American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020242

Moran process and Wright-Fisher process favor low variability

Jan Rychtář , and  Dewey T. Taylor

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  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.

Keywords: Wright-Fisher process, Moran process, birth-death process, death-birth process, finite population, frequency-independent model.
Mathematics Subject Classification: Primary: 92D15, 92D25, 60J80.
Citation: Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 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. Childs, C. 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

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

[14]

G. B. Fogel, P. 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. Fudenberg, M. A. Nowak, C. 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. McAvoy, N. Fraiman, C. Hauert, J. 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. Olofsson, J. 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. Pattni, M. 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. Ripa, H. 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. Taylor, D. Fudenberg, A. 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. Traulsen, M. 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. Childs, C. 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. Evans, A. 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. Fogel, P. 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. Fudenberg, M. A. Nowak, C. 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. McAvoy, N. Fraiman, C. Hauert, J. 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. Olofsson, J. 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. Pattni, M. 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. Ripa, H. 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. Taylor, D. Fudenberg, A. 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. Traulsen, M. 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
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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
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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
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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) $
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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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