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Generation Generation

MRCA Present MRCA Present

Figure 3.29 The effects of exponential population growth or shrinkage on coalescent genealogies. The upper panels show change in population size over time with exponential growth according to N(t) = N0e~rt with r = ±0.1, yielding relatively slow exponential population growth. The two genealogies illustrate examples of waiting times that might be seen under strong exponential population growth (left) and shrinkage (right). With strong exponential population growth coalescent times are longest in the present when the population is the largest, leading to genealogies characterized by long branches near the present and very short branches in the past around the time of the MCRA. With exponential population shrinkage, coalescence times are greatest in the past near the MRCA when the population was larger and shortest near the present when the population is at its smallest size. The genealogy on the left was obtained using equation 3.91 with r = 100.

times in the present are even shorter than they would be under constant population size. Similarly, when population size is shrinking coalescence times in the past are relatively greater than under constant population size because the probability of a coalescence event was greater (Fig. 3.29).

A common way to model growing or shrinking populations is to assume that population size is changing exponentially over time. Under exponential growth, the population size at time t in the past is a function of the initial population size in the present N0 and the rate of population growth r according to

Examples of population size over time under exponential population growth are shown in Fig. 3.29. With exponential growth, population size tends to change very rapidly.

The generalizations above regarding coalescent waiting times depend on rapid and sustained changes in population size over time such as under exponential population growth with a constant rate (r). In populations that are changing in size slowly over time, it is possible that coalescence waiting times differ very little from those expected under constant population size. Increases in population size over time cause the probability of coalescence to decrease toward the present. At the same time, the chance of a coalescence event increases toward the present simply due to a larger number of lineages (increasing k) available to coalesce. Only in populations with rapidly changing population size will the reduction in the probability of coalescence be great enough to overcome the effect of an increasing number of lineages available to coalesce toward the present. In addition, the variance in coalescence waiting times with constant N is large so that only very rapid and sustained change in population size will noticeably impact the distribution of coalescence waiting times.

Changing population size over time complicates finding the distribution of coalescent times. In the

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