*Demographic and Environmental Stochasticity in Population Processes*

**Abstract**

**Hans Metz:**

*Adaptive Dynamics*

**The canonical equation of adaptive dynamics for Mendelian diploids and haplo-diploids**

**Adaptive Dynamics**

**Slide set 1**

**Slide set 2**

Abstract: Adaptive dynamics (AD) is a mathematical framework for dealing with meso-evolution, seen as resulting from the continual ecology-driven substitution of new mutants. The core of AD is based on the following simplifying assumptions: clonal reproduction, rare mutations, smallish mutational effects, smoothness of the demographic parameters in the traits, and well-behaved community attractors. (That populations are largish is implicit in the focus on meso-evolution: on those time scales smaller populations go extinct.) However, often the results from AD models turn out to apply also under far less restrictive conditions.

The concept at the basis of AD is "fitness", defined as the asymptotic average relative rate of increase of a population in an ergodic environment. (The concept of environment is supposed to imply that given the course of the environment the individuals move through their state spaces and reproduce independently.) Fitness thus depends on two variables, the traits of the individuals, and the environment. In the case of a mutant the environment is set by the current resident types, thus making fitness a function of the mutant and resident trait vectors. Meso-evolutionary change can be analyzed largely in terms of the geometry of this function.

The first lecture will develop some of the geometrical AD tools for the case of scalar traits. The second lecture will consider an approximating differential equation for the trait vector of a focal type, derived on the assumption that mutational steps are small, and work out the unexpected equality of an effective population size that crops up in its derivation to the effective population size of population genetics.