As explained by my supervisor.

Wright Fisher

Take a population of \(N\) diploid, \(2N\) haploid, individuals. Each of these individuals has a parent, which is chosen completely randomly backwards in time. Occasionally, with a probability \(\frac{1}{N}\), two individuals will have the same parent, and their lineages become the same or “coalesce”. Overall, the probability of a coalescence event in any of the individuals in the population is the number of combinations of individuals possible multiplied by the probability that they will coalesce, or \(\binom{N}{2} \frac{1}{N}\). Kingman showed that in the limit, as \(N \rightarrow \infty\), the coalescent times \(T_i\) are exponentially and independantly distributed as \(\) f_{T_i} (t_i) = \binom{i}{2} e^{- \binom{i}{2}t_i} \(\) such that \(t_i \geq 0\) and \(i = 2, \dots, n\). Mean and variance accordingly follow. Note that this is valid for a particular time scaling, see Wakeley.

This is basically all of the background needed to introduce migration.

Migration

Consider that instead of $N$ diploid invididuals, we are interested in two populations from different geographic locations or “islands”. Individuals have the opportunity to move from island to island. Denote these as “red” and “blue” becuase these were the pen colours available.

Looking forward in time, we take an individual from one of the populations and track their lineage. Create a row vector $\mathbf{P}$ which denotes the population of the individual $\begin{pmatrix} 1 & 0 \end{pmatrix}$. In this case, our individual starts in the red population.

We know that the probability of the individual either staying in their current population or migrating to the other population is 1. Call the probability of being in a population at a particulular time $q(t) \in \mathbb{R}^2$. Following from the previous,\(q_1(t) + q_2(t) = 1\) Probability of being in a particular population changes from generation to generation according to a rate matrix $\mathbf{A}$ where the columns sum to 0.

\[\mathbf{A} = \begin{bmatrix} -r_{12} & r_{21} \\ r_{12} & -r_{21} \end{bmatrix}\]

Sidebar: How do we know that the columns sum to zero?

We want the probability $q(t)$ to sum to 1 each generation:

\[\begin{aligned} \begin{pmatrix} 1 & 1 \end{pmatrix} q(t) &= 1 \\ \frac{d}{dt} \begin{pmatrix} 1 & 1 \end{pmatrix} q(t) &= 0 \\ \begin{pmatrix} 1 & 1 \end{pmatrix} \mathbf{A} q(t) &= 0 \\ \\ \begin{pmatrix} 1 & 1 \end{pmatrix} \mathbf{A} &= \begin{pmatrix} 0 & 0 \end{pmatrix} \end{aligned}\]

So $\mathbf{A}$ must have columns which sum to zero in order for this property to hold.

This equation can now be solved numerically.