Journal of Mathematical Biology

, Volume 53, Issue 1, pp 15–60 | Cite as

A theory of Fisher's reproductive value



The formal Darwinism project aims to provide a mathematically rigorous basis for optimisation thinking in relation to natural selection. This paper deals with the situation in which individuals in a population belong to classes, such as sexes, or size and/or age classes. Fisher introduced the concept of reproductive value into biology to help analyse evolutionary processes of populations divided into classes. Here a rigorously defined and very general structure justifies, and shows the unity of concept behind, Fisher's uses of reproductive value as measuring the significance for evolutionary processes of (i) an individual and (ii) a class; (iii) recursively, as calculable for a parent as a sum of its shares in the reproductive values of its offspring; and (iv) as an evolutionary maximand under natural selection. The maximand is the same for all parental classes, and is a weighted sum of offspring numbers, which implies that a tradeoff in one aspect of the phenotype can legitimately be studied separately from other aspects. The Price equation, measure theory, Markov theory and positive operators contribute to the framework, which is then applied to a number of examples, including a new and fully rigorous version of Fisher's sex ratio argument. Classes may be discrete (e.g. sex), continuous (e.g. weight at fledging) or multidimensional with discrete and continuous components (e.g. sex and weight at fledging and adult tarsus length).

Key words or phrases

Reproductive Value R.A. Fisher Class-structured population Natural Selection Formal Darwinism Optimisation 


