The Migration Matrix Model

In the migration matrix model, there are \(D\) sub-populations or demes each of size \(N_i\) and the size of the total population is \(N=\sum_i N_i\). Between the demes there is gene flow mediated by propagules. The parameter \(m_{ij}\) defines the probability that a propagule produced by an individual in deme i is dispersed to deme j.

In the discrete non-overlapping generation version of the model the constituents of deme i in the next generation are chosen at random from among all the propagules received by that deme, subject to the constraint that the total number of individuals is still \(N_i\).

Individuals produce many more propagules than can possibly survive, and each deme receives propagules from many other demes of varying sizes. The so-called backwards migration probability is the probability that an individual in the next generation of deme j originated from a propagule produced by deme i. It is related to the forwards migration probability \(m_{ij}\) by the equation

\(b_{ji} = \dfrac{N_i m_{ij}}{\sum_k N_k m_{kj}}\)

This equation assumes all individuals, regardless of deme, are equally fit in the sense that they produce equally many propagules. The numerator is proportional to the total number of propagules produced by deme i and destined for deme j. The denominator is proportional to the total number of propagules destined for deme j from any source.

Weak migration approximation

The model can be simplified by assuming that the probability, per individual, of migration is very small, of the order of \(1/N\). This sort of approximation is commonly used, and is often applied to the probability of mutation per individual per generation, or the probability of recombination.

Under this assumption,

\(\begin{array}{ccc} b_{ji} & = & \dfrac{N_i m_{ij}}{N_j \left(1-\sum_{k\neq j}m_{jk}\right) +\sum_{k\neq j} N_k m_{kj}} \\
 & \approx & \dfrac{N_i m_{ij}}{N_j}\end{array}\)

Time reversibility

A useful property of the model is time reversibility. The migration matrix is time reversible if

\(\pi_i b_{ij} = \pi_j b_{ji}\),

where \(\pi\) is the equilibrium frequency distribution of the model. Time reversible models have the common property that for each pair of states i and j, there is a value s such that

\(b_{ij} = s \pi_j , b_{ji} = s \pi_i \).

This property is commonly used to construct time reversible models.

What are the conditions that would make the backwards migration matrix time reversible? From the definition of the backwards migration probabilities, a time-reversible model is obtained when

\(m_{ij} =  s N_j, m_{ji} = s N_i\).

This model has the appealing property that the equilibrium frequency distribution is proportional to the deme sizes. The interpretation of this model is that for each pair of demes, \(i\) and \(j\), the rate of migration from one to the other is proportional to the size of the destination deme. This might be considered a natural assumption if it can be argued either that larger demes attract more migrants, or that they are larger because they attract more migrants.

Further reading

Bodmer WF, Cavalli-Sforza LL (1968) A migration matrix model for the study of random genetic drift. Genetics 59: 565–592. pdf

Nordborg M (2007) Coalescent theory. In Handbook of Statistical Genetics, Third Edition. pp 843-877. Eds Balding DJ, Bishop M, Cannings C. John Wiley & Sons, Ltd.