My own research interests are in evolutionary aspects of genetics. Here, mathematical modelling of the behaviour of genetic variants in populations has been fundamental, starting soon after the rediscovery of Mendel’s work in 1900. In the first chapter of The Genetical Theory of Natural Selection, unquestionably the most important book on evolution after The Origin of Species, Fisher described how the particulate nature of Mendelian inheritance resolves the difficulties that Darwin encountered in explaining the origin of the variation that is needed for natural selection to work (Fisher 1930, 1999). Fisher argued that Darwin’s belief in blending inheritance led him to conclude that characters acquired during the development of the parents could frequently be transmitted to the offspring (what we now call Lamarckian inheritance—Darwin called it pangenesis), in order to offset the rapid loss of variability that occurs with blending (the genetic variance is halved every generation). If there is no blending of the material contributed to an offspring by the two parents, as with Mendelian inheritance, then this problem disappears, and there is no need to postulate a high rate of origin of new genetic variability in every generation.

The formal statement of this principle, for diploid randomly mating populations, is the Hardy–Weinberg law (Hardy 1908; Weinberg 1908). But the constancy of allele frequencies in an infinitely large population that underlies this law holds for any mating system, although there may be only an asymptotic approach to the equilibrium allele frequencies. In populations with age structure, like humans, the approach to equilibrium may take some time, even for an autosomal locus (Charlesworth 1974). Nonetheless, the basic conclusion, that the mechanics of inheritance conserves rather than destroys variation, is correct, with some qualifications with respect to phenomena such as the effects of biased gene conversion on allele frequencies (Charlesworth and Charlesworth 2010, p.35). We could not know this without analysing mathematical models.

However, we need to know where variation comes from in the first place. There is now a large body of work on the rate at which new variation in quantitative traits arises through mutation, to which my lab has made some modest contributions with respect to components of fitness (Houle et al. 1994; Houle et al. 1997; Charlesworth et al. 2004; Charlesworth 2015). The general picture is that there is only a low rate of increase in variance due to mutation in higher organisms like Drosophila melanogaster, of the order of 1000th the variance found in a natural population (Halligan and Keightley 2009; Walsh and Lynch, 2017). Current levels of variability are the joint product of mutation and other evolutionary forces, such as selection and genetic drift.

Important modelling work, initiated by Alan Robertson in 1960, has generated predictions about the ultimate change in the population mean of a trait that can be caused by selection on standing variation in a sexually reproducing population. This change is approximately equal to the product of twice the effective population size (N e ) and the response to selection in the first generation (Robertson 1960; Barton 2017; Walsh and Lynch 2017). In large, sexually reproducing populations, a very large and sustained response to selection in a quantitative trait can thus be produced from variation that was present in the initial generation, resulting in phenotypes far outside the range of variability in the original population, as is seen in many experiments on artificial selection (Hill 2010). Long-continued selection that exploits new mutations can have even more dramatic effects (Hill 2010), and the rate of evolution by the fixation of new selectively favourable mutations is also proportional to N e (Kimura and Ohta 1971, p.11).

The dependence of selection response on N e leads on to the question—what do we mean by the effective population size? This seminal concept was introduced by Wright (1931) as a way of describing the effect of genetic drift in a population that does not fit the assumptions of the simple model of a discrete generation population introduced by Fisher (1922). This model assumes N randomly mating diploid individuals with no sex differences, such that the 2N copies of a gene at a given locus that are present in the adults of the next generation are a random (binomial) sample from an infinite pool of gametes produced by the members of the preceding generation. The asymptotic rate of genetic drift under more complex scenarios (such as different numbers of breeding males and females) is equated to 1/(2N e ) instead of the 1/(2N) that appears in the simple “Wright-Fisher” model, as it has come to be termed (why is not it called the “Fisher model”?). N e is often much smaller than the number of breeding adults in the population (Crow and Morton 1955; Frankham 1995).

In the case of humans, for example, DNA sequence diversity and mutation rate estimates suggest a value of N e of ~25,000. This figure comes from the equation derived by Kimura (1971) for the equilibrium level of diversity per nucleotide site for selectively neutral variants: π = 4N e u, where u is the mutation rate per nucleotide site per generation. For sites that are likely to be close to neutrality, mean π in humans is approximately equal to 0.001, and u is around 10–8 (Kong et al. 2012), yielding N e = 25,000. This apparent conflict with the current human population size of 7.6 billion probably largely reflects the fact that the harmonic mean (the reciprocal of the mean of the reciprocals) of the effective population sizes in each generation gives the N e that is relevant to current levels of variability (Slatkin and Hudson 1991). With a large and rapid expansion in population size, the harmonic mean of N e will be much closer to the ancestral value of N e than the current value. Another contributory factor may be the effects of selection at linked sites in reducing variability, discussed in the next section.

This raises the questions of what we mean by generation time and effective population size in species like humans, which have overlapping generations and separate sexes. These questions have been answered by theoretical analyses of populations with age structure, to which I devoted a good deal of research time during the first part of my career (Charlesworth 1994). The first question was answered by showing that the generation time is given by the mean of the average ages of mothers and fathers at the time of conception. It can differ between autosomal, X-linked and mitochondrial genes because of differences among these components of the genome in the relative contributions of males and females to the offspring, combined with sex differences in survival rates and age-specific rates of reproduction. The importance of these life-history variables for the interpretation of data on rates of molecular evolution in different evolutionary lineages is now becoming recognised in studies of humans and their primate relatives (Amster and Sella 2016).