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  1. 1.
    Anderson, E.J., Nash, P.: Linear programming in infinite-dimensional spaces: theory and applications. Wiley, Chichester, 1987Google Scholar
  2. 2.
    Ash, R.B., Doleans-Dade, C.A.: Probability and measure theory. Academic Press, San Diego, CA 2000Google Scholar
  3. 3.
    Bishop, D.T., Cannings, C.: A generalized war of attrition. J. Theor. Biol. 70, 85–125 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boomsma, J.J., Grafen, A.: Colony-level sex ratio selection in the eusocial Hymenoptera. J. Evol. Biol. 4, 383–407 (1991)CrossRefGoogle Scholar
  5. 5.
    Caswell, H.: Matrix population models: construction, analysis and interpretation. Sinauer, Sunderland, Massachusetts, 1989Google Scholar
  6. 6.
    Charlesworth, B.: Evolution in age-structured populations. Cambridge University Press, 1994Google Scholar
  7. 7.
    Darwin, C.R.: The descent of man and selection in relation to sex. John Murray, London, 1871Google Scholar
  8. 8.
    Easterling, M.R., Ellner, S.P., Dixon, P.M.: Size-specific sensitivity: Applying a new structured population model. Ecology 81, 694–708 (2000)CrossRefGoogle Scholar
  9. 9.
    Edwards, A.W.F.: Natural selection and the sex ratio: Fisher's sources. Am. Natur. 151, 564–569 (1998)CrossRefGoogle Scholar
  10. 10.
    Edwards, A.W.F.: Carl Düsing on the regulation of the sex-ratio. Theor. Population Biol. 58, 255–257 (2000)CrossRefGoogle Scholar
  11. 11.
    Fisher, R.A.: The genetical theory of natural selection. Oxford University Press. (OUP published in 1999 a variorum edition of the 1930 and 1958 editions) 1930Google Scholar
  12. 12.
    Fretwell, S.: Populations in a seasonal environment. Princeton University Press, 1972Google Scholar
  13. 13.
    Grafen, A.: Split sex ratios and the evolutionary origins of eusociality. J. Theor. Biol. 122, 95–121 (1986)Google Scholar
  14. 14.
    Grafen, A.: The logic of divisively asymmetric contests: respect for ownership and the desperado effect. Animal Behaviour 35, 462–467 (1987)CrossRefGoogle Scholar
  15. 15.
    Grafen, A.: Sexual selection unhandicapped by the Fisher process. J. Theor. Biol. 144, 475–518 (1990)Google Scholar
  16. 16.
    Grafen, A.: Fertility and labour supply in Femina economica. J. Theor. Biol. 194, 429–455 (1998)CrossRefGoogle Scholar
  17. 17.
    Grafen, A.: Formal Darwinism, the individual-as-maximising-agent analogy, and bet-hedging. Proceedings of the Royal Society, Series B 266, 799–803 (1999)CrossRefGoogle Scholar
  18. 18.
    Grafen, A.: Developments of Price's equation and natural selection under uncertainty. Proceedings of the Royal Society, Series B 267, 1223–1227 (2000)CrossRefGoogle Scholar
  19. 19.
    Grafen, A.: A first formal link between the Price equation and an optimisation program. J. Theor. Biol. 217, 75–91 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grafen, A.: Optimisation of inclusive fitness. J. Theor. Biol. 238, 541–563 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hamilton, W.D.: The genetical evolution of social behaviour. J. Theor. Biol. 7, 1–52 (1964)CrossRefGoogle Scholar
  22. 22.
    Hamilton, W.D.: Selfish and spiteful behaviour in an evolutionary model. Nature 228, 1218–1220 (1970)CrossRefGoogle Scholar
  23. 23.
    Kanwal, R.P.: Generalized functions: theory and technique. Academic Press. Volume 171 of Mathematics in Science and Engineering, 1983Google Scholar
  24. 24.
    Krasnosel'skii, M.: Positive solutions of operator equations. P. Noordhoff Ltd, Groningen, The Netherlands. Translated from the Russian by Richard. E. Flaherty, 1964Google Scholar
  25. 25.
    Leimar, O.: Life-history analysis of the Trivers and Willard sex-ratio problem. Behaviour Ecology 7, 316–325 (1996)Google Scholar
  26. 26.
    Luenberger, D.G.: Optimization by vector space methods. Wiley, New York 1997Google Scholar
  27. 27.
    Moran, P.A.P.: On the nonexistence of adaptive topographies. Annals of Human Genetics 27, 383–393 (1964)MATHCrossRefGoogle Scholar
  28. 28.
    Phelps, R.R.: Lectures on Choquet's theorem. Springer, New York, USA 2001Google Scholar
  29. 29.
    Price, G.R.: Selection and covariance. Nature 227, 520–521 (1970)CrossRefGoogle Scholar
  30. 30.
    Price, G.R.: Extension of covariance selection mathematics. Ann. Human Genetics 35, 485–490 (1972)MATHCrossRefGoogle Scholar
  31. 31.
    Rosenblatt, M.: Markov processes: structure and asymptotic behavior. Springer-Verlag, New York, 1971Google Scholar
  32. 32.
    Schechter, E.: Handbook of analysis and its foundations. Academic Press, 1997Google Scholar
  33. 33.
    Seger, J., Stubblefield, J.W.: Models of sex ratio evolution. In: Hardy, I.C.W. (ed) Sex ratios: concepts and research methods. Chap. 1. Cambridge University Press, Cambridge, pp. 2–25 (2002)Google Scholar
  34. 34.
    Taylor, P.D.: Allele-frequency change in a class-structured population. Am. Natur. 135, 95–106 (1990)CrossRefGoogle Scholar
  35. 35.
    Taylor, P.D.: Inclusive fitness arguments in genetic models of behaviour. J. Math. Biol. 34, 654–674 (1996)MATHGoogle Scholar
  36. 36.
    Trivers, R.L., Willard, D.E.: Natural selection of parental ability to vary the sex of offspring. Science 179, 90–92 (1973)Google Scholar
  37. 37.
    Vilenkin, N.Y., Gorin, E. A., Kostyuchenko, A. G., Krasnosel'skii, M. A., Krein, S.G. (Editor), Maslov, V.P., Mityagin, B.S., Petunin, Y.I., Rutitski, Y.B., Sobolev, V.I., Stetsenko, V.Y., Faddeev, L.D., Tsitlanadze, E.S. Functional analysis. Groningen: Wolters-Noordhoff. Translated from the Russian by Richard E. Flaherty, 1972Google Scholar
  38. 38.
    Williams, K., Simon, C.: The ecology, behavior, and evolution of periodical cicadas. Ann. Rev. Entomol. 40, 269–295 (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.St John's CollegeOxfordUnited Kingdom

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