Foundations of a mathematical theory of darwinism
 Charles J. K. Batty,
 Paul Crewe,
 Alan Grafen,
 Richard Gratwick
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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.
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 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
 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