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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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[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.

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

[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.

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

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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

[38]

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

[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.

[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.

[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.

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

[53]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[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.

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

[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.

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

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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

[38]

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

[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.

[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.

[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.

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

[53]

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

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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