The Canonical Equation of Adaptive Dynamics: A Mathematical View

The Darwinian evolution of a quantitative adaptive character is described as a jump process. As the variance of the distribution of mutation steps goes to zero, this process converges in law to the solution of an ordinary differential equation. In the case where the mutation step distribution is symmetrical, this establishes rigorously the socalled canonical equation first proposed by Dieckmann and Law (1996). Our mathematical approach naturally leads to extend the canonical equation to the case of biased mutations, and to seek ecological and genetic conditions under which evolution proceeds either through punctualism or through radiation.