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July  2021, 26(7): 3491-3504. doi: 10.3934/dcdsb.2020242

Moran process and Wright-Fisher process favor low variability

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284-2014, USA

* Corresponding author: Jan Rychtář

Received  October 2019 Revised  May 2020 Published  July 2021 Early access  August 2020

We study evolutionary dynamics in finite populations. We assume the individuals are one of two competing genotypes, $ A $ or $ B $. The genotypes have the same average fitness but different variances and/or third central moments. We focus on two frequency-independent stochastic processes: (1) Wright-Fisher process and (2) Moran process. Both processes have two absorbing states corresponding to homogeneous populations of all $ A $ or all $ B $. Despite the fact that types $ A $ and $ B $ have the same average fitness, both stochastic dynamics differ from a random drift. In both processes, the selection favors $ A $ replacing $ B $ and opposes $ B $ replacing $ A $ if the fitness variance for $ A $ is smaller than the fitness variance for $ B $. In the case the variances are equal, the selection favors $ A $ replacing $ B $ and opposes $ B $ replacing $ A $ if the third central moment of $ A $ is larger than the third central moment of $ B $. We show that these results extend to structured populations and other dynamics where the selection acts at birth. We also demonstrate that the selection favors a larger variance in fitness if the selection acts at death.

Citation: Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242
References:
[1]

B. Allen and M. A. Nowak, Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.  doi: 10.4171/EMSS/3.  Google Scholar

[2]

B. Allen and C. E. Tarnita, Measures of success in a class of evolutionary models with fixed population size and structure, J. Math. Biol., 68 (2014), 109-143.  doi: 10.1007/s00285-012-0622-x.  Google Scholar

[3]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, John Wiley & Sons, 2000.  Google Scholar

[4]

F. A. Chalub and M. O. Souza, The frequency-dependent Wright-Fisher model: Diffusive and non-diffusive approximations, J. Math. Biol., 68 (2014), 1089-1133.  doi: 10.1007/s00285-013-0657-7.  Google Scholar

[5]

F. A. Chalub and M. O. Souza, On the stochastic evolution of finite populations, J. Math. Biol., 75 (2017), 1735-1774.  doi: 10.1007/s00285-017-1135-4.  Google Scholar

[6]

D. Z. ChildsC. J. E. Metcalf and M. Rees, Evolutionary bet-hedging in the real world: Empirical evidence and challenges revealed by plants, Proc. Roy. Soc. B: Biol. Sci., 277 (2010), 3055-3064.  doi: 10.1098/rspb.2010.0707.  Google Scholar

[7]

D. Cohen, Optimizing reproduction in a randomly varying environment, J. Theoret. Biol., 12 (1966), 119-129.  doi: 10.1016/0022-5193(66)90188-3.  Google Scholar

[8]

W. S. Cooper and R. H. Kaplan, Adaptive "coin-flipping": A decision-theoretic examination of natural selection for random individual variation, J. Theoret. Biol., 94 (1982), 135-151.  doi: 10.1016/0022-5193(82)90336-8.  Google Scholar

[9]

P. Czuppon and A. Traulsen, Fixation probabilities in populations under demographic fluctuations, J. Math. Biol., 77 (2018), 1233-1277.  doi: 10.1007/s00285-018-1251-9.  Google Scholar

[10]

R. Durrett, Probability Models for DNA Sequence Evolution, Springer Science & Business Media, 2008. doi: 10.1007/978-0-387-78168-6.  Google Scholar

[11]

S. N. EvansA. Hening and S. J. Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J. Math. Biol., 71 (2015), 325-359.  doi: 10.1007/s00285-014-0824-5.  Google Scholar

[12]

W. J. Ewens, Mathematical Population Genetics. I. Theoretical Introduction, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21822-9.  Google Scholar

[13]

R. A. Fisher, On the dominance ratio, Proc. Roy. Soc. Edinburgh, 42 (1923), 321-341.   Google Scholar

[14]

G. B. FogelP. C. Andrews and D. B. Fogel, On the instability of evolutionary stable strategies in small populations, Ecological Modelling, 109 (1998), 283-294.  doi: 10.1016/S0304-3800(98)00068-4.  Google Scholar

[15]

S. A. Frank and M. Slatkin, Evolution in a variable environment, The American Naturalist, 136 (1990), 244-260.  doi: 10.1086/285094.  Google Scholar

