Infinite Population Models and Random Drift

1.1 Goals

One purpose of this paper is to provide a clearer characterization of what biologists’ talk of infinite populations means in practice, and to explore different ideas and strategies associated with “infinite population” talk in evolutionary biology. Though philosophical discussions of so-called (countably) infinite population models correctly treat these models as assuming the absence of genetic drift, these discussions usually fail to make clear that an infinite population is neither a necessary nor sufficient condition for the absence of genetic drift. This follows from the claim below that infinite populations, as such, can play no role in evolutionary modeling. Given that it is not the idea of evolution in an infinite population that supports an assumption that drift is absent, it’s valuable to explore exactly what biologists do mean when they speak of infinite populations. It turns out that there are subtly different roles that assumptions about drift play in “infinite population” models. The case studies I present illustrate some of these roles. My general suggestion will be that “infinite population” talk among biologists is merely idiomatic—and that sophisticated evolutionary biologists implicitly know this. “Infinite population” assumptions have different consequences in different cases, but they at least function as ways of stating that particular effects of random drift are not represented in a given model. A further purpose of the paper is to illuminate and discuss ways of understanding different roles that assumptions about drift or its absence play in models of evolution. Though the idea of an infinite population as such is incompatible with even hypothetical evolution, other kinds of apparent inconsistency within models can be quite useful. For example, I’ll look at a model in which drift is described as both present and absent, though in different parts of the model. I’ll suggest that such modeling practices are fruitful, and that phrases such as “infinite population” need not have clear, unambiguous meanings in order to be useful in evolutionary biology.

1.2 Non-goals

Given (sometimes reasonable) misunderstandings or unfounded expectations about earlier versions of this paper, I suggest readers at least skim this section—which summarizes issues that are not part of the focus of the paper.

3.1 Why there is no evolution in (countably) infinite populations

Let’s start with some basic terminology. An allele⁹ is a genetic variant at a particular locus, a location on a chromosome. Some organisms in a population may have one or more other alleles at the same locus. We can also focus on genotypes, combinations of alleles at one or more loci, or even phenotypes—traits—if they can in some sense be transmitted from one organism to another. I’ll take natural selection to involve probabilities of changes in frequencies of such things in a population. A standard claim is that what is selected for has a higher probability of increasing in frequency than other mutually exclusive alleles/genotypes/phenotypes if no forces other than selection are acting. Compare natural selection with random drift, which is inversely related to population size:¹⁰ The smaller the population, the more that drift acts to add randomness to outcomes, for essentially the same reason that smaller samples have higher variance. Pure natural selection is known as a “deterministic” force that acts alone when a population is infinite; it is adulterated by drift when the population is finite.¹¹ Here is a quotation from a population genetics textbook that illustrates some of these ideas:

Unless some of the parameters, such as the selection coefficients and mutation rates, required to specify the above evolutionary forces fluctuate at random, these forces will be deterministic. In a finite population, however, allelic frequencies will vary probabilistically because of the random sampling of genes from one generation to the next. This process is called random genetic drift, random sampling drift, or simply random drift. (Nagylaki, 1992, 3, emphasis mine except in the last sentence)

Even without understanding all the technical terms, one can see that Nagylaki contrasts “deterministic” forces with what happens in a finite population, seemingly implying that deterministic forces would be what one would get in an infinite population. Nagylaki’s statement suggests that in an infinite population, natural selection (whose strength can be measured by selection coefficients) and mutation (whose strength is measured by mutation rates) act to change or maintain changing (relative) frequencies in a biological population in a deterministic way. When a population is finite, however, random drift introduces probabilistic variation in frequencies.

All the same, it requires only a little thought to see that the idea that of natural selection or mutation-caused changes in an infinite population is absurd. As I remarked earlier, this point has nothing to do with the fact that no actual biological population is infinite (e.g., Plutynski 2004; Giere 2009; Matthewson and Calcott 2011), nor that as a representation of a finite population, an infinite population model would attribute false properties to it (Potochnik, 2017). Since population genetic models are nearly always intended as approximate characterizations of their targets, and may involve a number of idealizations, the fact that an infinite quantity is used to model a large one need not in itself be problematic.

However, as Strevens (2019) and I (Abrams, 2006) implied earlier, the only relative frequencies definable in a (countably) infinite population are 0 and 1. For example, how many elements from an infinite set are needed to give us a relative frequency of 1/2? If we’re willing to consider arithmetic operations involving infinity at all, we should say that half of infinity is infinity, and that for any finite number $n$, $n/\mathrm{\text{infinity}}=0$. However, infinity/infinity is undefined, in general. So there is no subset of elements of an infinite set whose relative frequency is 1/2. The same point can be made about any number greater than 0 and less than 1. Yet evolution has to do with changes in frequencies of types—usually frequencies other than 0 and 1. Thus evolution in a countably infinite population makes no sense.¹² Again, the point is not simply that a model that supposedly represents evolution in an infinite population represents the world falsely. It’s not even that the idea of evolution in an infinite population is internally inconsistent, although that might be enough of a problem. The point is that there is simply no way to represent an absolutely central idea of natural selection—the possibility of change in relative frequency—in an infinite population. There is no way that a model in which there is supposed to be evolution in an infinite population, as such, can do any work in helping us to understand evolution. (“As such”, because I’ll suggest below that there may be less central, informal roles that ideas about infinite populations do play.)

