Elsevier

Les Houches

Volume 83, 2006, Pages 489-491, 493-545
Les Houches

Course 11 - Evolution in Fluctuating Populations

https://doi.org/10.1016/S0924-8099(06)80048-XGet rights and content

Introduction

Understanding the evolution of individuals which live in a structured and fluctuating population is of central importance in mathematical population biology. Two types of structure are important. First, individuals live in a particular spatial position and their rate of reproduction depends on where they are and who is living near them. Second, genes are embedded in different genetic backgrounds. Because genes are organised on chromosomes and these in turn are grouped into individuals, different genes do not evolve independently of one another. The evolution of a gene which is itself selectively neutral, in that it does not confer any particular advantage or disadvantage to the organism that carries it, can nonetheless be influenced by selection acting on other genes in that same organism.

Typically, data is used to make inferences about the genealogical relationships between individuals in a sample from a population and so it is these genealogical relationships, in other words the family trees that relate individuals in the sample, that we try to model. In 1982, Kingman [31], [32], introduced a process called the coalescent. This process provides a simple and elegant description of the genealogical relationships amongst a set of neutral genes in a large randomly mating (biologists would say panmictic) population of constant size. With the flood of DNA sequence data over the last decade, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like varying population size, natural selection, recombination and spatial structure of populations. This has been achieved with varying degrees of success.

Analytic results for these more general coalescent models are very hard to obtain, but they are relatively easy, at least in principle, to simulate and so they have become a fundamental tool in DNA sequence analysis. However, as the sophistication of the underlying population model increases, so does the computational complexity associated with simulating the corresponding coalescent. Even numerically, it is only really tractable to consider interactions of a very small number of genes. Substantial effects of natural selection are likely to be due to the cumulative effects of weak selection acting on many different interacting genes. This means that the number of genetic backgrounds that we must track is large. For example, if we are modelling the interactions between, say, ten different genes, each of which has just two possible forms, that is 210 different backgrounds and so it is certainly far from automatic that the number of individuals in each of those backgrounds is big enough to justify a coalescent approximation. Thus, not only do numerical computations become impractical, but we must also be careful to check that the theoretical basis for the coalescent approximation does not break down.

With a small number of genetic backgrounds, one often makes the assumption that the number of individuals in each background is a constant. This sidesteps the problem of actually modelling the population size in each background, thereby considerably reducing the complexity of the model. However, we shall see an example in which, although at first sight this approach seems eminently reasonable, fluctuations matter. In this same example we shall see why we don't necessarily require the population size to be large in all backgrounds all of the time in order to justify a coalescent approximation.

Similar problems arise with spatially structured populations. A species does not typically form a single random mating unit because the distance over which an individual migrates is usually much smaller than the whole species range. This phenomenon was dubbed isolation by distance by Wright, [51], and in particular means that we must take into account local effects. Many biological populations are distributed across continuous two-dimensional habitats and even subdivided populations may look approximately continuous when viewed over sufficiently large spatial scales. To describe the genealogical relationships in a sample (from well-separated locations) taken from such a population, we would like a simple model that approximates a wide variety of local structures, in other words an analogue of Kingman's coalescent, but as we shall see, there is no really satisfactory analogue of the coalescent in a two-dimensional continuum. Instead, we take as our starting point a formula due to Malécot, [37], which provides an expression for the generating function of the time to the most recent common ancestor of two individuals sampled from a certain two-dimensional population model. Although Malécot's assumptions are now known to be internally inconsistent, we shall see that a modification of his formula has the potential for fairly wide application, but there are significant practicalities to overcome, not least in finding explicit population models to which to apply the theory.

A popular alternative approach to spatial structure is to model such two-dimensional populations as though they are subdivided into demes arranged on a suitable lattice, typically ℤ2. Although generally the demes are assumed to be of constant size, the lesson from our example in the world of genetic structure is that we should not ignore fluctuations in the number of individuals in each deme and so once again we must find suitable models for the evolution of the spatial distribution of the population. Spatial population models, both lattice based and continuous, will be our final topic.

These notes can be viewed as falling into three broad sections. Although they are interrelated, the mathematics in each is of quite a different flavour and it should be possible to read each in isolation. In particular, a reader who finds the genetics heavy going should nonetheless be able to pick up the thread when we talk about spatial population models in the last section. In the first part we recall some classical coalescent theory and then present a simple example of genetic structure which illustrates some of the problems that face us when we try to tackle spatially structured populations. In the second, we present an approach to spatially continuous models that builds on classical results of Malécot. The focus here is on approximating the genealogical relationships between individuals in a very general spatial population model. Our analysis is backwards in time and makes no reference to the exact form of the model, but rather assumes that it has certain properties. The final section will be devoted to some explicit forwards in time population models. In particular, we shall be concerned with models which overcome the problem of ‘clumping and extinction’ which are inherent in branching process models in two dimensions.

Section snippets

Acknowledgement

These notes were written while the author was visiting the University of Edinburgh. She would like to thank everyone there for their hospitality. Several people have read preliminary versions. In particular, the author would like to thank Nick Barton and Aernout van Enter for their comments and Matthias Birkner for many corrections and clarifications, especially in §4.7.

Some classical coalescent theory

We begin by recalling one of the many derivations of Kingman's coalescent and some elementary extensions. These ‘back-of-the-envelope’ calculations will be a useful guide in what follows.

Spatial structure and the Malécot formula

We now turn to spatial structure. From now on we will consider only the very simplest form of selection in which our (haploid) population has just two genetic types with relative fitnesses 1 + s : 1, or neutral populations (s = 0).

Spatial models

In a truly continuous population, we cannot separate population genetics and population dynamics. Necessarily there will be local interactions between nearby ancestral lineages, reflecting fluctuations in the local population density. Even in subdivided populations, we need adequate models for the fluctuations in population sizes in different demes. In this section, we therefore turn to models for the population dynamics. We shall shamelessly flip between models in discrete and continuous

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