Abstract
Ancestral processes are fundamental to modern population genetics and spatial structure has been the subject of intense interest for many years. Despite this interest, almost nothing is known about the distribution of the locations of pedigree or genetic ancestors. Using both spatially continuous and stepping-stone models, we show that the distribution of pedigree ancestors approaches a travelling wave, for which we develop two alternative approximations. The speed and width of the wave are sensitive to the local details of the model. After a short time, genetic ancestors spread far more slowly than pedigree ancestors, ultimately diffusing out with radius ∼t rather than spreading at constant speed. In contrast to the wave of pedigree ancestors, the spread of genetic ancestry is insensitive to the local details of the models.
Highlights
There has long been interest in the flow of genes through spatially structured populations, and in the ancestral relationships between genes—classically, through the concept of identity by descent (Wright, 1943; Jacquard, 1974), and more recently, through the coalescent (Kingman, 1982; Hudson, 1983b)
If we wish to trace the history of the pedigree ancestors of our sample, we always follow both parents of each event in which at least one of the lineages we are following is an offspring
We wish to quantify the local size of the population of pedigree ancestors at a given time, and so we let N(t, x) be a random variable counting the number of individuals within distance 1 of position x at time t
Summary
There has long been interest in the flow of genes through spatially structured populations, and in the ancestral relationships between genes—classically, through the concept of identity by descent (Wright, 1943; Jacquard, 1974), and more recently, through the coalescent (Kingman, 1982; Hudson, 1983b). We examine a different structure, in which dispersal is local, so that tracing backwards in time, ancestral lineages diffuse out from the location of a sampled individual This process was first analysed by Wright (1943), for single genes. We expect to see about l independent lineages diffusing according to symmetric random walks, √giving rise to a wave of genetic ancestors expanding at speed t instead of linearly in time This is what we shall study, mostly through simulations. We develop two approximations: one, based on Wright’s (1943) idea that ancestors diffuse out in a Gaussian distribution, and the other, based on a partial differential equation that approximates the continuum model Both lead to a travelling wave for pedigree ancestry.
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