3.2 Norton on idealization

Norton’s distinction between approximation and idealization in physics provides a useful perspective on talk of infinite populations in evolutionary biology. For example, Norton writes:

An approximation is an inexact description of a target system

e.g. of a real-world system. On the other hand, an idealization (in a different sense than defined above)

is a real or fictitious system, distinct from the target system, …

such that

an exact description of the [real or] fictitious system for aspects of interest turn out to provide an inexact description of the target system. (2014, 199, emphasis in original)

Thus, an approximation is a statement or a collection of statements that describe(s) a system in a way that is not fully accurate, while an idealization, in Norton’s sense, is a second system, perhaps imaginary, whose precise description would provide an approximation for a real system. Norton points out that the fact that a system has an idealization implies that it has a corresponding approximation, but that the reverse implication need not hold.

One of Norton’s illustrations of this distinction uses a model of a reversible process in thermodynamics, which represents heat as being transferred without an increase in entropy (Norton, 2014).¹³ In the model, one can let the temperature difference between two objects go to zero while simultaneously letting the elapsed time go to infinity. The model is reasonable for all nonzero values of temperature difference and all finite values of time as the limits are approached, but the idea of a system in which these limits are actually achieved is “nonsense” according to Norton (2014, 202). In such a system, basic physical principles would imply that the amount of heat transferred was represented by zero times infinity, an undefined quantity. This is why Norton calls the idea of a system in which the limit is actually achieved is nonsense.

We might apply Norton’s distinction to countably infinite populations in this way: An infinitely large population of organisms in which allele frequencies (for example) can change over time might be an idealization, if it could possibly exist. However, it can’t. It would have to have an impossible combination of properties.¹⁴ We could still say that phrases like “infinitely large population” provide ways of identifying models that are approximations to large populations. These models include, for example, those in which a real population is large enough that it can treated as if frequencies of alleles in one generation directly determine the frequencies of alleles in some later stage of a model.

3.3 Other “infinite population” models

Some models could easily be confused with the ones referenced above and discussed in the rest of the paper, so it’s worth briefly characterizing these other models in order to put them aside. Given the large number of modeling strategies that I want to exclude at this point, I hope readers will allow me to make broad generalizations about existing (and sometimes merely possible) evolutionary models in this section without referencing any models explicitly. The next sections of the paper will return to particular models of the kind that are my focus, discussing some of them in detail.

Note, first, that most of the points in the previous two sections could apply to an uncountably infinite population as well, but in this case the idea of the relative frequency of organisms with a particular trait has even less of a direct connection to real populations. For example, suppose that a model were to treat the number of organisms in a population as the number of points between 0 and 1, viewing the number of organisms with a trait $A$ as the number of points between 0 and 0.4. Of course 0.4/1.0 is well-defined, but this hypothetical model doesn’t represents the number of organisms with $A$ as the number 0.4; it represents that number of organisms as uncountably infinite. Dividing the uncountably infinite number of points between 0 and 0.4 by the uncountably infinite number of points between 0 and 1 is of course undefined.

However, biologists can and do create create useful models in which it’s stipulated that real numbers between 0 and 1 (inclusive) represent frequencies in a population. For example, 0.4 can represent the idea that 400 out of $N=1000$ organisms in a population have trait $A$. In this case the number 0.4 doesn’t represent a subset of points in $[0,1]$. The number 0.4 is also not a relative frequency per se: it’s not defined as a ratio between an integer representing a number of organisms and a population size. It’s just a real number stipulated to represent relative frequencies indirectly, and often approximately. Such real numbers can then be mathematically manipulated according to rules specified in the model. Though there is an (uncountable) infinity of points between 0 and 1, representing frequencies in this manner doesn’t imply that one is modeling the population as an infinite set of organisms.¹⁵

Let’s consider randomly sampling from a population. This is relevant to some examples described below, and it’s of interest because models of random drift are often conceptualized as involving random sampling from some set of organisms. Let’s start by considering the idea of sampling from either a finite or a countably infinite population. Here is one way to randomly choose an element from a finite set of size $N$: give each element equal probability $1/N$, and select elements with that probability. With a countably infinite set, $N$ is infinity, and as noted above, $1/\infty =0$, so the simple rule for random selection that works with a finite population would give each element of an infinite set zero probability of being chosen. That is, the obvious idea that we can sample organisms by giving each of them an equal probability doesn’t work when $N$ is infinity.

However, a model can also stipulate that there is a probability distribution over traits shared by many organisms, and then choose an organism with trait $A$ with probability $𝖯\left(A\right)$. (This stipulated distribution need not be derived from relative frequencies over a population size in the manner illustrated in the previous paragraph.) One can use these probabilities to model trait frequencies more directly. One way to do this is to view $𝖯\left(A\right)$ as the probability that an organism is randomly sampled from the population with probability $𝖯\left(A\right)$. This makes sense as a representation of relative frequency, because if one chooses a member of a finite population, giving each organism an equal probability of being sampled, the probability $𝖯\left(A\right)$ of choosing an organism with $A$ would be equal to the relative frequency of $A$ in that population. By starting with the probability distribution $P$, a model can abstract from details about particular population size, however.

Notice, now, that probabilities such as $𝖯\left(A\right)$ can be allowed to vary continuously from 0 to 1. So an uncountably infinite number of possible population states can be represented using such probabilities. However, noted above, the infinity of possible states doesn’t imply that the modeled population is infinitely large. The model simply uses a form of mathematical representation with an uncountably infinite number of possible values, to provide approximate representations of relative frequencies in a population that in real cases will have finite size.¹⁶

The important point is that while in general (a) it can be entirely legitimate to model a population using a continuously varying quantity that represents trait frequencies, (b) doing so does not have to force the assumption that random drift is absent. This is because though such models represent changes in frequency using a countably infinite number of values, this representation has no intrinsic connection to population size. It abstracts from population size. In fact, in diffusion models of evolution, a common kind of model with continuous variation in population states, drift is required by the formalism except in degenerate cases. In diffusion models, it is, rather, directional influences such as natural selection that are optional additions.¹⁷

In the rest of the paper, except where noted, I’ll use “infinite population” to refer to countably infinite populations that are infinite in size because they include an infinite number of discrete organisms. This is the idea of an infinite population that’s associated with the idea that drift fails to influence the population in some respect.

