Understanding Populations

From the earliest days of population genetics, mathematics has played an important role in the field. Until the 1960s, most population genetics theory focused on deductive analysis, and the models were generally focused on following the evolution of populations that were presumed to have originally been located in just one or two places. Early investigators showed how evolution would proceed under plausible models of genetic inheritance and natural selection. These analyses illuminated the dynamics of allele frequencies in populations, and they showed with what speed evolution could occur and how this speed depended on various parameters. Both deterministic models and models with random genetic drift were examined. Diffusion approximations to Markov chains were particularly important (Kimura, 1983). These diffusion processes could be analyzed by solving simple ordinary differential equations to obtain important quantities such as the probability of fixation of a new variant and the mean time for such fixation. These analyses strongly shaped our current understanding of natural selection in large, but finite, populations and guided experimental work. In recent years the emphasis has shifted from these deductive activities to inductive or retrospective approaches that address the question of what we can infer about evolutionary history and the nature of the evolutionary process from current patterns of genetic variation.

The primary goal of population and evolutionary geneticists today is to understand patterns of genetic variation within populations and pat-

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6
Understanding Populations
POPULATION GENETICS
From the earliest days of population genetics, mathematics has played
an important role in the field. Until the 1960s, most population genetics
theory focused on deductive analysis, and the models were generally fo-
cused on following the evolution of populations that were presumed to
have originally been located in just one or two places. Early investigators
showed how evolution would proceed under plausible models of genetic
inheritance and natural selection. These analyses illuminated the dynam-
ics of allele frequencies in populations, and they showed with what speed
evolution could occur and how this speed depended on various param-
eters. Both deterministic models and models with random genetic drift
were examined. Diffusion approximations to Markov chains were par-
ticularly important (Kimura, 1983). These diffusion processes could be
analyzed by solving simple ordinary differential equations to obtain im-
portant quantities such as the probability of fixation of a new variant and
the mean time for such fixation. These analyses strongly shaped our cur-
rent understanding of natural selection in large, but finite, populations
and guided experimental work. In recent years the emphasis has shifted
from these deductive activities to inductive or retrospective approaches
that address the question of what we can infer about evolutionary history
and the nature of the evolutionary process from current patterns of ge-
netic variation.
The primary goal of population and evolutionary geneticists today is
to understand patterns of genetic variation within populations and pat-
99

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100 MATHEMATICS AND 21ST CENTURY BIOLOGY
terns of genetic divergence between species. Population geneticists have
asked, What are the important forces that determine the amount and na-
ture of genetic variation in populations, the spatial distribution of this
variation, the distribution of variation across the genome, and the evolu-
tionary changes that occur over short and long timescales? The process
that has shaped this variation within and between species is a complex
one involving a complex genome and a complex, spatially and temporally
varying environment. It is certain that stochasticity is an important aspect
of the process. The rapidly growing database of DNA polymorphism and
divergence studies from a variety of organisms, including humans and
other primates, provide an exciting opportunity to learn about the evolu-
tionary history of populations and the evolutionary processes that have
resulted in the patterns of variation that we observe in extant popula-
tions. The difficulty is that even very simple models of this process lead to
challenging mathematical problems.
Some examples of current approaches and the mathematical chal-
lenges facing us are described here. To be concrete and to avoid an overly
vague description of the problems, a very specific population genetic
model of sequence evolution will be described. The particular model, the
Wright-Fisher model, has a long and rich history, but it is not necessarily
the most realistic or tractable for every purpose, and it is only one of many
models that might have been considered here.
The Wright-Fisher model assumes discrete generations (as opposed
to a model with distinct age classes and overlapping generations, which
would be more realistic for some populations, including humans). The
focus is on a particular segment of the genome, referred to as a gene, and
it is first assumed that no recombination or mutation occurs. To begin, it is
assumed as well that population size (N) is constant and that there is no
spatial structure. A haploid model is also assumed, which means that each
individual carries just one copy of the gene. (Humans are in fact diploid,
which means that each individual carries two copies of each gene, a ma-
ternal and a paternal copy.)
In the Wright-Fisher model, successive generations are produced as
follows. Each of the N individuals of the offspring generation is produced
by replicating, without error, the gene sequence of a randomly drawn
individual of the parental generation. Each offspring individual is as-
sumed to be generated independently from the parental population in
this manner. The number of offspring of any particular individual of the
parent generation is thus a random variable, being the result of N inde-
pendent Bernoulli trials, with probability of success equal to 1/N. In large
populations, the number of offspring of any individual would approxi-
mately follow a Poisson distribution with mean 1. If it is supposed that
the parents do not all have identical gene sequences, then their distinct