[16]

D. FudenbergM. A. NowakC. Taylor and L. A. Imhof, Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoret. Popul. Biol., 70 (2006), 352-363.  doi: 10.1016/j.tpb.2006.07.006.  Google Scholar

[17]

J. H. Gillespie, Natural selection for within-generation variance in offspring number, Genetics, 76 (1974), 601-606.   Google Scholar

[18]

C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations, J. Theoret. Biol., 299 (2012), 106-112.  doi: 10.1016/j.jtbi.2011.06.010.  Google Scholar

[19]

J. Hofbauer and W. H. Sandholm, Evolution in games with randomly disturbed payoffs, J. Econ. Theory, 132 (2007), 47-69.  doi: 10.1016/j.jet.2005.05.011.  Google Scholar

[20] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[21]

L. A. Imhof and M. A. Nowak, Evolutionary game dynamics in a wright-fisher process, J. Math. Biol., 52 (2006), 667-681.  doi: 10.1007/s00285-005-0369-8.  Google Scholar

[22]

M. Kandori, G. J. Mailath and R. Rob, Learning, mutation, and long run equilibria in games, Econometrica: Journal of the Econometric Society, 29-56. doi: 10.2307/2951777.  Google Scholar

[23] S. Karlin, A First Course in Stochastic Processes, Academic press, 2014.   Google Scholar
[24]

E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs, Nature, 433 (2005), 312. doi: 10.1038/nature03204.  Google Scholar

[25]

N. Masuda, Directionality of contact networks suppresses selection pressure in evolutionary dynamics, J. Theoret. Biol., 258 (2009), 323-334.  doi: 10.1016/j.jtbi.2009.01.025.  Google Scholar

[26]

A. McAvoyN. FraimanC. HauertJ. Wakeley and M. A. Nowak, Public goods games in populations with fluctuating size, Theoret. Popul. Biol., 121 (2018), 72-84.  doi: 10.1016/j.tpb.2018.01.004.  Google Scholar

[27]

M. Mesterton-Gibbons and T. N. Sherratt, Information, variance and cooperation: Minimal models, Dynamic Games and Applications, 1 (2011), 419-439.  doi: 10.1007/s13235-011-0017-4.  Google Scholar

[28] P. A. P. Moran, The Statistical Process of Evolutionary Theory, Clarendon Press, Oxford, 1962.   Google Scholar
[29]

K. Nishimura and D. Stephens, Iterated prisoner's dilemma: Pay-off variance, J. Theoret. Biol., 188 (1997), 1-10.  doi: 10.1006/jtbi.1997.0439.  Google Scholar

[30]

M. A. Nowak, A. Sasaki, C. Taylor and D. Fudenberg, Emergence of cooperation and evolutionary stability in finite populations, Nature, 428 (2004), 646. doi: 10.1038/nature02414.  Google Scholar

[31]

H. OlofssonJ. Ripa and N. Jonzén, Bet-hedging as an evolutionary game: The trade-off between egg size and number, Proc. Roy. Soc. B: Biol. Sci., 276 (2009), 2963-2969.  doi: 10.1098/rspb.2009.0500.  Google Scholar

[32]

H. J. Park, Y. Pichugin, W. Huang and A. Traulsen, Population size changes and extinction risk of populations driven by mutant interactors, Phys. Rev. E, 99 (2019), 022305. doi: 10.1103/PhysRevE.99.022305.  Google Scholar

[33]

K. Pattni, Evolution in Finite Structured Populations with Group Interactions, Ph.D thesis, City, University of London, 2017. Google Scholar

[34]

K. PattniM. Broom and J. Rychtář, Evolving multiplayer networks: Modelling the evolution of cooperation in a mobile population, Discrete Contin. Dyn. Syst. B, 23 (2018), 1975-2004.  doi: 10.3934/dcdsb.2018191.  Google Scholar

[35]

K. Pattni, M. Broom, J. Rychtář and L. J. Silvers, Evolutionary graph theory revisited: When is an evolutionary process equivalent to the moran process?, Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 471 (2015), 20150334. doi: 10.1098/rspa.2015.0334.  Google Scholar

[36]

T. Philippi and J. Seger, Hedging one's evolutionary bets, revisited, Trends in Ecology & Evolution, 4 (1989), 41-44.  doi: 10.1016/0169-5347(89)90138-9.  Google Scholar

