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Cumulant Dynamics in a Finite Population: Linkage Equilibrium Theory


  • Magnus Rattray
  • Jonathan L. Shapiro


The evolution of a finite population at linkage equilibrium is described in terms of the dynamics of phenotype distribution cumulants. This provides a powerful method for describing evolutionary transients and we elucidate the relationship between the cumulant dynamics and the diffusion approximation. A separation of time-scales between the first and higher cumulants for low mutation rates is demonstrated in the diffusion limit and provides a significant simplification of the dynamical system. However, the diffusion limit may not be appropriate for strong selection as the standard Fisher-Wright model of genetic drift can break down in this case. Two novel examples of this effect are considered: we show that the dynamics may depend on the number of loci under strong directional selection and that environmental variance results in a reduced effective population size. We also consider a simple model of a changing environment which cannot be described by a diffusion equation and we derive the optimal mutation rate for this case.

Suggested Citation

  • Magnus Rattray & Jonathan L. Shapiro, 1999. "Cumulant Dynamics in a Finite Population: Linkage Equilibrium Theory," Working Papers 99-07-055, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:99-07-055

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    References listed on IDEAS

    1. Myerson, Roger B. & Pollock, Gregory B. & Swinkels, Jeroen M., 1991. "Viscous population equilibria," Games and Economic Behavior, Elsevier, vol. 3(1), pages 101-109, February.
    2. Ellison, Glenn, 1993. "Learning, Local Interaction, and Coordination," Econometrica, Econometric Society, vol. 61(5), pages 1047-1071, September.
    3. Durlauf, S.N., 1997. "The Memberships Theory of Inequality: Ideas and Implications," Working papers 9711, Wisconsin Madison - Social Systems.
    4. Avner Shaked & Larry Samuelson & George J. Mailath, 1997. "Correlated equilibria and local interactions (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(3), pages 551-556.
    5. George A. Akerlof, 1997. "Social Distance and Social Decisions," Econometrica, Econometric Society, vol. 65(5), pages 1005-1028, September.
    6. Blume Lawrence E., 1993. "The Statistical Mechanics of Strategic Interaction," Games and Economic Behavior, Elsevier, vol. 5(3), pages 387-424, July.
    7. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    8. Brock, W.A. & Durlauf, S.N., 1995. "Discrete Choice with Social Interactions I: Theory," Working papers 9521, Wisconsin Madison - Social Systems.
    9. repec:cdl:ucsbec:2-98 is not listed on IDEAS
    10. Eshel, I. & Samuelson, L. & Shaked, A., 1996. "Altruists, Egoists and Hooligans in a Local Interaction Model," Working papers 9612, Wisconsin Madison - Social Systems.
    11. Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January.
    12. Ulrich Schwalbe & Siegfried K. Berninghaus, 1996. "Conventions, local interaction, and automata networks," Journal of Evolutionary Economics, Springer, vol. 6(3), pages 297-312.
    13. Leigh TESFATSION, 1995. "A Trade Network Game With Endogenous Partner Selection," Economic Report 36, Iowa State University Department of Economics.
    14. Eshel, Ilan & Samuelson, Larry & Shaked, Avner, 1998. "Altruists, Egoists, and Hooligans in a Local Interaction Model," American Economic Review, American Economic Association, vol. 88(1), pages 157-179, March.
    15. Anderlini, Luca & Ianni, Antonella, 1996. "Path Dependence and Learning from Neighbors," Games and Economic Behavior, Elsevier, vol. 13(2), pages 141-177, April.
    16. George J. Mailath & Larry Samuelson & Avner Shaked, 1997. "Endogenous Interactions," CARESS Working Papres endo-one, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    17. Samuelson Larry, 1994. "Stochastic Stability in Games with Alternative Best Replies," Journal of Economic Theory, Elsevier, vol. 64(1), pages 35-65, October.
    18. Vega-Redondo Fernando, 1993. "Competition and Culture in an Evolutionary Process of Equilibrium Selection: A Simple Example," Games and Economic Behavior, Elsevier, vol. 5(4), pages 618-631, October.
    19. Vega-Redondo Fernando, 1995. "Expectations, Drift, and Volatility in Evolutionary Games," Games and Economic Behavior, Elsevier, vol. 11(2), pages 391-412, November.
    20. repec:cdl:ucsbec:02-98 is not listed on IDEAS
    21. Boylan, Richard T., 1992. "Laws of large numbers for dynamical systems with randomly matched individuals," Journal of Economic Theory, Elsevier, vol. 57(2), pages 473-504, August.
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    Evolutionary dynamics; mutation-selection dynamics; theoretical population biology; diffusion approximation;

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