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UNDERSTANDING POPULATIONS
gene sequences are known as haplotypes. Given the frequencies of the
different haplotypes in the parental generation, the numbers of the differ-
ent haplotypes in the offspring generation will be multinomially distrib-
uted. Regardless of how much variation existed in the founding popula-
tion, the population under this model will eventually arrive at a state in
which every individual carries the same sequence. This process of ran-
dom change in the frequencies of the different haplotypes is referred to as
genetic drift, and it eventually results in the population becoming mono-
morphic.
Next, mutation is introduced into the model. Let it be supposed that
the replication process that generates an offspring copy of the gene from
its parent has some error rate, so that each offspring differs from its par-
ent at a Poisson-distributed number of sites in the gene sequence. If this
model is run for many generations, the pattern of genetic variation
asymptotically approaches a stationary distribution resulting from a sto-
chastic balance between mutation, which generates variation in the popu-
lation, and genetic drift, which tends to eliminate variation. Many prop-
erties of this stationary distribution are known. Also, many properties of
samples drawn from this stationary distribution are known. In the model
as it has been defined here, all individuals are in some sense equivalent.
For example, all individuals have the same distribution of offspring num-
ber with expectation equal to 1. All the genetic variation is said to be
selectively neutral, and the model is referred to as a neutral model. In
generalizations of this model, some sequence variants may have a sys-
tematic tendency to produce more offspring than others, and the fre-
quency of such variants will tend to increase. These are models of evolu-
tion with natural selection.
The Wright-Fisher neutral model is a particular case of a more gen-
eral class of neutral models in which all parents are equivalent; these are
referred to as “exchangeable models.” In these models, the distribution of
offspring number need not be Poissonian. In the limit of large popula-
tions and a low mutation rate, the models’ stationary properties depend
on a single compound parameter, Nu/v, where u is the mutation rate and
v is the variance of offspring number. Despite the simplicity of this model,
in which there is no selection, no geographic structure, no variation in
population size, and no recombination, the probabilities of sample con-
figurations of sequences under this model are difficult to calculate.
Strobeck (1983) first described recursions for these probabilities for the
case where only two or three haplotypes are present in a sample. The
difficulty of obtaining sample configuration probabilities led to the use of
summaries of the data, with an inevitable loss of information. Only in the
last 10 years have full likelihood approaches been developed. Griffiths
and Tavaré (1994a, 1995) were the first to find a practical method to esti-

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102 MATHEMATICS AND 21ST CENTURY BIOLOGY
mate full likelihoods for this simple neutral model using a method based
on importance sampling. Kuhner et al. (1995) described a Markov chain
Monte Carlo (MCMC) method for obtaining quantities proportional to
the sample probabilities. The main point here is that the sampling proper-
ties of sequence data under even this simplest, one-parameter, neutral
model lead to recursions that are not analytically tractable. Monte Carlo
methods have provided a way forward. Much of the progress in under-
standing these models is based on analyzing properties of the genealogi-
cal relationships of sampled copies rather than analyzing the dynamics of
population frequencies of haplotypes. The population genetics theory of
sample genealogies has come to be known as calescent theory. The early
important work in this area was done by Watterson (1975), Kingman
(1982), and Tajima (1983). (See Chapter 2 for more information on this.)
For models without recombination, it is possible to extend these
Monte Carlo methods to the case in which population size is not held
constant. For some special cases, it is possible to infer past population size
changes (Griffiths and Tavaré, 1994b; Kuhner et al., 1995). Additional
Monte Carlo methods for demographic inference using other types of ge-
netic data (microsatellite data, for example) have also been developed
(Beaumont, 1999). Much remains to be done in this area.
Models with geographic structure are more difficult. Historically,
simple “island” models have been employed. These subdivide the popu-
lation, but with a special structure in which each subdivision is assumed
to communicate equally with all other subdivisions. More realistic step-
ping-stone models are more difficult, but some results are known (Durrett,
2002). Wakeley (2004) recently investigated a class of models with large
numbers of subdivisions and obtained elegant results for this model. This
work has capitalized on results for coalescent processes that operate on
different timescales. Bayesian Monte Carlo methods have again begun to
play an important role in analyzing data of several types (Pritchard et al.,
2000; Beaumont, 1999).
If recombination is added to the model, the difficulties increase enor-
mously. With recombination, each offspring produced in the model has
some small probability, r, of being the product of two parent individuals,
one parent contributing a part of the gene on the left and the other parent
contributing the rest of the gene, the boundary between the two contribu-
tions being random. In models with recombination, complex statistical
dependencies between sites arise. Sample configuration probabilities are
very difficult to obtain. For a model with just two sites between which
recombination can occur, a relatively simple recursion can be written
down for sample probabilities (Golding, 1984). These recursions are in-
tractable for all but very small samples, as the state space becomes enor-
mous. For more than two sites the situation quickly gets much worse.

