Abstract
This paper pursues the ‘formal darwinism’ project of Grafen, whose aim is to construct formal links between dynamics of gene frequencies and optimization programmes, in very abstract settings with general implications for biologically relevant situations. A major outcome is the definition, within wide assumptions, of the ubiquitous but problematic concept of ‘fitness’. This paper is the first to present the project for mathematicians. Within the framework of overlapping generations in discrete time and no social interactions, the current model shows links between fitness maximization and gene frequency change in a class-structured population, with individual-level uncertainty but no uncertainty in the class projection operator, where individuals are permitted to observe and condition their behaviour on arbitrary parts of the uncertainty. The results hold with arbitrary numbers of loci and alleles, arbitrary dominance and epistasis, and make no assumptions about linkage, linkage disequilibrium or mating system. An explicit derivation is given of Fisher’s Fundamental Theorem of Natural Selection in its full generality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allison AC (1954) Notes on sickle-cell polymorphism. Ann Hum Genet 19:39–57
Billingsley P (1995) Probability and measure, 3rd edn. Wiley series in probability and mathematical statistics. Wiley, New York
Darwin CR (1859) The origin of species. John Murray, London
Davies NB, Krebs JR, West SA (2012) An introduction to behavioural ecology. Wiley-Blackwell, London
Diestel J, Uhl JJ Jr (1977) Vector measures. American Mathematical Society, Providence
Dunford N, Schwartz JT (1958) Linear operators. I. General theory. Interscience Publishers, Inc., New York
Edwards AWF (1994) The fundamental theorem of natural selection. Biol Rev 69:443–474
Ewens WJ (1979) Mathematical population genetics. Springer, Berlin
Ewens WJ (1989) An interpretation and proof of the fundamental theorem of natural selection. Theor Popul Biol 36:167–180
Ewens WJ (2004) Mathematical population genetics I. Theoretical introduction. Springer, Berlin
Ewens WJ (2011) What is the gene trying to do? Br J Philos Sci 62:155–176
Falconer DS (1981) Introduction to quantitative genetics, 2nd edn. Longman, London
Fisher RA (1930) The genetical theory of natural selection. Oxford University Press, Oxford [see Fisher (1999) for a version in print]
Fisher RA (1941) Average excess and average effect of a gene substitution. Ann Eugen 11:53–63
Fisher RA (1999) The genetical theory of natural selection. Oxford University Press, Oxford [a Variorum edition of the 1930 and 1958 editions, edited by J.H. Bennett]
Frank SA (2011a) Natural selection. I. Variable environments and uncertain returns on investment. J Evol Biol 24(11):2299–2309
Frank SA (2011b) Natural selection. II. Developmental variability and evolutionary rate. J Evol Biol 24(11):2310–2320
Frank SA (2012a) Natural selection. III. Selection versus transmission and the levels of selection. J Evol Biol 25(2):227–243
Frank SA (2012b) Natural selection. IV. The Price equation. J Evol Biol 25(6):1002–1019
Frank SA (2012c) Natural selection. V. How to read the fundamental equations of evolutionary change in terms of information theory. J Evol Biol 25(12):2377–2396
Frank SA (2013a) Natural selection. VI. Partitioning the information in fitness and characters by path analysis. J Evol Biol 26(3):457–471
Frank SA (2013b) Natural selection. VII. History and interpretation of kin selection theory. J Evol Biol 26(6):1151–1184
Frank SA, Slatkin M (1990) Evolution in a variable environment. Am Nat 136:244–260
Frank SA, Slatkin M (1992) Fisher’s fundamental theorem of natural selection. Trends Ecol Evol 7:92–95
Grafen A (2000) Developments of Price’s equation and natural selection under uncertainty. Proc R Soc Ser B 267:1223–1227
Grafen A (2002) A first formal link between the Price equation and an optimization program. J Theor Biol 217:75–91
Grafen A (2006a) Optimization of inclusive fitness. J Theor Biol 238:541–563
Grafen A (2006b) A theory of Fisher’s reproductive value. J Math Biol 53:15–60
Halmos PR (1950) Measure theory. D. Van Nostrand Company, Inc., New York
Hamilton WD (1964) The genetical evolution of social behaviour. J Theor Biol 7:1–52
Kechris AS (1995) Classical descriptive set theory. Graduate texts in mathematics, vol 156. Springer, New York
Kingman JFC, Taylor SJ (1966) Introduction to measure and probability. Cambridge University Press, London
Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33(3):183–212
Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4):213–245
Lessard S (1997) Fisher’s fundamental theorem of natural selection revisited. Theor Popul Biol 52:119–136
Lewis EG (1942) On the generation and growth of a population. Sankhyā Indian J Stat (1933–1960) 6(1):93–96
Lewontin RC (1974) The genetic basis of evolutionary change. Columbia University Press, New York
Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, Cambridge
Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18
Okasha S (2008) Fisher’s fundamental theorem of natural selection: a philosophical analysis. Br J Philos Sci 59:319–351
Price GR (1970) Selection and covariance. Nature 227:520–521
Price GR (1972a) Extension of covariance selection mathematics. Ann Hum Genet 35:485–490
Price GR (1972b) Fisher’s ‘fundamental theorem’ made clear. Ann Hum Genet 36:129–140
Rao MM (2004) Measure theory and integration, monographs and textbooks in pure and applied mathematics, vol 265, 2nd edn. Marcel Dekker Inc., New York
Rosenblatt M (1971) Markov processes, structure and asymptotic behavior. Springer, New York
Rudin W (1966) Real and complex analysis. McGraw-Hill Book Co., New York
Taylor PD (1990) Allele-frequency change in a class-structured population. Am Nat 135:95–106
Taylor PD (1996) Inclusive fitness arguments in genetic models of behaviour. J Math Biol 34:654–674
Wagner DH (1977) Survey of measurable selection theorems. SIAM J Control Optim 15(5):859–903
Acknowledgments
The authors wish to thank Alain Goriely and two anonymous referees for numerous helpful comments on earlier versions of this manuscript, and the participants of the reading seminar on Fisher (1999) held in the St John’s College Research Centre in 2012, in particular organizer Jean-Baptiste Grodwohl, for the opportunity to explore Fisher’s work and the topics central to this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is part of the ‘Formal Darwinism’ project funded by St John’s Research Centre, St John’s College, Oxford, grant to CJKB and AG.
Rights and permissions
About this article
Cite this article
Batty, C.J.K., Crewe, P., Grafen, A. et al. Foundations of a mathematical theory of darwinism. J. Math. Biol. 69, 295–334 (2014). https://doi.org/10.1007/s00285-013-0706-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-013-0706-2