Bulletin of Mathematical Biology

, Volume 79, Issue 9, pp 2132–2148 | Cite as

Modelling the Evolution of Traits in a Two-Sex Population, with an Application to Grandmothering

  • Matthew H. ChanEmail author
  • Kristen Hawkes
  • Peter S. Kim
Original Article


We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness trade-off associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price equation to include two sexes, age structure and multiple traits. We use this formula to examine the effect of grandmothering, whereby post-fertile females subsidise their daughter’s fertility by provisioning grandchildren. Grandmothering can drive a shift towards increasingly male-biased mating sex ratios due to a post-fertile life stage in females, while male fertility continues to older ages. Our fitness landscapes show a net increase in fitness for both males and females at longer lifespans, and as a result, we find that grandmothering alone provides an evolutionary trajectory to higher longevities.


Population dynamics Evolutionary dynamics Sexual conflict Grandmother hypothesis Human evolution 



MHC and PSK were supported by the Australian Research Council, Discovery Project (DP160101597).


  1. Batty CJ, Crewe P, Grafen A, Gratwick R (2014) Foundations of a mathematical theory of darwinism. J Math Biol 69(2):295–334MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chan MH, Hawkes K, Kim PS (2016) Evolution of longevity, age at last birth and sexual conflict with grandmothering. J Theor Biol 393:145–157CrossRefzbMATHGoogle Scholar
  3. Chapman T, Arnqvist G, Bangham J, Rowe L (2003) Sexual conflict. Trends Ecol Evol 18(1):41–47CrossRefGoogle Scholar
  4. Charnov EL (1993) Life history invariants: some explorations of symmetry in evolutionary ecology. Oxford University Press, OxfordGoogle Scholar
  5. Coxworth JE, Kim PS, McQueen JS, Hawkes K (2015) Grandmothering life histories and human pair bonding. Proc Natl Acad Sci 112(38):11806–11811CrossRefGoogle Scholar
  6. Frank SA (2012) Natural selection. IV. The Price equation. J Evol Biol 25(6):1002–1019CrossRefGoogle Scholar
  7. Grafen A (2015) Biological fitness and the Price Equation in class-structured populations. J Theor Biol 373:62–72MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hawkes K, Coxworth JE (2013) Grandmothers and the evolution of human longevity: a review of findings and future directions. Evol Anthropol 22(6):294–302CrossRefGoogle Scholar
  9. Hawkes K, Kim PS, Kennedy B, Bohlender R, Hawks J (2011) A reappraisal of grandmothering and natural selection. Proc R Soc Lond B Biol Sci 278(1714):1936–1938CrossRefGoogle Scholar
  10. Hawkes K, O’Connell JF, Blurton Jones NG, Alvarez H, Charnov EL (1998) Grandmothering, menopause, and the evolution of human life histories. Proc Natl Acad Sci 95(3):1336–1339CrossRefGoogle Scholar
  11. Hawkes K, Smith KR, Robson SL (2009) Mortality and fertility rates in humans and chimpanzees: how within-species variation complicates cross-species comparisons. Am J Hum Biol 21(4):578–586CrossRefGoogle Scholar
  12. Kachel AF, Premo LS, Hublin JJ (2011) Grandmothering and natural selection. Proc Biol Sci 278(1704):384–391CrossRefGoogle Scholar
  13. Kim PS, Coxworth JE, Hawkes K (2012) Increased longevity evolves from grandmothering. Proc Biol Sci 279(1749):4880–4884CrossRefGoogle Scholar
  14. Kim PS, McQueen JS, Coxworth JE, Hawkes K (2014) Grandmothering drives the evolution of longevity in a probabilistic model. J Theor Biol 353:84–94CrossRefGoogle Scholar
  15. Loo SL, Chan MH, Hawkes K, Kim PS (2017) Further mathematical modelling of mating sex ratios & male strategies with special relevance to human life history. Bull Math Biol. doi: 10.1007/s11538-017-0313-2 MathSciNetGoogle Scholar
  16. Mineau G, Bean L, Skolnick M (1979) Mormon demographic history II: the family life cycle and natural fertility. Population Studies 33(3):429–446CrossRefGoogle Scholar
  17. Parker GA (2006) Sexual conflict over mating and fertilization: an overview. Philos Trans R Soc B Biol Sci 361(1466):235–259CrossRefGoogle Scholar
  18. Pavard S, Branger F (2012) Effect of maternal and grandmaternal care on population dynamics and human life-history evolution: a matrix projection model. Theor Popul Biol 82(4):364–376CrossRefzbMATHGoogle Scholar
  19. Price GR (1970) Selection and covariance. Nature 227(5257):520–521CrossRefGoogle Scholar
  20. Price GR (1972) Extension of covariance selection mathematics. Ann Hum Genet 35(4):485–490CrossRefzbMATHGoogle Scholar
  21. Schacht R, Bell AV (2016) The evolution of monogamy in response to partner scarcity. Sci Rep 6:32472CrossRefGoogle Scholar
  22. Süli E, Mayers DF (2003) An introduction to numerical analysis. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  23. te Velde ER, Pearson PL (2002) The variability of female reproductive ageing. Hum Reprod Update 8(2):141–154CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  • Matthew H. Chan
    • 1
    Email author
  • Kristen Hawkes
    • 2
  • Peter S. Kim
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  2. 2.Department of AnthropologyUniversity of UtahSalt Lake CityUSA

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