4.1 Terminological background

My first example is just a little bit complicated, but I believe it’s important to get a sense of how concepts are used in research, and not only in contexts where pedagogical constraints might distort normal practices. Vagne et al. (2015) investigated the evolution of genes at two loci when those genes interact to produce phenotypes in a manner known as “reciprocal sign epistasis”. These authors were interested in cases in which the resulting phenotypes’ fitnesses exhibited a pattern known as truncation selection. I’ll start by introducing two terms, “epistasis” and “linkage disequilibrium”. The latter will come up repeatedly in the case studies below. Readers already familiar with these concepts should feel free to skip to the next section.

4.2 Case 1: fixed-effect frequency

Vagne et al.’s models are of populations of sexually reproducing organisms with haploid genomes. This means that each organism has only one copy of each chromosome, and Vagne et al. assume that when two organisms mate, recombination can occur: corresponding pieces of chromosomes from each parent may be probabilistically swapped to produce each offspring’s chromosomes. The researchers specified relationships between phenotypes generated by the four genotypes $AB$, $ab$, $Ab$, and $aB$, so that $AB$’s phenotype was fitter than $ab$’s, which in turn was fitter than either $aB$’s or $Ab$’s phenotype. (One could abbreviate these relationships as $AB>ab>aB,Ab$, taking “$\underset{\_}{}>\underset{\_}{}$” to mean “$\underset{\_}{}$ has a fitter phenotype than does $\underset{\_}{}$”.) This implements a kind of sign epistasis with respect to fitness since, for example, a change from $a$ to $A$ would produce a more fit or less fit organism depending on whether there was a $b$ or $B$ allele at the B locus. It is reciprocal sign epistasis because the $AB$ and $ab$ genotypes are considered opposite extremes, and they are fitter than either of the intermediate genotypes $aB$ and $Ab$.

For each (discrete, nonoverlapping) generation in Vagne et al.’s models, the researchers calculated the frequencies of the four genotypes $AB$, $ab$, $Ab$, $aB$ after selection on possible parents, and subsequent random mating between those organisms they allowed to reproduce. That is, Vagne et al. modeled truncation selection in which a fixed percentage of organisms was allowed to reproduce; these organisms had the genotypes that produced the fittest phenotypes. Frequencies were calculated using specific values for a parameter $r$ that measures the probability of recombination between the A and B loci.¹⁹ Calculation of frequencies in the next generation also involved linkage disequilibrium, $D$, between $A$ and $B$. That is, in a given generation, it could turn out that $A$ and $B$ or $a$ and $b$ occur together more often or less often than would be the case if their combinations were determined randomly. For example, $D=0$ when the frequency of $AB$ is equal to the frequency of $A$ times the frequency of $B$; that is, there is no linkage disequilibrium in this case.

The researchers repeatedly iterated this process of calculating frequencies from the previous generation’s frequencies, and did this with variety of model parameters. Their goal was to determine the conditions that would allow certain outcomes, and to determine properties of the evolving population. For example, one property of interest was the number of generations until one of the four genotypes went to fixation, i.e. until all members of the population had that genotype.

Vagne et al. explicitly described their models as involving infinite populations:

A three-step process was applied to gain insight into the evolution of a population selected for a trait subject to reciprocal sign epistasis. First, we … determined the equilibrium states. … Secondly, times to fixation were estimated by numerical calculations in random mating in populations of infinite size. Finally, simulations of finite populations enabled us to investigate the role of genetic drift.

That is, after analyzing mathematical models with “populations of infinite size”, Vagne et al. performed simulations using populations of finite size. I’ll focus on the former.²⁰

Examination of Vagne et al.’s mathematical models shows that what makes the populations into ones “of infinite size” is that the models include terms for relative frequencies of genotypes $AB$, $aa$, $Ab$ and $aB$, and these terms enter into the calculation of which organisms are selected in a very direct way, as does the parameter $r$: the calculation of which combinations of alleles have which frequencies in the next generation is performed without the kind of stochastic fluctuation that would be expected in a finite population. For example, in one of Vagne et al.’s models, the frequency $p_{ab}^{n+1}$ of the $ab$ individuals in generation $n+1$ is (Vagne et al., 2015, 47, under “Configuration 1”):

$p_{ab}^{n+1}=\frac{p_{ab}^n}{p}-r{D}^{n\prime }$(1)

where $p_{ab}^n$ is the frequency of genotype $ab$ in generation $n$, and $p$ is the proportion of individuals selected to be parents of the next generation. $D^{n\prime }$ is the linkage disequilibrium, the (frequency-based) correlation between $A$ and $B$, in generation $n$, after selection. The formula for $D^{n\prime }$ is a little bit complicated, but I’ll point out what is important to notice about it:

$D^{n\prime }=\frac{p_{AB}^n{p}_{ab}^n}{p^2}-p_{aB}^n{p}_{Ab}^n{\left(\frac{p-p_{AB}^n-p_{ab}^n}{p(p_{aB}^n+p_{Ab}^n)}\right)}^2$(2)

The new terms here, $p_{AB}^n$, $p_{aB}^n$, and $p_{Ab}^n$, are frequencies of $AB$, $aB$, and $Ab$ respectively in generation $n$. The thing to notice about the calculation of $p_{ab}^{n+1}$ (the frequency of $ab$ in generation $n+1$) from (1) and (2), is that $p_{ab}^{n+1}$ is calculated solely from frequencies in generation $n$, plus a fixed proportion $p$ of individuals who reproduce, along with a constant recombination rate $r$. There is nothing stochastic in such a model. In particular, it does not include anything that could count as modeling genetic drift. In a real population, or in a model with drift, the relative frequencies in the current generation would typically influence the frequencies in the next generation, but the precise frequencies in the next generation would depend on how many individuals of each genotype happened to be selected stochastically from the current generation.

