# Correct migration rate expression in infinite island model

In Sewall Wright's infinite island model, where all demes exchange migrants each generation, I have seen the migration rate stated variously as $m$ or $\frac{m}{d - 1}$ (as in Matthew B. Hamilton's Population Genetics textbook, chapter 4, p. 132, figure 4.12), where d is the number of demes.

My question is, which expression for the migration rate is the correct one?

If it's the latter, then where does the "-1" come from?

Definition of $m$

There is no correct or wrong expression. It all depends what exactly you are interested in and what is the definition of m in a specific paper.

Def. 1

Some models assume that $m$ is the migration rate from a given population to another given population.

Def. 2

Some models assume that $m$ is the probability of migrating to any other population. In a two population case, this definition and the above are the same. In other case, the probability of migration to a given population is $\frac{m}{d-1}$

Def. 3

Some models assume a migration pool. With probability $m$ an individual is a migrant, goes to the migrant-pool and is then redistributed among all other populations (only works for island models). In such model, a migrant could be an individual that actually come back to the same population. So, the probability of really migrating is $\frac{m (d-1)}{d}$ and assuming equal migration rate among all populations, the probability to migrate to a given population is $\frac{m}{d-1}$.

Fst

If you're dealing with $F_{ST}$ in a finite island model, then you will have some $\frac{m d}{d-1}$ terms. I talk about it in this post but it is not complete, so you might want to read Slatkin (1991) and Charlesworth (1998)

• Thanks! Though this is a bit more challenging. I am building an R simulation where I have $K$ subpopulations and a migration rate of individuals. Initially, I just set the migration rate equal to $m$ (or $\frac{m}{2}$ for Kimura's linear stepping-stone model), but now I see that it should also depend on $K$. Nov 16 '17 at 20:07
• Well, it depends upon what definition you want to give to $m$. If you want to match your simulations with results of some other paper, then make sure to copy their definition of $m$. Otherwise, most people call $m$ as being the migration rate between any two populations (without migrant pool complication or anything). Note however, that if you have populations of different sizes (or if migration depends upon local population mean fitness aka. hard selection), then the forward and backward mutation rates may differ and you'll have to make your definition clear regarding this issue again. Nov 16 '17 at 20:18
• R is extremely slow. You'll find plenty of simulation platforms written in C++ that will be infinitely faster that what you'll code in R. So, if you need speed, you will find advice here. SimBit is a simulation platform I wrote myself if you are interested. Nov 16 '17 at 20:21
• Thanks! I will take a look at SimBit. It sounds very interesting. Nov 19 '17 at 14:53
• Do you know where the -1 in the equation for the migration probability comes from? My guess is that it is used as a correction factor to make the estimator unbiased, but I am uncertain. Nov 19 '17 at 14:55