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Technical Background

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The Statistical Physics of Fixation and Equilibration in Individual-Based Models

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In this chapter I will outline the mathematical and numerical techniques used throughout this thesis to analyse stochastic systems.

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Notes

  1. 1.

    Rev. Thomas Bayes (1701–1761).

  2. 2.

    Pierre-Simon Laplace (1749–1827).

  3. 3.

    Specifically of the matrix \(\mathbb {W}\) shown in Eq. (2.15).

  4. 4.

    After Carl Gustav Jacob Jacobi (1804–1851), who will reappear in Chaps. 5 and 6.

  5. 5.

    Adriaan Fokker (1877–1972) and Max Planck (1858–1947).

  6. 6.

    Hendrik Kramers (1894–1952) and José Enrique Moyal (1910–1998).

  7. 7.

    Nico van Kampen (1921–2013).

  8. 8.

    Kiyoshi Itō (1915–2008) and Ruslan Stratanovich (1930–1997).

  9. 9.

    Enrico Fermi (1901–1954).

  10. 10.

    Pat Moran (1917–1988).

References

  1. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 2007)

    MATH  Google Scholar 

  2. N. Goel, N. Richter-Dyn, Stochastic Models in Biology (Academic Press, New York, 1974)

    Google Scholar 

  3. T. Antal, I. Scheuring, Fixation of strategies for an evolutionary game in finite populations. Bull. Math. Biol. 68, 1923 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. W.J. Ewens, Mathematical Population Genetics. I. Theoretical Introduction (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  5. A. Traulsen, C. Hauert, Stochastic evolutionary game dynamics, in Reviews of Nonlinear Dynamics and Complexity, vol. II, ed. by H.G. Schuster (Wiley-VCH, Weinheim, 2009)

    Google Scholar 

  6. R. Shankar, Principles of Quantum Mechanics (Springer, New York, 1994)

    Book  MATH  Google Scholar 

  7. S. Karlin, H.M. Taylor, A Second Course in Stochastic Processes (Academic Press, New York, 1981)

    MATH  Google Scholar 

  8. M.A. Nowak, Evolutionary Dynamics (Harvard University Press, Cambridge, MA, 2006)

    MATH  Google Scholar 

  9. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, UK, 2001)

    Book  MATH  Google Scholar 

  10. P.M. Altrock, A. Traulsen, F.A. Reed, Stability properties of underdominance in finite subdivided populations. PLoS Comput. Biol. 7, e1002260 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  11. D.A. Levin, Y. Peres, E.L. Wilmer, Markov Chains and Mixing Times (AMS Publishing, Providence, RI, 2009)

    MATH  Google Scholar 

  12. A.J. Black, A. Traulsen, T. Galla, Mixing times in evolutionary games. Phys. Rev. Lett. 109, 028101 (2012)

    Article  ADS  Google Scholar 

  13. S. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, Boulder, CO, 2000)

    Google Scholar 

  14. C.W. Gardiner, Handbook of Stochastic Methods (Springer, New York, 2009)

    MATH  Google Scholar 

  15. R. Pawula, Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Phys. Rev. 162, 186 (1967)

    Article  ADS  Google Scholar 

  16. H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  17. H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284 (1940)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. J. Moyal, Stochastic processes and statistical physics. J. R. Stat. Soc. 11, 151 (1949)

    MathSciNet  MATH  Google Scholar 

  19. J.W. Weibull, Evolutionary Game Theory (MIT Press, Cambridge, MA, 1995)

    MATH  Google Scholar 

  20. J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, UK, 1998)

    Book  MATH  Google Scholar 

  21. H. Gintis, Game Theory Evolving (Princeton University Press, Princeton, NJ, 2009)

    MATH  Google Scholar 

  22. W.H. Sandholm, Population Games and Evolutionary Dynamics (MIT Press, Cambridge, MA, 2010)

    MATH  Google Scholar 

  23. J. Maynard Smith, G.R. Price, The logic of animal conflict. Nature 246, 15 (1973)

    Google Scholar 

  24. M. Milinski, Tit For Tat in sticklebacks and the evolution of cooperation. Nature 325, 433 (1987)

    Article  ADS  Google Scholar 

  25. R.C. MacLean, A. Fuentes-Hernandez, D. Greig, L.D. Hurst, I. Gudelj, A mixture of “cheats” and “co-operators” can enable maximal group benefit. PLoS Biol. 8, e1000486 (2010)

    Article  Google Scholar 

  26. A. Traulsen, F.A. Reed, From genes to games: cooperation and cyclic dominance in meiotic drive. J. Theor. Biol. 299, 120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. P.D. Taylor, L. Jonker, Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Lenormand, D. Roze, F. Rousset, Stochasticity in evolution. Trends Ecol. Evol. 24, 157 (2009)

    Article  Google Scholar 

  29. A.J. Black, A.J. McKane, Stochastic formulation of ecological models and their applications. Trends Ecol. Evol. 27, 337 (2012)

    Article  Google Scholar 

  30. P.M. Altrock, A. Traulsen, T. Galla, The mechanics of stochastic slowdown in evolutionary games. J. Theor. Biol. 311, 94 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Arnoldt, M. Timme, S. Grosskinsky, Frequency-dependent fitness induces multistability in coevolutionary dynamics. J. R. Soc. Interface 9, 3387 (2012)

    Article  Google Scholar 

  32. J. Du, B. Wu, P.M. Altrock, L. Wang, Aspiration dynamics of multi-player games in finite populations. J. R. Soc. Interface 11, 20140077 (2014)

    Article  Google Scholar 

  33. J.F. Nash, Equilibrium points in \(n\)-person games. Proc. Natl. Acad. Sci. USA 36, 48 (1950)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. P.M. Altrock, A. Traulsen, Fixation times in evolutionary games under weak selection. New J. Phys. 11, 013012 (2009)

    Article  ADS  Google Scholar 

  35. B. Wu, P.M. Altrock, L. Wang, A. Traulsen, Universality of weak selection. Phys. Rev. E 82, 046106 (2010)

    Article  ADS  Google Scholar 

  36. B. Wu, B. Bauer, T. Galla, A. Traulsen, Fitness-based models and pairwise-comparison models of evolutionary games are typically different even in unstructured populations. New J. Phys. 17, 023043 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  37. A. Traulsen, N. Shoresh, M.A. Nowak, Analytical results for individual and group selection of any intensity. Bull. Math. Biol. 70, 1410 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. P.M. Altrock, A. Traulsen, Deterministic evolutionary game dynamics in finite populations. Phys. Rev. E 80, 011909 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  39. P.A.P. Moran, The Statistical Processes of Evolutionary Theory (Clarendon Press, Oxford, UK, 1962)

    MATH  Google Scholar 

  40. C. Taylor, D. Fudenberg, A. Sasaki, M.A. Nowak, Evolutionary game dynamics in finite populations. Bull. Math. Biol. 66, 1621 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. D.T. Gillespie, Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340 (1977)

    Article  Google Scholar 

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  1. Institute for Integrative Biology, ETH Zürich, Zürich, Switzerland

    Peter Ashcroft

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Ashcroft, P. (2016). Technical Background. In: The Statistical Physics of Fixation and Equilibration in Individual-Based Models. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41213-9_2

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