Coalescence theory for a general class of structured populations with fast migration

Abstract

In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population size N/c are obtained.

On the eigenvalue effective size of structured populations

“…Other applications of Theorem 5 with includes a single deme with age classes. Explicit formulas for the constant in can be found under general assumptions on how reproductivity varies randomly between and within age classes, thereby extending results of Felsenstein ( 1971 ), Sagitov and Jagers ( 2005 ) and Hössjer ( 2011 ).…”

“…(Combined age and spatial structure) . Age structured models have been studied by Felsenstein ( 1971 ), Hill ( 1972 ), Kaj et al ( 2001 ), Sagitov and Jagers ( 2005 ) and Hössjer ( 2011 ). Here we consider a population with that has two demes with age classes each.…”

“…Felsenstein ( 1971 ) seems to have been first to use ( 89 ) for weighting pairs of subpopulations. Hössjer ( 2011 ) studied models with fixed backward migration, and showed that satisfies ( 83 ) when , with as in ( 90 ).…”

“…It is equivalent to the eigenvalue effective size , defined in terms of the largest non-unit eigenvalue of a Markov chain of allele frequencies (Ewens 1982 , 2004 ). The nucleotide diversity or mutation effective size is essentially the expected coalescence time of a pair of haploid individuals (Ewens 1989 ; Durrett 2008 ), whereas the coalescent effective size is defined for populations such that the ancestral tree of any finite number of individuals converges to a Kingman coalescent in the limit of large populations (Kingman 1982 ; Nordborg and Krone 2002 ; Sjödin et al 2005 ; Wakeley and Sargsyan 2009 ; Hössjer 2011 ).…”

Do estimates of contemporary effective population size tell us what we want to know?

“…First, the original, mathematically elegant definition of N eCo requires convergence of an ancestral tree towards Kingman's coalescent (Nordborg & Krone, ; Sjödin et al, ; Wakeley & Sargsyan, ) for any number of ancestral lines. This definition is quite restrictive, and therefore N eCo rarely exists for subdivided populations unless the system is in equilibrium and the migration rate is large (Hössjer, ). Second, whenever N eCo exists it equals N eE (Hössjer, ), an effective size we already included in our numerical examples.…”