Thus, an infinitely large population as such plays absolutely no role in Vagne et al.’s “infinite population” model. Infinity plays no role in the mathematics. What is implied by “populations of infinite size” in Vagne et al.’s models is simply that frequencies in one generation can be calculated directly from frequencies in previous generations along with constant parameters, without any stochastic effects corresponding to genetic drift.²¹ (Later, when introducing their simulations, Vagne et al. say that “Drift was taken into account by setting a finite population size ($N=1000$ and $N=5000$)” (Vagne et al., 2015, 48).) We don’t have to worry about paradoxes concerning relative frequencies other than 0 and 1 in infinite populations, because infinity plays no role in the models’ calculations. Vagne et al.’s models, as they are used, involve no inconsistencies of the kind described in section 3.1.²² (This is not to say that we have to understand models as merely mathematical. At the very least, the variables in the Vagne et al. use are interpreted from the start.)

4.3 Case 2: equilibria²³

Zhao and Charlesworth also discussed a model having to do with linkage disequilibrium, in which a locus A that is not under selection can appear to be under selection if, by chance, it becomes correlated with a locus B that is under selection. Again the model assumes that there are two alleles at each locus. Zhao and Charlesworth write:

In the present case, the starting point is assumed to be a population of infinite size, at equilibrium under mutation and selection at the B locus and with no LD [linkage disequilibrium] between the two loci. It is thereafter maintained at a population size of $N$ breeding individuals each generation. (Zhao and Charlesworth, 2016, 1317, emphasis added)

In contrast to Vagne et al.’s model, “infinite population” here does not imply that frequencies will be applied directly in calculating the makeup of the next generation. The “infinite population” assumption only applies to the initial state in this model—before calculations are iterated to model subsequent generations with finite size $N$. One reason for describing initial population as having infinite size here is that it is only in large populations that linkage disequilibrium will be near zero, since linkage disequilibrium can be generated by genetic drift (e.g., Charlesworth and Charlesworth 2010, 380, 383; cf. quote from Wang below). So “infinite population” implies that LD is equal to zero. However, setting LD to zero does not required modeling evolution in an infinite population per se. If we were to take the infinite size claim literally, it would be difficult to make sense of the claim that mutation and selection have been at equilibrium in the initial population. Mutation is something that happens in individuals, and the overall rate of mutation in a population is thus a function of the number of organisms in it (e.g., Gillespie 2004, 29, 33; Charlesworth and Charlesworth 2010, 43). Selection, similarly, depends on frequencies of offspring per individual. However, as in the preceding section, there is no infinite population as such in this model. Rather, the model simply includes assumptions—such as there being no linkage disequilibrium—that would be unlikely to hold if there were drift.

4.4 Case 3: fixed-effect frequency with nonzero minimum value

The examples in the two preceding sections were from theoretical—but empirically oriented—articles. Examples from textbooks are useful as well when they reflect experience and established wisdom from modelers. In this section I consider an example from a well-known textbook that at first glance may seem to be in tension with some of my preceding remarks. Gillespie (2004) seems to model populations without drift in roughly the same way as in the example that I described in my section 4.2, yet Gillespie describes the population as finite. It’s worth looking closely at the text to see what he means.

I consider a model from Gillespie (2004, sec. 4.2). Gillespie does not describe this model as involving an infinite population; in fact the only reference to population size that does any work occurs in a description of parameters of a computer simulation used to generate a plot illustrating the model (110, cf. 117).²⁴ Gillespie says on page 110 that in the simulation, $N$ (population size) was set equal to 5000. By contrast, near the end of the next section of the book (4.3), which discusses models closely related to the one discussed in Gillespie’s section 4.2, he explicitly says that a new model will be “for an infinite population ($N=\infty$) in order to remove all effects of genetic drift” (Gillespie, 2004, 115f). Nevertheless, although Gillespie doesn’t describe the model in section 4.2 as involving an infinite population, it includes no randomness—in particular, no genetic drift—and in that respect is like the model near the end of Gillespie’s section 4.3.²⁵

For example, consider this equation from section 4.2 (Gillespie, 2004, 107),

$x{\prime }_1=\frac{x_1{\bar{w}}_1-w_{14}rD}{\bar{w}}.$(3)

This uses notation that’s a little bit different from Vagne et al.’s. It gives the next-generation frequency $x{\prime }_1$ of organisms with allele $A_1$ at the A locus and allele $B_1$ at the B locus. This frequency $x{\prime }_1$ is a function of the frequency $x_1$ of $A_1{B}_1$ organisms in the current generation, where $w_{14}$ is a fixed fitness parameter, and ${\bar{w}}_1$ and $\bar{w}$ are average fitnesses calculated from (i) fitness parameters and (ii) genotype frequencies in the current generation. $D$ is again linkage disequilibrium, although its calculation is simpler than in the Vagne example:

$D=x_1-p_1{p}_2$

Here $p_1$ and $p_2$ are the overall frequencies of the $A_1$ and $B_1$ alleles, respectively (p. 102).

