## Effective population sizes and the canonical equation of adaptive dynamics

### Johan Metz (October 24, 2011)

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### Abstract

Deterministic population dynamical models connect to reality through their interpretation as limits for systems size going to infinity of stochastic processes in which individuals are represented as discrete entities. In structured population models individuals may be born in different states (e.g. locations in space) after which they proceed through their h(eterogeneity)-state space, e.g. spanned by their i(dividual)-state and location. On such models one can graft evolutionary processes like random genetic drift or adaptive evolution by rare repeated substitutions of mutants in heritable traits affecting the state transition and reproduction processes of individuals. From this general perspective I will derive the so-called Canonical Equation of adaptive dynamics, a differential equation for evolutionary trait change derived under the additional assumption that mutations have small effect. In the CE approximation the rate of evolution is found to correspond to the product of a parameter $n_{e,A}$, equal to the population size times a dimensionless product of life history parameters (including spatial movements), times the gradient of the invasion fitness of potential mutants with respect to their trait vector. From a heuristic connection with the diffusion approximation for genetic drift it follows that $n_{e,A} = n_{e,D}$, the effective population size from population genetics.