[37]

S. H. Rice, The expected value of the ratio of correlated random variables, https://www.depts.ttu.edu/biology/people/Faculty/Rice/home/ratio-derive.pdf, 2015 Google Scholar

[38]

S. H. Rice and A. Papadopoulos, Evolution with stochastic fitness and stochastic migration, PloS One, 4. doi: 10.1371/journal.pone.0007130.  Google Scholar

[39]

J. RipaH. Olofsson and N. Jonzén, What is bet-hedging, really?, Proc. Roy. Soc. B: Biol. Sci., 277 (2009), 1153-1154.  doi: 10.1098/rspb.2009.2023.  Google Scholar

[40]

M. E. Schaffer, Evolutionarily stable strategies for a finite population and a variable contest size, J. Theoret. Biol., 132 (1988), 469-478.  doi: 10.1016/S0022-5193(88)80085-7.  Google Scholar

[41]

P. H. Schimit, K. Pattni and M. Broom, Dynamics of multiplayer games on complex networks using territorial interactions, Phys. Rev. E, 99 (2019), 032306. doi: 10.1103/PhysRevE.99.032306.  Google Scholar

[42]

S. J. Schreiber, The evolution of patch selection in stochastic environments, The American Naturalist, 180 (2012), 17-34.  doi: 10.1086/665655.  Google Scholar

[43]

S. J. Schreiber, Unifying within-and between-generation bet-hedging theories: An ode to J.H. Gillespie, The American Naturalist, 186 (2015), 792-796.  doi: 10.1086/683657.  Google Scholar

[44]

J. Seger and H. Brockmann, Oxford surveys in evolutionary biology, Oxford Surveys in Evolutionary Biology, 4 (1987), 182-211.   Google Scholar

[45]

H. Seltman, Approximations for mean and variance of a ratio, http://www.stat.cmu.edu/ hseltman/files/ratio.pdf. Google Scholar

[46]

J. Starrfelt and H. Kokko, Bet-hedging - a triple trade-off between means, variances and correlations, Biol. Rev., 87 (2012), 742-755.  doi: 10.1111/j.1469-185X.2012.00225.x.  Google Scholar

[47]

C. TaylorD. FudenbergA. Sasaki and M. A. Nowak, Evolutionary game dynamics in finite populations, Bull. Math. Biol., 66 (2004), 1621-1644.  doi: 10.1016/j.bulm.2004.03.004.  Google Scholar

[48]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[49]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, Rev. Nonlin. Dyn. Complex., 2 (2009), 25-61.   Google Scholar

[50]

A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation, Phys. Rev. E, 74 (2006), 011909. doi: 10.1103/PhysRevE.74.011909.  Google Scholar

[51]

A. TraulsenM. A. Nowak and J. M. Pacheco, Stochastic payoff evaluation increases the temperature of selection, J. Theoret. Biol., 244 (2007), 349-356.  doi: 10.1016/j.jtbi.2006.08.008.  Google Scholar

[52]

C. Wallace and H. P. Young, Stochastic evolutionary game dynamics, in Handbook of Game Theory with Economic Applications, vol. 4, Elsevier, 2015, 327-380. doi: 10.1016/B978-0-444-53766-9.00006-9.  Google Scholar

[53]

S. Wright, Evolution in Mendelian populations, Genetics, 16 (1931), 97. Google Scholar

show all references

References:
[1]

B. Allen and M. A. Nowak, Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.  doi: 10.4171/EMSS/3.  Google Scholar

[2]

B. Allen and C. E. Tarnita, Measures of success in a class of evolutionary models with fixed population size and structure, J. Math. Biol., 68 (2014), 109-143.  doi: 10.1007/s00285-012-0622-x.  Google Scholar

[3]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, John Wiley & Sons, 2000.  Google Scholar

[4]

F. A. Chalub and M. O. Souza, The frequency-dependent Wright-Fisher model: Diffusive and non-diffusive approximations, J. Math. Biol., 68 (2014), 1089-1133.  doi: 10.1007/s00285-013-0657-7.  Google Scholar

[5]

F. A. Chalub and M. O. Souza, On the stochastic evolution of finite populations, J. Math. Biol., 75 (2017), 1735-1774.  doi: 10.1007/s00285-017-1135-4.  Google Scholar

[6]

