# Island model vs. stepping stone model

Wright's Island model and Kimura's stepping stone model are classic explanations for migration/gene flow o individuals between subpopulations within the field of population genetics.

My question concerns the quantity Nm, the (effective) number of migrants per generation.

In the Island model, where all demes are exchanging migrants at a given time, individuals migrate with a frequency of m, as seen in Wright's approximation for $F_{ST}$

$F_{ST} = \frac{1}{4Nm + 1}$

In Kimura's stepping stone model, individuals can only migrate to an immediately neighbouring subpopulation with a frequency of $\frac{m}{2}$.

My question is:

In the stepping stone model, are there still only $Nm$ individuals migrating, or is it now $\frac{Nm}{2}$?

My thinking is that an individual can either move left or right, each with a frequency of $\frac{m}{2}$, for a total of $\frac{m}{2}$ + $\frac{m}{2}$ = $m$, so that there are still $Nm$ total migrants. Am I correct in my thinking or should it really be $\frac{Nm}{2}$?

• It all depends upon how $m$ is defined and that might vary from a paper to another. – Remi.b Jan 31 '18 at 21:44
• I thought that may be the case. Was SimBit designed assuming $Nm$ or $\frac{Nm}{2}$? – compbiostats Jan 31 '18 at 21:50

It all depends upon how $m$ is defined and that might vary from a paper to another. The rest, I find, is very much just a question of logic.

In the infinite island model (the model from which Wright's approximation was computed), the total number of migrants is of course, infinite (if $m≠0$). With $d$ demes (or islands), the total number of migrants is $dNm$ if $m$ is defined as the migration rate between any two patches.

Under a linear stepping stone model with only 1 step (because there exist more complex stepping stones models), with $d$ demes (or stones), the total number of migrants is $dNm$ (minus small approximation for the extremities of the world) if $m$ is defined as the migration rate from one deme to an adjacent deme. Half of $dNm$ are migrating toward the left and half of them are migrating toward the right.

I thought that may be the case. Was SimBit designed assuming NmNm or Nm2Nm2?

In SimBit, you can use multiple modes to enter migration information. You can specify the migration rate from any patch to any other patch (mode A).

You can use island where you specify the probability of not migrating (which might arguably be a little unintuitive). If the probability of not migrating is $x$, and there are $d$ demes, then the number of migrants from one deme which patch size is $N$ to a single other deme of interest is $\frac{N(1-x)}{d}$. The number of migrants from this deme to any other deme is $N(1-x)$ and the total number of migrants in the whole (meta)population is $dN(1-x)$.

The mode LSS (Linear Stepping Stone), is a little complex as it is very flexible. You can specify the migration rate from a focal patch to any other patch considering they are place in a linear order. SimBit LSS indeed allows for stepping several stones apart if you want. If for example you input LSS 3 0.1 0.8 0.1 2 it means that the probably to migrate from any patch to a given adjacent patch is 0.1. The probability of migrating is 0.2 and the probability of not migrating is 0.8. With such model, the number of individuals exiting a given patch is $0.2N$ ($0.1N$ on each side) for a patch of size $N$. Assuming all patches have the same size, the total number of migrants for a (meta)population of $d$ patches is $0.2dN$.

Of course, if you allow for patch size to vary with carrying capacity or for migration rate to be affected by population mean fitness, then this expected number of migrants will vary among generations.