The canonical equation of adaptive dynamics in individual-based models with power law mutation rates

TL;DR

This paper derives the canonical equation of adaptive dynamics from individual-based models with power law mutation rates, analyzing its behavior in one and higher-dimensional trait spaces and providing explicit trait evolution paths.

Contribution

It extends the canonical equation framework to models with power law mutation probabilities and offers explicit trait evolution paths in higher dimensions.

Findings

In one-dimensional trait space, results align with established theory.

Mutation restrictions slow the evolution speed in higher dimensions.

Explicit paths for dominant traits are derived without solving the canonical equation.

Abstract

In this paper, we consider an individual-based model with power law mutation probability. In this setting, we use the large population limit with a subsequent ``small mutations'' limit to derive the canonical equation of adaptive dynamics. For a one-dimensional trait space this corresponds to well established results and we can formulate a criterion for evolutionary branching in the spirit of Champagnat and M\'el\'eard (2011). In higher dimensional trait spaces, we find that the speed at which the solution of the canonical equation moves through space is reduced due to mutations being restricted to the underlying grid on the trait space. However, as opposed to the canonical equation with rare mutations, we can explicitly calculate the path which the dominant trait will take without having to solve the equation itself.