Gillespie’s mathematical model has the same character as those that Vagne et al. use for “populations of infinite size” (quoted above): frequencies in the next generation are calculated directly from frequencies in the previous generation along with constant parameters. So it seems as if Gillespie’s model can be viewed as an infinite population model. Yet the computer simulation that generates a plot illustrating this model is said (p. 110) to use a finite population size ($N=5000$). So in what sense can this be a drift-free model? Gillespie gives sample computer source code at the end of the chapter as a solution to an exercise in which the reader is supposed to recreate the data that produced his figure.²⁶ In this code, a variable N is set to 5000. This variable appears exactly once more in the program, in the next line, where it’s used to initialize a variable eps (epsilon) with the value 1/(2$\times$N). The variable eps is in turn used for only one purpose, which is to determine when the frequency $p_1$ of the $A_1$ allele—the allele being selected for in the model—becomes close enough to 1.0 that the computer program should stop. The idea is that if the population size is 5000 and each organism has the A locus on two chromosomes, the number of $A_1$ alleles in the population would be 10,000. Thus if the frequency $p_1$ of $A_1$ in the population differs from 1.0 by less than eps $=1/10000$, that means that the number of organisms that lack the $A_1$ allele is less than 1, so the population has gone to fixation: all organisms have $A_1$ on both chromosomes. In that case, the simulation can be stopped.²⁷

Apparently, this is the only role that finite population size plays in the model described in Gillespie’s section 4.2—the model that is is embodied in the simulation code in his appendix. In other respects, the model is similar in character to models that are characterized as involving an infinite population, including Gillespie’s section 4.3 model. So we might view Gillespie’s model as both assuming both infinite population size and finite population size. That sounds contradictory. However, if we understand “infinite population size” in models as Vagne et al. seem to do—as implying merely that the model calculates next-generation frequencies directly from current generation frequencies—then there is no contradiction.

4.5 Case 4: drift and no drift

Gillespie’s model did not allow drift, though it depended in a narrow sense on assuming that the population was finite. Charlesworth and Charlesworth describe a more complex hitchhiking model that also includes what might, initially, be thought of as a theoretical conflict. The model first assumes that there is no drift, then adds an assumption that the population is finite, and subsequently considers a particular effect of drift in a finite population. The Charlesworths’ presentation is based on Barton’s version of a model introduced by Maynard Smith and Haigh (1974).²⁸ Most of the details of this Barton/Charlesworth model are a bit too complex to summarize here, but certain aspects of the model provide a good illustration of modeling practices that I want to highlight.

In modeling the effect of the introduction by mutation of a new, beneficial allele $B_2$ at one locus, Charlesworth and Charlesworth (2010, 410f, box 8.7) gradually develop and simplify an expression for the change $\Delta q_A$ of the frequency $q_A$ of an allele $A_2$ at a different locus. For more than half of the presentation, the Charlesworths’ model involves none of the stochastic effects that would usually be present in a realistic population of finite size. So this again is an “infinite population” model in the same sense that Vagne et al.’s model was. However, about two-thirds of the way through the presentation (p. 411, middle), the Charlesworths introduce a new element to the model. They note that since there is usually only one copy of a new allele when it is introduced by mutation,²⁹ the frequency of such an allele would be $1/2N$, where $N$ is population size—assuming that each organism has two copies of each chromosome, and hence of each locus. The Charlesworths then plug this value $1/2N$ into the equation for $\Delta q_A$ that they have derived without any explicit reference to population size, and without any drift-like effects represented. As the analysis continues, the authors note that most new mutations are lost by chance—essentially, because of drift—and they give an estimate for the probability that a new mutation persists in the population—i.e. that it is not lost. This estimate uses a function of population size and other factors ($s{N}_e/N$). The estimate is used to derive a new formula for the expected value of $\Delta q_A$. (Until the introduction of the chance that a new allele would be lost, bare quantities themselves were estimated, but once stochasticity was introduced into the model, the Charlesworths switch to a focus on expectations of certain values such as $\Delta q_A$.)

Thus in estimating the change in frequency of $A_2$ from one generation to the next, the Charlesworths take into account certain probabilistic consequences of finite population size in one part of the derivation, even though they ignore other probabilistic consequences of finite population size in earlier parts of the derivation. Like Gillespie’s model, this model can be described as treating a population as infinite in some respects (in Vagne et al.’s sense), but finite in others. We can also describe it as involving both drift and its absence. I don’t think there’s anything surprising here for biologists. Neither Barton (2000) nor the Charlesworths (2010) remark on what might seem, superficially, like a contradiction in the model. And given the model’s pedigree from Maynard-Smith and Haigh’s (1974) paper, to Barton’s (2000) revised model, subsequently enshrined the Charlesworths’ (2010) textbook, there is no reason to think that the appearance of contradiction is the result of a mistake, or that the model represents a problematic departure from mainstream evolutionary biology.

It’s valuable to see that there are no mathematical contradictions in the model. It’s true that there are assumptions associated with the mathematics that are inconsistent with each other: some parts of the mathematics are those associated with a drift-free evolutionary process, and others are associated with a process that incorporates drift. However, the mathematics itself is consistent. When the Charlesworths assume that a new mutation can be lost through drift in an otherwise “deterministic” model, they are simply adding a new premise—concerning the probability of loss of a mutation—to a mathematical derivation that had not previously incorporated terms for mutation. At each point in the Charlesworths’ derivation, it is clear what it is about possible real populations that is being approximated.

This kind of modeling “inconsistency” seems common. A look at many extended analytical treatments of population genetical models shows that, even after a model’s initial simplifying assumptions are stated, various other modifications are made in the course of the analysis.³⁰ That is, it’s not simply that different models of the same system may incorporate incompatible assumptions but—as in the Barton/Charlesworth model—the same model may incorporate incompatible assumptions. For example, a modeler might include a squared fraction in one part of a derivation, but treat it as equal to zero at another point because it will be small relative to other terms.³¹ Or a modeler might replace an expression with its mathematical expectation, and use the resulting value as if it were the original expression. This kind of piecemeal, flexible, jury-rigged, empirically based mathematical inference is not unique to population genetics, but it seems common in both applied and theoretical population genetical modeling.