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Griffiths and Marjoram (1996) present recursions for sample configura-
tion probabilities under the infinite-sites version of this model with re-
combination. These are not analytically tractable, but Monte Carlo meth-
ods have been described for estimating these sampling probabilities under
this model. However, unlike the case without recombination, it appears
that these Monte Carlo methods are computationally infeasible for
samples of interesting size because convergence, while difficult to assess,
appears to take inordinate amounts of computer time. As a consequence,
approximate methods, ad hoc methods, and methods based on summary
statistics are still the rule when analyzing data from genes with recombi-
nation (Stephens, 2001). Much interest has focused on making inferences
about recombination rates and gene conversion rates under models in
which the rates vary across the genome (McVean et al., 2004). Improved
methods could contribute to understanding the genetic mechanisms of
recombination and also help in the mapping of disease genes via associa-
tion studies.
There is also great interest in assessing the importance of natural se-
lection in shaping patterns of variation within populations and the diver-
gence between populations. Many ad hoc tests have been developed over
the years (Kreitman, 2000; Bustamante et al., 2003) to explore these ques-
tions. Devising methods that make more efficient use of the data and com-
bine information from many loci should be a priority. These inferences
about selection must be made in the context of realistic models of popula-
tion structure and demographic history. More realistic models of muta-
tion and recombination are also needed, building on results shown in re-
cent work such as (Hwang and Green, 2004; Meunier and Duret, 2004).
This recent work points out how much there is still to learn about molecu-
lar evolution and how rich the mathematical models will need to be to
capture it.
The focus thus far in this chapter has been on variation within species,
but comparisons of sequences from different species can also be very in-
formative, both about evolutionary relationships of species (the phyloge-
netic inference problem) and about evolutionary processes. Again, recent
years have seen important progress on likelihood methods (Felsenstein,
1981) and, most recently, on the use of Monte Carlo approaches
(Huelsenbeck and Ronquist, 2001; Wong et al., 2004). Combining between-
and within-species data can be very useful, as is well exemplified by the
analysis of Poisson random field models (Bustamante et al., 2003). A re-
maining challenge in phylogenetic inference includes the problem of per-
forming multiple alignments and phylogenetic reconstruction simulta-
neously. Currently, alignment is carried out while ignoring phylogenetic
relationships, and phylogenetic reconstruction is carried out only with

