Adaptive Dynamics: Branching Phenomena and the Canonical Equation

Abstract

In this paper we provide an introduction to the field of Adaptive Dynamics. We present derivations for two of the fundamental components of the theory: “canonical equation ” and the classification of singular strategies. We supplement the existing theory with a derivation of the variance associated with the canonical equation. We then consider a common ecological model (an instance of the logistic equation) that has been used to explore branching in the context of Adaptive Dynamics. We show that the branching properties of this model are maintained in a much more general form of which the familiar example is a particular instance. We then determine the expected evolutionary trajectory of a population in this model using the canonical equation, and find the associated variance of this trajectory. We also examine the stability of the dimorphic, branched population. Having determined each of these components analytically, we then confirm these predictions by implementing a model