5.1 Impossible worlds?

McLoone notes that biologists sometimes approximate the growth of a population (e.g. of rabbits) using a logistic equation model:

$\frac{dN}{dt}=aN(1-\frac{N}{K}).$(4)

Here $N$ is population size, $a$ is a growth rate, and $K$ is the carrying capacity of the environment, i.e. the number of organisms that it can support (cf. Roughgarden 1979, 303, 306; Hartl and Clark 1989, 518). McLoone remarks that mathematically, this treats population size as a continuously varying quantity, with an uncountably infinite number of states between 0 and the population size $N$. McLoone specifies, as an assumption I view as separate from the continuity assumption implied by the mathematical formalism, that “the population of rabbits is infinitely large ($N\to \infty$)” (McLoone, 2021, 12157).³² In section 3.3 I explained that a model with uncountable states needn’t be treated as embodying the idea of an infinite population. So the kind of model that McLoone discusses doesn’t seem to require an infinite population in the sense that is my focus here. Nevertheless, McLoone proposes a novel strategy for thinking about models with infinite states, and while this isn’t the place for a full discussion of his paper, it’s worth considering whether his approach could be relevant to some of the models I discussed above.

McLoone notes that while the model he discusses represents population size as a continuous quantity, no rabbit population size varies continuously. He argues that in order to understand practices involving such models we should use a semantics for scientific statements that allows impossible worlds as well as possible worlds. McLoone’s paper goes on to describe such a semantics based on the common philosophical idea of defining possible worlds as sets of propositions.³³ Treating worlds as sets of propositions makes it possible to define “impossible” worlds, ones that contain inconsistent propositions, by including in a world (i.e. a set of propositions) some propositions that are inconsistent with each other.

McLoone’s argument for using impossible world semantics is based on some authors’ arguments that models should be understood as specifying counterfactual worlds. McLoone suggests that a semantics involving impossible worlds would be needed to capture the meaning of counterfactual statements such as

If a population of rabbits satisfied the assumptions of the logistic equation, then its size would eventually equal the carrying capacity. (McLoone, 2021, 12161)

However, McLoone’s argument doesn’t seem to depend on models for continuous quantities per se. Since many models idealize in the sense of misrepresenting what is modeled (e.g., Weisberg 2013; Potochnik 2017), any statement that conjoins assumptions of a model with truths about what is modeled is likely to involve contradiction.

McLoone may be right about such model-world conjunctive statements, and he may be right that philosophical views that take models to specify counterfactual worlds mean that such statements are implied by scientific models like the logistic rabbit model. However, I don’t believe we need to take such conjunctions as central to modeling in biology. One common view is that a model based on the logistic equation uses continuous quantities to represent discrete numbers of rabbits (e.g., Weisberg 2013; Morrison 2015; Gelfert 2016; Potochnik 2017), rather than identifying discrete and continuous numbers (cf. §3.3). We needn’t assume that conjunctions of model assumptions and what’s true in the world must play a role in modeling practices in biology. We only have to assume that the model is used to represent the world. So I don’t see this aspect of McLoone’s approach as needed for cases like those discussed in section 4.

However, my argument in section 3.1 was not that (countably) infinite populations give rise to a paradox in the model-world relation. The paradox was in the model itself. If semantics involving impossible worlds was potentially valuable in some contexts, one might wonder whether they could be used to make sense of biologists’ talk of infinite populations. Let us consider the idea of a non-actual world in which there was evolution in a countably infinite population. Such a world would be one, I suppose, that contained both the proposition $\alpha$ that the number of organisms of type $T$ was infinite, and a proposition $\beta$ that the relative frequency of organisms of type $T$ was equal to some finite number strictly between 0 and 1. These statements are inconsistent, as I argued, but that seems OK for a semantics that allows impossible worlds. The thing to notice, however, is that the infinite population claim would not do any work in a model corresponding to this picture. For modeling purposes, we would need propositions related to $\beta$ that stipulated how relative frequencies of trait types changed over time (§3.3), but propositions such as $\alpha$ concerning the infinite size of the population would play no role in inferences about changes in frequencies. So an impossible worlds semantics doesn’t add anything of immediate use for thinking about evolution in an infinite population.

On the other hand, perhaps it could be argued that a semantics involving impossible worlds would provide a useful way to think about the Gillespie and Barton/Charlesworth models of sections 4.4 and 4.5, since in these models, there is both a finite population, or drift, in one sense, and an infinite population, or no drift, in another. Note that this would be a claim only about the model itself, and not about populations represented by the model or the mathematics of the model. I don’t think we need impossible world semantics to understand such cases, and I’ll describe what what I see as better alternatives. However, I won’t rule out the possibility that some views about modeling might have a role for such a strategy.

5.2 Limits

The examples I discussed in section 4 concern models that, I argued, incorporate particular assumptions that have sometimes been described as consequences of assuming an infinite population. Specifically, the models assume that frequencies at time $t$ directly determine subsequent changes (§§4.2, 4.4, 4.5), or they assume the absence of linkage disequilibrium (§4.3). Given usual ways of thinking about evolutionary processes, both of these assumptions imply the absence of stochastic effects of random drift—effects whose probability would go to zero as population size increased. So we might suggest that statements about infinitely large populations are “justified as shorthands for claims about limits as population size is increased without bound” (Abrams, 2006, 264f), or that as Sober suggested, “the meaning behind the idea” (1984, 44) of an infinite population should be given in terms of limits (see §2). Strevens (2019) argued that reasoning using such limit statements should be viewed as “a rational reconstruction of what is going through population geneticists’ minds when they advance the infinite population idealization”.³⁴ The general idea here is that probabilities of effects of drift lessen as population size is increased, and these effects have a probability of zero in the limit.³⁵ Mathematical models that ignore drift would be literally true of what happens in the limit. This is consistent with the argument in section 3.1 that evolution in an infinite population per se is nonsensical. As Norton (2014) notes (§3.2), something can be true in the limit as $N$ increases toward infinity without it being true for $N$ equal to infinity.