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104 MATHEMATICS AND 21ST CENTURY BIOLOGY
fully aligned sequences produced prior to the reconstruction. This clearly
is not an optimal solution.
Genetic data are becoming available at an ever-increasing rate. More
loci, more species, and more individuals within species will be surveyed.
More and more frequently, essentially complete genomes will be
compared. These advances result in new opportunities and new math-
ematical and computational challenges. Different biological questions,
different organisms, and different types of genetic markers will require
somewhat different models, different methods, and different approxi-
mations. With more and richer data sets, researchers will be able to fruit-
fully consider somewhat more complex models, with increased demo-
graphic complexity (bottlenecks and expansions, more complex spatial
structure) and increased genetic complexity (heterogeneous recombina-
tion and mutation rates), and with more complex types of natural selec-
tion (interaction between sites and spatial and temporal variation in se-
lection coefficients). These complexities present significant mathematical
challenges. Stochastic models, possibly with many parameters, and com-
plex, nonindependent data make the computational difficulties substan-
tial. Insight about the models and mathematical skill will be needed to
make progress. While a wide range of approaches will clearly contribute
to advances, the important roles that Monte Carlo methods and Bayesian
approaches have recently played seem likely to continue.
ECOLOGICAL ASPECTS OF POPULATIONS
Population growth with density dependence was formulated math-
ematically by Verhulst (1838), who developed a number of models to
investigate the consequences of deviations from unrestricted growth.
One of the models, known as the logistic equation, remains a standard
model for population growth. It is described by the following differential
equation:
N
dN
= rN 1 –
K
dt
where N(t) is the population size at time t, r is the intrinsic rate of growth,
and K is the carrying capacity. This model continues to serve as an impor-
tant illustration of the effects of negative density dependence, as encoded
by the term in parentheses. It is an example of a phenomenological model,
one that is intended to embody, in a concise form, some of the observed
behaviors of populations, but it has also been used as a predictive model:
Based on data on the U.S. population from 1790 to 1910, Pearl and Reed

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UNDERSTANDING POPULATIONS
(1920) fitted the logistic equation to the observed growth of the U.S. popu-
lation, estimating that it would level off at 197 million.
A discrete-time version of the logistic growth equation exhibits sur-
prising properties, from periodic behavior to chaos. This latter behavior
was introduced to ecology by May (1974, 1976). Although a known phe-
nomenon in mathematics, the idea that deterministic models can exhibit
unpredictable behavior was new to ecologists at that time and still spawns
new research in ecology (Cushing et al., 2002). Although difficult to verify
in nature, experimental systems have been developed to test for chaos
(Costantino et al., 1997). The theoretical study of single populations in a
purely ecological context remains challenging. Much research is being
devoted to anthropogenic impacts on natural and managed systems.
Mathematical challenges include modeling and analysis of spatial aspects,
temporal and spatial variation, demographic stochasticity (in particular
when dealing with small populations), and nonequilibrium dynamics. The
committee highlights two areas of interest: species extinction and food
supply.
The threat of species extinction arising from either habitat fragmen-
tation or species invasion has resulted in much theoretical work. The need
here is for both conceptual models, which will give us a better under-
standing of the underlying mechanisms, and predictive models, which
can be used for management. Much of the theoretical work focuses on
single-species models. On the empirical side, observational studies domi-
nate, and data are not always unequivocal (Debinski and Holt, 2000).
There are few controlled experiments of habitat fragmentation and
species invasions because of the difficulties associated with these
experiments.
Reliable food supply depends on the ability to manage this renew-
able resource. Fisheries management is an example that has enjoyed so-
phisticated modeling by both economists and biologists. Conceptual and
highly species-specific models are widespread, but many focus only on
single populations. Large, spatially explicit data sets exist, although there
are troublesome uncertainties associated with some of the data. More
realistic models also must deal with the uncertainties associated with
management strategies and inherently stochastic processes, such as birth
and death. They also need to take food web structure into account. A
comprehensive model that includes visualization tools was developed
by James Kitchell and co-workers for the Central North Pacific ecosys-
tem to assess the effects of fishing on productivity (Hinke et al., 2004).
Visualizations are useful to convey the impact of different management
scenarios to managers. The development of such complex models re-
quires a thorough understanding of the ecological interactions in addi-
tion to long-term data to parameterize the models and test scenarios.

