# Population Structure

## Wahlund effect

Mixing two populations without reproduction causes there to be a deficiency of heterozygotes.

This reduction in the number of heterozygotes is proportional to 2pqFST.

The two-locus Wahlund effect includes linkage disequilibrium.

## F-statistics

### FST is a standardized measure of the genetic variance among populations.

where Var(p) is the variance among populations in the allele frequency p of some allele, and the bar above the p represents the mean of p.

### F-statistics can be defined hierarchically

FIS is the same as the simple inbreeding coefficient of a sub-population.

F-statistics can be defined by ANOVA models (see Weir and Cockerham [1984])

## Island Model

### Definitions

#### FST is often related to the number of migrants per generation by the formula

where N is the local population size and m is the migration rate among populations.

#### This formula depends on the assumptions of the ISLAND MODEL

The assumptions of the island model:

All populations are created equal, with N individuals and equal contributions to the migrant pool

There is NO spatial structure: in effect all populations are equally close to all other populations (no isolation by distance)

Everything is at equilibrium, nothing is changing.

No selection

no mutation

Derivations

Setting F'ST equal to FST:

when m is small.

## Why do we care?

### BUT..... there are two problems:

* The real world is not like the island model

* Even the island model is not always like the island model: statistical problems

## The real world is not like the Island model-- take the assumptions one by one:

* All populations are created equal, with N individuals and equal contributions to the migrant pool -

=> Population sizes are extremely variable, both in space and time

=> Populations are variable in their contributions to the migrant pool (e.g. sources and sinks)

=> populations vary through time in migration rates

=> populations fission and fuse

* There is NO spatial structure: in effect all populations are equally close to all other populations (no isolation by distance)

=> Dispersal is almost always distance related; there is isolation by distance

=> Dispersal is also often affected by other factors: rivers, roads, mountains, etc.

* Everything is at equilibrium, nothing is changing.

=> Populations often go extinct, and new ones form by colonization

=> History matters -- often the circumstances which determine the current population structure are the conditions of the past, which may have changed

=> There may be migration in from outside the study system, changing allele frequencies over time

* No selection

=> There's ALWAYS selection

* no mutation

=> Mutation can be at very fast rates, for example in microsatellites

## The island model is not always like the island model

* For mitochondrial markers (or others inherited uniparentally) FST = 1/(2Nm+1)

* The statistical properties of FST are not well worked out, but they're ugly - see the figures

* Dispersal rates for genetic purposes are often quite different than what is needed fro ecological studies

=> Genetic dispersal only counts if the migrants reproduce effectively

=> Genetic dispersal only counts if the reproduction of migrant individuals is equal to resident individuals (i.e., migrants have to move before their reproductive life starts)

=> Selection can over-amplify migrant genetic contributions

* Problems of scale : Genetic analysis only tells you about migration at the geographical scale at which the samples are drawn from.