Journal of Mathematical Biology

, Volume 69, Issue 2, pp 295–334

Foundations of a mathematical theory of darwinism


  • Charles J. K. Batty
    • St John’s College
  • Paul Crewe
    • St John’s College
  • Alan Grafen
    • St John’s College
    • St John’s College

DOI: 10.1007/s00285-013-0706-2

Cite this article as:
Batty, C.J.K., Crewe, P., Grafen, A. et al. J. Math. Biol. (2014) 69: 295. doi:10.1007/s00285-013-0706-2


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.


Formal darwinismReproductive valueFitness maximization Price equation

Mathematics Subject Classification (2000)


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© Springer-Verlag Berlin Heidelberg 2013