D. Z. ChildsC. J. E. Metcalf and M. Rees, Evolutionary bet-hedging in the real world: Empirical evidence and challenges revealed by plants, Proc. Roy. Soc. B: Biol. Sci., 277 (2010), 3055-3064.  doi: 10.1098/rspb.2010.0707.  Google Scholar

[7]

D. Cohen, Optimizing reproduction in a randomly varying environment, J. Theoret. Biol., 12 (1966), 119-129.  doi: 10.1016/0022-5193(66)90188-3.  Google Scholar

[8]

W. S. Cooper and R. H. Kaplan, Adaptive "coin-flipping": A decision-theoretic examination of natural selection for random individual variation, J. Theoret. Biol., 94 (1982), 135-151.  doi: 10.1016/0022-5193(82)90336-8.  Google Scholar

[9]

P. Czuppon and A. Traulsen, Fixation probabilities in populations under demographic fluctuations, J. Math. Biol., 77 (2018), 1233-1277.  doi: 10.1007/s00285-018-1251-9.  Google Scholar

[10]

R. Durrett, Probability Models for DNA Sequence Evolution, Springer Science & Business Media, 2008. doi: 10.1007/978-0-387-78168-6.  Google Scholar

[11]

S. N. EvansA. Hening and S. J. Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J. Math. Biol., 71 (2015), 325-359.  doi: 10.1007/s00285-014-0824-5.  Google Scholar

[12]

W. J. Ewens, Mathematical Population Genetics. I. Theoretical Introduction, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21822-9.  Google Scholar

[13]

R. A. Fisher, On the dominance ratio, Proc. Roy. Soc. Edinburgh, 42 (1923), 321-341.   Google Scholar

[14]

G. B. FogelP. C. Andrews and D. B. Fogel, On the instability of evolutionary stable strategies in small populations, Ecological Modelling, 109 (1998), 283-294.  doi: 10.1016/S0304-3800(98)00068-4.  Google Scholar

[15]

S. A. Frank and M. Slatkin, Evolution in a variable environment, The American Naturalist, 136 (1990), 244-260.  doi: 10.1086/285094.  Google Scholar

[16]

D. FudenbergM. A. NowakC. Taylor and L. A. Imhof, Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoret. Popul. Biol., 70 (2006), 352-363.  doi: 10.1016/j.tpb.2006.07.006.  Google Scholar

[17]

J. H. Gillespie, Natural selection for within-generation variance in offspring number, Genetics, 76 (1974), 601-606.   Google Scholar

[18]

C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations, J. Theoret. Biol., 299 (2012), 106-112.  doi: 10.1016/j.jtbi.2011.06.010.  Google Scholar

[19]

J. Hofbauer and W. H. Sandholm, Evolution in games with randomly disturbed payoffs, J. Econ. Theory, 132 (2007), 47-69.  doi: 10.1016/j.jet.2005.05.011.  Google Scholar

[20] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[21]

L. A. Imhof and M. A. Nowak, Evolutionary game dynamics in a wright-fisher process, J. Math. Biol., 52 (2006), 667-681.  doi: 10.1007/s00285-005-0369-8.  Google Scholar

[22]

M. Kandori, G. J. Mailath and R. Rob, Learning, mutation, and long run equilibria in games, Econometrica: Journal of the Econometric Society, 29-56. doi: 10.2307/2951777.  Google Scholar

[23] S. Karlin, A First Course in Stochastic Processes, Academic press, 2014.   Google Scholar
[24]

E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs, Nature, 433 (2005), 312. doi: 10.1038/nature03204.  Google Scholar

[25]

N. Masuda, Directionality of contact networks suppresses selection pressure in evolutionary dynamics, J. Theoret. Biol., 258 (2009), 323-334.  doi: 10.1016/j.jtbi.2009.01.025.  Google Scholar

[26]

A. McAvoyN. FraimanC. HauertJ. Wakeley and M. A. Nowak, Public goods games in populations with fluctuating size, Theoret. Popul. Biol., 121 (2018), 72-84.  doi: 10.1016/j.tpb.2018.01.004.  Google Scholar

[27]

M. Mesterton-Gibbons and T. N. Sherratt, Information, variance and cooperation: Minimal models, Dynamic Games and Applications, 1 (2011), 419-439.  doi: 10.1007/s13235-011-0017-4.  Google Scholar

