Island Population Model

The Island Population Model

PhyDyn provides a ready-made BEAST class for defining island population models. The island population model is characterised by constant migration and birth rates. We further assume constant population sizes. The benefits of using this class, instead of using PopModelODE, are succinctness and performance: no ODEs need to be specified and solved. Furthermore, the resultant time-series is made of two points (start and end) and a single FGY object, thus taking advantage of the assumption of constant population and rates.

An m-island population model is defined by an m-vector of effective population sizes (Ne) and a m-by-m matrix of migration rates (M). The birth (F) and migration (G) rates used by our framework are defined by the following formulas:

The value of x_F does not affect the value of the structured tree likelihood.

A PhyDyn XML island model takes the following parameters:

• demes: Deme names.
• Ne: Effective population sizes.
• migRates: A vector denoting the migration rates matrix (from the point of view of the coalescent process). Note that this rates correspond to values in the matrix M in the formula above (backward rates), as opposed to G, which correspond to the forward migration rates of the population process.
  Rates are entered row-wise, skipping the diagonal since M_ll=0.
• t0: Trajectory start time.
• t1 (optional): Trajectory end time. t1 must be specified if the population model is not associated with a tree (in which case t1 takes the value of the date of the tip furthest from the root). If the migration matrix is symmetric, we only need to enter the values of the upper triangle of the matrix.
• xF (optional, default=1000): the factor x_F is defined as the relation between the real and effective deme population sizes. In most cases it's safe to work with the default (1000) since the value of the value of Y_F does not affect the value of the structured coalescent likelihood.

The following XML code extract defines an island population model with demes I0 and I1, effective population sizes 0.5 and 0.25, and backward migration rates M_I0_I1 = 0.001 and M_I1_I0 = 0.01 :

<model spec='PopModelIsland' id='islandtwodeme' demes="I0 I1"
   Ne = "0.5 0.25" migRates="0.001 0.01"
   t0="-1" t1="42.8" >
</model>

A three-deme model with symmetric rates M_I0_I1 = M_I1_I0 = 0.01, M_I0_I2 = M_I2_I0 = 0.02 and M_I1_I2 = M_I2_I1 = 0.03 would be defined as follows:

<model spec='PopModelIsland' id='islandthreedeme' demes="I0 I1 I2"
   Ne = "0.5 0.25" migRates="0.01 0.02 0.03"
   t0="-1" t1="42.8" >
</model>