: RePEc:eee:thpobi:v:84:y:2013:i:c:p:46-55
    DOI: 10.1016/j.tpb.2012.11.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0040580912001232
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.tpb.2012.11.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Samir Okasha, 2010. "Altruism researchers must cooperate," Nature, Nature, vol. 467(7316), pages 653-655, October.
    2. Erez Lieberman & Christoph Hauert & Martin A. Nowak, 2005. "Evolutionary dynamics on graphs," Nature, Nature, vol. 433(7023), pages 312-316, January.
    3. Hisashi Ohtsuki & Christoph Hauert & Erez Lieberman & Martin A. Nowak, 2006. "A simple rule for the evolution of cooperation on graphs and social networks," Nature, Nature, vol. 441(7092), pages 502-505, May.
    4. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
    5. Martin A. Nowak & Corina E. Tarnita & Edward O. Wilson, 2010. "The evolution of eusociality," Nature, Nature, vol. 466(7310), pages 1057-1062, August.
    6. Lehmann, Laurent & Rousset, François, 2009. "Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill–Robertson effect and to the joint evolution of helping and punishment," Theoretical Population Biology, Elsevier, vol. 76(1), pages 35-51.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Chaoqian & Szolnoki, Attila, 2023. "Inertia in spatial public goods games under weak selection," Applied Mathematics and Computation, Elsevier, vol. 449(C).
    2. Du, Faqi & Fu, Feng, 2013. "Quantifying the impact of noise on macroscopic organization of cooperation in spatial games," Chaos, Solitons & Fractals, Elsevier, vol. 56(C), pages 35-44.
    3. Benjamin Allen & Christine Sample & Robert Jencks & James Withers & Patricia Steinhagen & Lori Brizuela & Joshua Kolodny & Darren Parke & Gabor Lippner & Yulia A Dementieva, 2020. "Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs," PLOS Computational Biology, Public Library of Science, vol. 16(1), pages 1-20, January.
    4. McAvoy, Alex & Fraiman, Nicolas & Hauert, Christoph & Wakeley, John & Nowak, Martin A., 2018. "Public goods games in populations with fluctuating size," Theoretical Population Biology, Elsevier, vol. 121(C), pages 72-84.
    5. Liu, Xuesong & Pan, Qiuhui & He, Mingfeng & Liu, Aizhi, 2019. "Promotion of cooperation in evolutionary game dynamics under asymmetric information," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 258-266.
    6. Sarkar, Bijan, 2021. "The cooperation–defection evolution on social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 584(C).
    7. Li, Bin-Quan & Wu, Zhi-Xi & Guan, Jian-Yue, 2022. "Critical thresholds of benefit distribution in an extended snowdrift game model," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    8. Flávio L Pinheiro & Jorge M Pacheco & Francisco C Santos, 2012. "From Local to Global Dilemmas in Social Networks," PLOS ONE, Public Library of Science, vol. 7(2), pages 1-6, February.
    9. Kroumi, Dhaker & Lessard, Sabin, 2015. "Evolution of cooperation in a multidimensional phenotype space," Theoretical Population Biology, Elsevier, vol. 102(C), pages 60-75.
    10. Matthijs van Veelen & Benjamin Allen & Moshe Hoffman & Burton Simon & Carl Veller, 2016. "Inclusive Fitness," Tinbergen Institute Discussion Papers 16-055/I, Tinbergen Institute.
    11. Cheng, Jiangjiang & Mei, Wenjun & Su, Wei & Chen, Ge, 2023. "Evolutionary games on networks: Phase transition, quasi-equilibrium, and mathematical principles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 611(C).
    12. Charles G Nathanson & Corina E Tarnita & Martin A Nowak, 2009. "Calculating Evolutionary Dynamics in Structured Populations," PLOS Computational Biology, Public Library of Science, vol. 5(12), pages 1-7, December.
    13. Van Cleve, Jeremy, 2015. "Social evolution and genetic interactions in the short and long term," Theoretical Population Biology, Elsevier, vol. 103(C), pages 2-26.
    14. Li, Bin-Quan & Wu, Zhi-Xi & Guan, Jian-Yue, 2022. "Alternating rotation of coordinated and anti-coordinated action due to environmental feedback and noise," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    15. Dhaker Kroumi, 2021. "Aspiration Can Promote Cooperation in Well-Mixed Populations As in Regular Graphs," Dynamic Games and Applications, Springer, vol. 11(2), pages 390-417, June.
    16. Dario Madeo & Chiara Mocenni, 2018. "Self-regulation promotes cooperation in social networks," Papers 1807.07848, arXiv.org.
    17. Alex McAvoy & Christoph Hauert, 2015. "Asymmetric Evolutionary Games," PLOS Computational Biology, Public Library of Science, vol. 11(8), pages 1-26, August.
    18. Swami Iyer & Timothy Killingback, 2016. "Evolution of Cooperation in Social Dilemmas on Complex Networks," PLOS Computational Biology, Public Library of Science, vol. 12(2), pages 1-25, February.
    19. Mahdi Hajihashemi & Keivan Aghababaei Samani, 2022. "Multi-strategy evolutionary games: A Markov chain approach," PLOS ONE, Public Library of Science, vol. 17(2), pages 1-17, February.
    20. Yao Meng & Sean P. Cornelius & Yang-Yu Liu & Aming Li, 2024. "Dynamics of collective cooperation under personalised strategy updates," Nature Communications, Nature, vol. 15(1), pages 1-11, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:thpobi:v:84:y:2013:i:c:p:46-55. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/intelligence .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.