[28] P. A. P. Moran, The Statistical Process of Evolutionary Theory, Clarendon Press, Oxford, 1962.   Google Scholar
[29]

K. Nishimura and D. Stephens, Iterated prisoner's dilemma: Pay-off variance, J. Theoret. Biol., 188 (1997), 1-10.  doi: 10.1006/jtbi.1997.0439.  Google Scholar

[30]

M. A. Nowak, A. Sasaki, C. Taylor and D. Fudenberg, Emergence of cooperation and evolutionary stability in finite populations, Nature, 428 (2004), 646. doi: 10.1038/nature02414.  Google Scholar

[31]

H. OlofssonJ. Ripa and N. Jonzén, Bet-hedging as an evolutionary game: The trade-off between egg size and number, Proc. Roy. Soc. B: Biol. Sci., 276 (2009), 2963-2969.  doi: 10.1098/rspb.2009.0500.  Google Scholar

[32]

H. J. Park, Y. Pichugin, W. Huang and A. Traulsen, Population size changes and extinction risk of populations driven by mutant interactors, Phys. Rev. E, 99 (2019), 022305. doi: 10.1103/PhysRevE.99.022305.  Google Scholar

[33]

K. Pattni, Evolution in Finite Structured Populations with Group Interactions, Ph.D thesis, City, University of London, 2017. Google Scholar

[34]

K. PattniM. Broom and J. Rychtář, Evolving multiplayer networks: Modelling the evolution of cooperation in a mobile population, Discrete Contin. Dyn. Syst. B, 23 (2018), 1975-2004.  doi: 10.3934/dcdsb.2018191.  Google Scholar

[35]

K. Pattni, M. Broom, J. Rychtář and L. J. Silvers, Evolutionary graph theory revisited: When is an evolutionary process equivalent to the moran process?, Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 471 (2015), 20150334. doi: 10.1098/rspa.2015.0334.  Google Scholar

[36]

T. Philippi and J. Seger, Hedging one's evolutionary bets, revisited, Trends in Ecology & Evolution, 4 (1989), 41-44.  doi: 10.1016/0169-5347(89)90138-9.  Google Scholar

[37]

S. H. Rice, The expected value of the ratio of correlated random variables, https://www.depts.ttu.edu/biology/people/Faculty/Rice/home/ratio-derive.pdf, 2015 Google Scholar

[38]

S. H. Rice and A. Papadopoulos, Evolution with stochastic fitness and stochastic migration, PloS One, 4. doi: 10.1371/journal.pone.0007130.  Google Scholar

[39]

J. RipaH. Olofsson and N. Jonzén, What is bet-hedging, really?, Proc. Roy. Soc. B: Biol. Sci., 277 (2009), 1153-1154.  doi: 10.1098/rspb.2009.2023.  Google Scholar

[40]

M. E. Schaffer, Evolutionarily stable strategies for a finite population and a variable contest size, J. Theoret. Biol., 132 (1988), 469-478.  doi: 10.1016/S0022-5193(88)80085-7.  Google Scholar

[41]

P. H. Schimit, K. Pattni and M. Broom, Dynamics of multiplayer games on complex networks using territorial interactions, Phys. Rev. E, 99 (2019), 032306. doi: 10.1103/PhysRevE.99.032306.  Google Scholar

[42]

S. J. Schreiber, The evolution of patch selection in stochastic environments, The American Naturalist, 180 (2012), 17-34.  doi: 10.1086/665655.  Google Scholar

[43]

S. J. Schreiber, Unifying within-and between-generation bet-hedging theories: An ode to J.H. Gillespie, The American Naturalist, 186 (2015), 792-796.  doi: 10.1086/683657.  Google Scholar

[44]

J. Seger and H. Brockmann, Oxford surveys in evolutionary biology, Oxford Surveys in Evolutionary Biology, 4 (1987), 182-211.   Google Scholar

[45]

H. Seltman, Approximations for mean and variance of a ratio, http://www.stat.cmu.edu/ hseltman/files/ratio.pdf. Google Scholar

[46]

J. Starrfelt and H. Kokko, Bet-hedging - a triple trade-off between means, variances and correlations, Biol. Rev., 87 (2012), 742-755.  doi: 10.1111/j.1469-185X.2012.00225.x.  Google Scholar

[47]

