Foundations of a mathematical theory of darwinism
 Charles J. K. Batty,
 Paul Crewe,
 Alan Grafen,
 Richard Gratwick
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 classstructured population, with individuallevel 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.
Inside
Within this Article
 Introduction
 Biological motivation
 Notation and concepts
 Reproductive value and the Price equation
 Optimization
 Links between gene dynamics and optimization
 Interpretation, significance, and context of our results
 An example with an agestructured population
 References
 References
Other actions
 Allison AC (1954) Notes on sicklecell polymorphism. Ann Hum Genet 19:39–57 CrossRef
 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. WileyBlackwell, London
 Diestel J, Uhl JJ Jr (1977) Vector measures. American Mathematical Society, Providence CrossRef
 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 CrossRef
 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 CrossRef
 Ewens WJ (2004) Mathematical population genetics I. Theoretical introduction. Springer, Berlin CrossRef
 Ewens WJ (2011) What is the gene trying to do? Br J Philos Sci 62:155–176 CrossRef
 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 CrossRef
 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 CrossRef
 Frank SA (2011b) Natural selection. II. Developmental variability and evolutionary rate. J Evol Biol 24(11):2310–2320 CrossRef
 Frank SA (2012a) Natural selection. III. Selection versus transmission and the levels of selection. J Evol Biol 25(2):227–243 CrossRef
 Frank SA (2012b) Natural selection. IV. The Price equation. J Evol Biol 25(6):1002–1019 CrossRef
 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 CrossRef
 Frank SA (2013a) Natural selection. VI. Partitioning the information in fitness and characters by path analysis. J Evol Biol 26(3):457–471 CrossRef
 Frank SA (2013b) Natural selection. VII. History and interpretation of kin selection theory. J Evol Biol 26(6):1151–1184 CrossRef
 Frank SA, Slatkin M (1990) Evolution in a variable environment. Am Nat 136:244–260 CrossRef
 Frank SA, Slatkin M (1992) Fisher’s fundamental theorem of natural selection. Trends Ecol Evol 7:92–95 CrossRef
 Grafen A (2000) Developments of Price’s equation and natural selection under uncertainty. Proc R Soc Ser B 267:1223–1227 CrossRef
 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 CrossRef
 Halmos PR (1950) Measure theory. D. Van Nostrand Company, Inc., New York CrossRef
 Hamilton WD (1964) The genetical evolution of social behaviour. J Theor Biol 7:1–52 CrossRef
 Kechris AS (1995) Classical descriptive set theory. Graduate texts in mathematics, vol 156. Springer, New York CrossRef
 Kingman JFC, Taylor SJ (1966) Introduction to measure and probability. Cambridge University Press, London CrossRef
 Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33(3):183–212 CrossRef
 Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4):213–245 CrossRef
 Lessard S (1997) Fisher’s fundamental theorem of natural selection revisited. Theor Popul Biol 52:119–136 CrossRef
 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 CrossRef
 Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18 CrossRef
 Okasha S (2008) Fisher’s fundamental theorem of natural selection: a philosophical analysis. Br J Philos Sci 59:319–351 CrossRef
 Price GR (1970) Selection and covariance. Nature 227:520–521 CrossRef
 Price GR (1972a) Extension of covariance selection mathematics. Ann Hum Genet 35:485–490 CrossRef
 Price GR (1972b) Fisher’s ‘fundamental theorem’ made clear. Ann Hum Genet 36:129–140 CrossRef
 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 CrossRef
 Rudin W (1966) Real and complex analysis. McGrawHill Book Co., New York
 Taylor PD (1990) Allelefrequency change in a classstructured population. Am Nat 135:95–106 CrossRef
 Taylor PD (1996) Inclusive fitness arguments in genetic models of behaviour. J Math Biol 34:654–674 CrossRef
 Wagner DH (1977) Survey of measurable selection theorems. SIAM J Control Optim 15(5):859–903 CrossRef
 Title
 Foundations of a mathematical theory of darwinism
 Journal

Journal of Mathematical Biology
Volume 69, Issue 2 , pp 295334
 Cover Date
 20140801
 DOI
 10.1007/s0028501307062
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Formal darwinism
 Reproductive value
 Fitness maximization
 Price equation
 28B99
 49N99
 60J99
 92D15
 Industry Sectors
 Authors

 Charles J. K. Batty ^{(1)}
 Paul Crewe ^{(1)}
 Alan Grafen ^{(1)}
 Richard Gratwick ^{(1)}
 Author Affiliations

 1. St John’s College, Oxford, OX1 3JP, UK