The following statement from a recent scientific paper can be seen to lend some support to this limits view of infinite population claims. (It’s not necessary to understand all of the details of the quotation.)

… with [population size] $N=64$, linkage disequilibrium is important [has a significant effect] and does not decrease much with genome size. Even with an infinite genome size (i.e. free recombination between loci), the correlations are still 0.95 between $r_G$ and $r_M$ and 0.88 between $F_G$ and $F_M$. When the population size is increased to 512, these correlations reduce to 0.92 and 0.66. It is predictable that these correlations reduce to zero for an infinite genome size in an infinite population. (Wang, 2016, 8, emphasis added)

Here $F_G$, $F_M$, $r_G$, and $r_M$ are different ways of measuring genetic relationships that would be influenced by linkage disequilibrium. The quotation shows that Wang is envisioning a progression as population size $N$ (and genome size) increases. The last sentence shows that the terminal step of this progression is an infinite population (of organisms with infinite genomes). However, the justification for this point comes solely from a claim about limits. Thus, one might reinterpret claims about evolution in infinite populations as being about limits as $N\to \infty$, but not about populations where $N=\infty$ (since that makes no sense).

However, if the limit interpretation were correct about all references to infinite populations in evolutionary biology, the Gillespie model and the Barton/Charlesworth model discussed above show that this claim might have to be understood in a nuanced way: It may be correct that in one part of the mathematical development of a model, the assumption that there is no drift is justified by allowing population size $N$ to go to infinity. However, as we’ve seen, the model can assume a finite $N$ at another step. So the limit justification could not apply to the entire model in such cases.

5.3 Large numbers

Another problem with understanding infinite population talk solely in terms of limits is that though papers such as those of Vagne et al. (2015) and Zhao and Charlesworth (2016) are quite theoretical, they seem to be oriented toward possible empirical applications—where population sizes do not increase toward infinity. Zhao and Charlesworth’s interest in empirical applications is shown by their remark that their “results shed light on experiments on the loss of variability at marker loci in laboratory populations” (Zhao and Charlesworth, 2016, 3). Vagne et al.’s empirical orientation is apparent from the article’s title itself: “When is recombination favorable in a pre-breeding program with a selfing species?”. Here the “term ‘pre-breeding’ refers to the transfer of genes from related wild ancestors or from ancient varieties to breeding material” (Vagne et al., 2015, 45). That is, the models in the paper are intended to illuminate what happens to genetic patterns when animal or plant breeders practice artificial selection on a population of animals, at least some of whom were formerly wild. Similar methods are sometimes used for experiments on evolution (e.g., Alexander et al. 2014). Gillespie’s and the Charlesworths’ textbooks also repeatedly highlight connections to empirical work.

Why would it matter for empirical studies what happens as $N$ increases without bound? Why discuss “infinite population” models at all, if they concern what would be the case in the limit? One reason is that if we understand infinite population size in terms of limits, that implies that if a population is sufficiently large, its behavior will be close to the behavior of a population as size goes to the limit.

So perhaps all talk of infinite populations is simply a way of talking about large, finite populations. Infinite population models are often intended to give us insights about the approximate behavior of large populations. In fact many textbooks and articles fail to talk about infinite populations where they might have been expected to do so. Earlier, in section 3, I quoted a passage from the introduction to a textbook by Nagylaki that seemed to allude to infinite populations. However, when Nagylaki gets down to work and discusses actual models, he avoids talk of infinite populations, writing, for example:

The total number of offspring,

$N=\sum _i{n}_i\left(2.1\right)$

must be sufficiently large to allow us to neglect random drift. (Nagylaki, 1992, 5)

Thus one can ignore random drift, which involves changes whose probability goes to zero as $N$ goes to infinity, as long as $N$ is large enough that those probabilities are small. Another illustration can be found in an article by Waxman and Loewe:

Within the life cycle we assume that a very large number zygotes of all genotypes are produced, so that viability selection is essentially deterministic in character. (Waxman and Loewe, 2010, 246)

That is, a “large number” of organisms (in the form of fertilized eggs) are produced, so selection is “essentially deterministic”, i.e., it occurs without the interference of drift. In the Charlesworths’ textbook, there is an explicit statement of this equation of the “infinite” with the merely large:

… we assume that … the population size is so large that random fluctuations in genotype frequencies can be disregarded (for convenience, this is called an “infinite” population) (Charlesworth and Charlesworth, 2010, 50).

In the previous section, we considered the possibility that “infinite population” meant “in the limit as $N$ goes to infinity.” The examples in this section, by contrast, suggest that “infinite population” sometimes means “large population”—i.e. large enough that the influence of drift can be ignored.

5.4 Beyond limits and large numbers

It’s somewhat reasonable to think that “infinite population” always means “large population” or “in the limit as population size increases”. However, if the point of large populations or limiting size claims is that they are what justify leaving various effects of drift out of a model, how are we to understand the Gillespie and Barton/Charlesworth models? The first ignores effects of drift without assuming the population is large,³⁶ and the second would seem to need its assumptions justified both by assuming a population is large and by assuming it is not large. (A population’s size might be large enough to ignore for one purpose, but not for the other. However, it doesn’t seem as if these two models are supposed to be to be restricted to carefully tailored ranges of populations sizes in this way.) Similar remarks can be formulated in terms of limits as $N\to \infty$.

