Models of Adaptive Dynamics: Their Derivations and Relations

Ulf Dieckmann

Adaptive Dynamics Network, International Institute for Applied Systems Analysis, Laxenburg, Austria

Abstract

A unifying framework is presented for describing the phenotypic coevolutionary dynamics of a general ecological community. We start from an individual-based approach allowing for the interaction of an arbitrary number of species. The adaptive dynamics of species' trait values is derived from the underlying population dynamics operating in the community; in consequence, the evolutionary process is driven by ecological change.

We present a hierarchy of four dynamical models for the investigation of coevolutionary systems. The necessity of stochastic treatment is demonstrated and deterministic approximations are derived where appropriate. The mathematical framework advanced here is the first to combine the individual-based, stochastic perspective with a fully dynamical analysis of the phenotypic coevolutionary process.

Deductions are given to derive various well-known equations from the literature of (co)evolutionary modeling as special cases of our approach. In particular, equations central to the fields of evolutionary game theory, adaptive dynamics (in the narrow sense), replicator dynamics and reaction- diffusion systems are recovered. In consequence, the different domains of validity for these models are delineated and several ad-hoc assumptions can be removed.