Journal of Mathematical Biology

, Volume 69, Issue 2, pp 295-334

First online:

Foundations of a mathematical theory of darwinism

  • Charles J. K. BattyAffiliated withSt John’s College
  • , Paul CreweAffiliated withSt John’s College
  • , Alan GrafenAffiliated withSt John’s College
  • , Richard GratwickAffiliated withSt John’s College Email author 

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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 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 darwinism Reproductive value Fitness maximization Price equation

Mathematics Subject Classification (2000)

28B99 49N99 60J99 92D15