I think the most reasonable view is that in such cases, the modeler(s) simply idealize away effects of finite population size in one part of the model, but not in another. Or more specifically, they assume the absence of drift in one part of the model, but allow drift or other effects of finite population size in another part of the model. Recall that in neither the Gillespie model nor the Barton/Charlesworth model was there any mathematical inconsistency. There is no need to think of population size as both large and small, or as taking $N$ to the limit while keeping $N$ bounded.

Now, neither Gillespie nor the Charlesworths used the term “infinite population” to describe the models I’m currently discussing. However, either there are drift-free models that involve infinite populations in some sense, and also drift-free models that don’t; or it’s incorrect to identify claims about the “infinite size” of a population with the absence of drift within a model. The former alternative seems incorrect, because there appears to be no meaningful difference between models in which drift is absent that are associated with phrases like “infinite population”, and those that aren’t. Describing a model as involving an infinite population can aid communication, but it’s inessential. My guess is that Gillespie and the Charlesworths avoided using “infinite population” for these two models simply because they recognized that using “infinite population” would have confused readers. They did use the phrase elsewhere, in places where it would not have been confusing (§§4.4, 5.3).

Thus it seems most plausible to view “infinite population” talk as an informal way to convey that some effects of drift that might have been thought relevant to a model are excluded from it. This means that an infinite population model does not incorporate an idealization in the form of an infinite population assumption. That is, the model does not make a false assumption that a represented population is infinite. On the other hand, such models do incorporate idealizations in the sense that they falsely represent real populations as if they were free from one or more effects of drift.³⁷ But it’s wrong to view “infinite population” talk as always meaning (Sober, 1984), as justified by (Abrams, 2006), or as rationally reconstructible as (Strevens, 2019) statements about limits.³⁸ Further, contrary to what Potochnik wrote, it seems clear that an infinite population is not a “requisite assumption whenever genetic drift is not taken into account” (Potochnik, 2017, 55).

Virtues of loose language

The case studies and quotations above (§§4, 5.2–5.4) suggest that sometimes biologists think about infinite population assumptions as involving limits, while at other times they think of these assumptions as merely concerned with large populations. I suggest that at a minimum, “infinite population” means nothing more than “we’re leaving out such and such particular effects of drift”. Some readers may want an analysis of “infinite population” that gives it a more specific, determinate meaning. But in each of the examples I discussed in section 4, it was always clear from context what assumptions about drift or its absence were to be used. When authors used phrases like “infinite population”, it was clear what the implications of the phrasing was. Shouldn’t that be enough? I see no reason for assuming that biological language must express meanings with more precision than is needed for successful research. There is little reason to think that “infinite population” has a single, precisely specifiable meaning (even putting aside models that use uncountable numbers of population states, such as those I discussed in section 3.3).

Consider Keller’s response to the idea that scientists should adhere to the “harsh mandate” (p. 118) of always working to replace figurative language with precisely defined terms:

The difficulty is obvious: scientific research is typically directed at the elucidation of entities and processes about which no clear understanding exists, and to proceed, scientists must find ways of talking about what they do not know—about that which they as yet have only glimpses, speculations. To make sense of their day-to-day efforts, they need to invent words, expressions, forms of speech that can indicate or point to phenomena for which they have no literal descriptors. … Making sense of what is not yet known is thus necessarily an ongoing and provisional activity, a groping in the dark; and for this, the imprecision and flexibility of figurative language is indispensable. (Keller, 2002, 118)

On my view, the term “infinite population” is optional figurative language. Rather than a being used to grope toward an understanding of new phenomena, as in Keller’s quotation, the flexible associations of “infinite population” may aid communication about similar but different modeling patterns in related contexts. I suggest that the variation in uses of “infinite population”, and the deployment of different assumptions about drift and population size that I’ve described are consequences of this flexibility.

This is not to say that all important terms and concepts in evolutionary biology have the same flexibility. Some terms may come to have single, precise meanings, or may have a series of more or less precisely defined variants. For example, Ereshefsky (1992) may be right that there are only a few alternative species concepts, and the few gene concepts that Griffith and Stotz (2014) described may capture many common uses of “gene”. On the other hand Waters argues that

Sometimes it is useful to be vague, and in such contexts biologists invoke a blunt concept akin to the gene concept of classical genetics …. In other contexts it is important to be precise. When precision is important, biologists employ what I have called the molecular gene concept …. (Waters, 2017, 94, emphasis in original)

Waters characterizes the molecular gene concept as a parameterized family of ways of narrowing its application:

The molecular gene concept can be specified as follows:

A gene $g$ for linear sequence $l$ in product $p$ synthesized in cellular context $c$ is a potentially replicating nucleotide sequence, $n$, usually contained in DNA, that determines the linear sequence $l$ in product $p$ at some stage of DNA expression.

The reference of any gene, $g$ is a specific sequence of nucleotides. The exact sequence to which a $g$ refers depends on how the placeholders $l$, $p$, and $c$ are filled out. (Waters, 2017, 95)

So according to Waters, while some gene concepts are vague, the molecular gene concept is not. It is a parameterized family of more precise applications of the concept (see also Waters 2019).

Somewhat like the molecular gene concept, “infinite population” language is associated with a family of closely related ways of thinking about and applying ideas about population size and drift. “Infinite population” language per se is so vague, however, that it is just these associations that constitute what the term amounts to in practice. Although the associated ideas about population size (size in the limit, large size, etc.) and about effects of drift (frequency change, loss of mutations, linkage equilibrium, etc.) are logically and mathematically related, they can’t be organized in terms of a few parameters as in the case of the molecular gene concept. The applications of ideas concerning drift in modeling remain very flexible.