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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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |





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