Some notes on continuous approximations

Abstract

In the previous chapter, we derived the probability of fixation of an allele A by treating the allele frequency as a continuous variable. We noted that this approach gave a very accurate result, provided that the intensity of natural selection was not too great. In the present chapter, we consider some problems that arise when allele frequency, changing in discrete generations, is treated as a continuous variable changing in continuous time. We shall be especially concerned with the reliability of results obtained using these continuous approximations. By considering quite a simple biological problem, it will be shown that the continuous approach can, in some circumstances, yield distinctly inaccurate results. This should not, however, be interpreted as a general criticism of this approach, which, as we have seen, often works very well. Unfortunately, it is often difficult to identify in advance whether the approach will lead to an accurate solution to a given problem — see, for example, our brief discussion in Chapter 4. Frequently, the best that can be done is to check results obtained by the continuous approach against exact results obtainable in special cases — we shall give an example of this. Our main point is that results obtained from diffusion methods cannot be accepted uncritically and that every formula has, of necessity, to be checked individually. Of course, with the advent of computers, such checking has been carried out very extensively.

The diffusion equation approach has been used extensively in the study of gene frequency change because of its extreme usefulness. But it is an approximation based on rather intuitive arguments. Therefore, to investigate the conditions under which such an approximation may be valid, is an important task for mathematicians.

M. Kimura (1964)