C. TaylorD. FudenbergA. Sasaki and M. A. Nowak, Evolutionary game dynamics in finite populations, Bull. Math. Biol., 66 (2004), 1621-1644.  doi: 10.1016/j.bulm.2004.03.004.  Google Scholar

[48]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[49]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, Rev. Nonlin. Dyn. Complex., 2 (2009), 25-61.   Google Scholar

[50]

A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation, Phys. Rev. E, 74 (2006), 011909. doi: 10.1103/PhysRevE.74.011909.  Google Scholar

[51]

A. TraulsenM. A. Nowak and J. M. Pacheco, Stochastic payoff evaluation increases the temperature of selection, J. Theoret. Biol., 244 (2007), 349-356.  doi: 10.1016/j.jtbi.2006.08.008.  Google Scholar

[52]

C. Wallace and H. P. Young, Stochastic evolutionary game dynamics, in Handbook of Game Theory with Economic Applications, vol. 4, Elsevier, 2015, 327-380. doi: 10.1016/B978-0-444-53766-9.00006-9.  Google Scholar

[53]

S. Wright, Evolution in Mendelian populations, Genetics, 16 (1931), 97. Google Scholar

Figure 1.  Fixation probabilities for the Wright-Fisher process (left) and the Moran process (right). Darker areas correspond to higher fixation probabilities; $ 0.1 $ corresponds to a random drift. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 11 $ that have a fitness $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness of $ G_i $ is $ 2 $, the variance is $ i-1 $. For every pair of $ i,j $, we run $ 10^5 $ simulations starting with a single $ G_i $ individual among $ G_j $ individuals. Note that the individuals' fitness is sometimes more than double the expected value
Figure 2.  Left: A graphical representation of the genotypes $ G_p $ for $ p\in\{0.1, 0.2, \ldots, 0.9\} $. A fitness of $ G_p $ is $ \mu-x_p $ with probability $ p $ and $ \mu+y_p $ with probability $ 1-p $ where $ \mu = 4 $, $ x_p = (1-p)y_p/p $ and $ y_p = \sqrt{((1-p)^2/p + (1-p))^{-1}} $. The average fitness of $ G_i $ is $ \mu $ represented by the horizontal dotted line, the variance is $ 1 $. The genotypes are color coded by $ p $. The center of a disc corresponds to the fitness, the area of a disc corresponds to the probability of attaining such a fitness. The thick black curve is the third central moment $ s_p $ of the genotype $ G_p $ with values of the right $ y $ axis. Right: Fixation probabilities for the Moran process. The darker the color, the larger the fixation probability; $ 0.2 $ corresponds to a random drift. For $ N = 5 $ and every pair of $ p_i,p_j $, we run $ 10^7 $ simulations starting with a single $ G_{p_i} $ individual among $ G_{p_j} $ individuals
Figure 3.  Fixation probabilities for the Db-Moran process. At every step, an individual is selected to die with the probability inversely proportional to their fitness and is then replaced by a copy of a randomly selected remaining individuals. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 11 $ as in Figure 1; a fitness of $ G_i $ is $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness of $ G_i $ is $ 2 $, the variance is $ i-1 $. For every pair of $ i,j $, we run $ 10^5 $ simulations starting with a single $ G_i $ individual among $ G_j $ individuals
Figure 4.  Errors in approximation of $ 1/y $ by Taylor polynomials $ P_n(y) $ at $ y_0 $. The approximation by a higher degree polynomial is better only on $ (0,2y_0) $ and worse on $ (2y_0,\infty) $
Figure 5.  The mean square errors (MSE) of the estimate of $ \mathbb{E}[\pi_A'|\pi_A]-\pi_A $ by $ E_2 $ (left) and by $ E_3 $ (right). The darker the color, the larger the error. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 21 $; a fitness of genotype $ G_i $ is $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness is $ 2 $, the variance is $ i-1 $. For every $ \pi_A\in\{1/N, \ldots, (N-1)/N\} $, we run $ 10^6 $ simulations to estimate $ \mathbb{E}[\pi_A'|\pi_A]-\pi_A $ numerically by the average different $ \widehat{E} $. We then calculated the MSE as $ \frac{1}{N-1}\sum_{\pi_A\in\left \{\frac1N, \ldots, \frac{N-1}{N}\right\}} \left(\widehat{E}-E_2\right)^2 $
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