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106 MATHEMATICS AND 21ST CENTURY BIOLOGY
These models are typically so complex that simulation is the only tool
currently available for investigation.
In the two cases discussed here, the need is for models that incorpo-
rate the complex interactions of the target species with its surroundings,
often in spatially heterogeneous habitats and under nonequilibrium con-
ditions. Models that are used for management purposes often include
social and economic aspects. The importance of developing methods to
study nonequilibrium dynamics cannot be overemphasized (see, for ex-
ample, Hastings, 2004). Some of the mathematical theory has been devel-
oped, in particular when different timescales are involved. Many eco-
logical interactions occur far away from equilibrium, and large-scale
anthropogenic perturbations, such as land use change, species invasions,
or alterations of nutrient or carbon cycles, exacerbate this problem. Land
use change and the invasion of exotic species often result in rapid changes
and thus have the potential to move a system far from equilibrium, with
consequences for both ecological and evolutionary processes (see below).
Even experimental systems are probably not in equilibrium: Most field
experiments are studied over only short timescales, even if the dynamics
are slow.
A SYNTHESIS OF ECOLOGY AND EVOLUTION
During ecology’s early years, evolutionary thinking was prevalent.
However, as ecology focused more and more on abiotic and biotic causes
of diversity and species abundance, evolutionary thinking became less
prominent (Collins, 1986). Much of ecology now operates under the
premise that ecological and evolutionary processes act on different time-
scales. Evolutionary processes are often thought to take hundreds of gen-
erations before their effects can be measured, whereas ecological pro-
cesses often show effects after a few generations. This has led to the
intellectual separation of ecology and evolution. For systems that are
under strong selection, however, this may not be the case. There are clas-
sic examples, such as melanism among moths as a response to air pollu-
tion (Kettlewell, 1955) or the heavy-metal tolerance of plants (Bradshaw,
1952). As a consequence, purely ecological or purely genetic models are
often inadequate when strong selection is acting (Neuhauser et al., 2003).
An increasing number of studies are combining ecological and evolu-
tionary models to meet this challenge of understanding the consequences
of ecological and evolutionary forces acting on similar timescales
(Antonovics, 1992; Thompson, 1999; Whitman et al., 2003).
The mathematical challenges when both ecological and evolutionary
processes are considered simultaneously are numerous. First, the dimen-

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sionality increases because additional parameters must be introduced to
model both ecological and evolutionary processes. Second, these processes
are frequently both spatial and stochastic. The study of spatial stochastic
systems is an active area of mathematical research, but at this point, only
the simplest models seem to be tractable in a rigorous way. Third, the
interplay between ecological and evolutionary processes is most pro-
nounced during transient dynamics. Since there are no readily available
analytical methods for nonequilibrium processes that are spatial and sto-
chastic, most studies resort to simulations as the primary way to gain in-
sights. The following example illustrates a biological problem and the
mathematical challenges it brings.
As an example of why one might wish to couple ecological and evolu-
tionary processes, and of the mathematical challenges that result, con-
sider the evolution of resistance. This is of importance, for instance, in
understanding the ramifications of the use of transgenic Bt crops, which
have been engineered to express a toxin from the soil bacterium Bacillus
thuringiensis (Bt). Engineered versions are available for a number of crop
plants, such as maize, potatoes, cotton, and soybeans. In maize, the toxin
is expressed at high levels and is toxic to the European corn borer, Ostrinia
nubilalis (Hübner), the key herbivore insect pest. An important concern is
the pest’s development of resistance to the toxin (Tabashnik, 1994; Gould,
1998). Current practice is to plant a “refuge” of non-Bt maize alongside Bt
maize to allow sufficient numbers of susceptible European corn borers to
be available as mates if resistant types emerge from the Bt field. To model
the evolution of resistance, two things are needed: nonequilibrium mod-
els for at least two types of patches (Bt field and non-Bt field) and the
ability to study the time-varying genetic composition of European corn
borer populations throughout these modeled patches. One of the first
models that incorporated these aspects was developed by Comins (1977).
Since then, other models have been developed, and each seems to reveal
additional complexities. A consistent theory is still lacking, and it will
need to also take into account the community dynamics of associated en-
emies of the insect pest (see, for example, Neuhauser et al., 2003).
The field of population biology focuses largely on single populations.
Except under controlled experimental situations, populations rarely live
in isolation. Populations are typically embedded in communities, and
their dynamics are strongly influenced by other members of the commu-
nity. These feedbacks greatly complicate our understanding of the dy-
namics and present great challenges. The next chapter will discuss com-
munities.

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