Macroscopic and microscopic structures of the family tree for a decomposable branching process

Last modified: 2014-04-01

#### Abstract

A decomposable critical Galton-Watson branching process with N types of particles labeled 1,2,...,N is considered inwhich a type i particle may produce individuals of types j=i,...,N only. This model may be viewed as a stochastic model for the sizes of ageographically structured population occupying N islands, the location ofa particle being considered as its type. The newborn particles of island i=1,...,N-1 either stay at the same island ormigrate, just after their birth to the islands i+1,i+2,...,N. Particles ofisland N do not migrate.

We investigate the structure of the family tree for this process, thedistributions of the birth moment and the type of the MRCA.

It is shown, in particular, that if the population survives up toa distant moment n, then all surviving individuals are located at thismoment on island N and, moreover, at each moment in the past theirancestors were (asymptotically) located not more than on two specificislands.

Besides, we demonstrate that there are time-intervals ofincreasing orders within each of which the probability to find the MRCA ofthe population survived up to a distant moment n is negligiblecompared to the probability for the population to survive up to this moment.Moreover, these time-intervals are separated from each other by thetime-intervals of increasing orders within each of which the probability tofind the MRCA of the population survived up to a distant moment n is strictlypositive. Such a phenomenon has no analogues for the indecomposableGalton-Watson processes.

We investigate the structure of the family tree for this process, thedistributions of the birth moment and the type of the MRCA.

It is shown, in particular, that if the population survives up toa distant moment n, then all surviving individuals are located at thismoment on island N and, moreover, at each moment in the past theirancestors were (asymptotically) located not more than on two specificislands.

Besides, we demonstrate that there are time-intervals ofincreasing orders within each of which the probability to find the MRCA ofthe population survived up to a distant moment n is negligiblecompared to the probability for the population to survive up to this moment.Moreover, these time-intervals are separated from each other by thetime-intervals of increasing orders within each of which the probability tofind the MRCA of the population survived up to a distant moment n is strictlypositive. Such a phenomenon has no analogues for the indecomposableGalton-Watson processes.

#### Keywords

decomposable branching process; family tree; MRCA