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.
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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 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 $
[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.
![]() |
Fixation probabilities for the Wright-Fisher process (left) and the Moran process (right). Darker areas correspond to higher fixation probabilities;
Left: A graphical representation of the genotypes
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
Errors in approximation of
The mean square errors (MSE) of the estimate of