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Keywords:

  • isolation by distance;
  • kin selection;
  • population structure;
  • probability of fixation;
  • relatedness

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

The analysis of kin selection in subdivided populations has been hampered by the lack of well-defined measures of genealogical relatedness in the presence of localized dispersal. Furthermore, the usual arguments underlying the definition of game-theoretical measures of inclusive fitness are not exact under localized dispersal. We define such measures to give the first-order effects of selection on the probability of fixation of an allele. The derived measures of kin selection and relatedness are valid in finite populations and under localized dispersal. For the infinite island model, the resulting measure of kin selection is equivalent to a previously used measure. In other cases its definition is based on definitions of relatedness which are different from the usual ones. To illustrate the approach, we reanalyse a model with localized dispersal. We consider sex ratio evolution under sex-specific dispersal behaviour, and the results confirm the earlier conclusion that the sex ratio is biased towards the sex with the dispersal rate closer to the optimal dispersal rate in the absence of sex-specific dispersal behaviour.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Hamilton (1964) formulated his famous rule according to which, if an actor expresses a deviant behaviour which costs c offspring to itself and increases by b the number of offspring of individuals related to the actor, this behaviour is selected for if br − c > 0 where r is a measure of relatedness. There has been much debate over the interpretation of the fitness costs c, benefits b, and of the relatedness r which make Hamilton’s rule work.

Tools such as the direct fitness method ( Taylor & Frank, 1996) clarify such issues. In this method, the fitness (expectation of number of offspring) of a focal individual is given as a function of its own behaviour and of the behaviour of all individuals whose gametes compete with the focal individual’s gametes. Thus, one considers how different ‘actors’ affect the fitness of a single focal ‘recipient’. Derivatives of the fitness function with respect to variables representing the phenotype of actors are used to measure the effects of a small change in the actors’ phenotypes on the focal individual, and ‘relatedness coefficients’rj are used to measure the similarity of the focal’s and neighbours’ phenotypes. Application of this method leads to a measure of the form ∑j bjrj in which the bj are obtained as partial derivatives of the fitness function. In this sum, the effect bj of some actor on the focal individual is weighted by rj which measures the extent to which the actor has a deviant behaviour given the focal individual transmits the allele for the deviant behaviour. The measure of selection is thus a sum of the effects of all actors on the focal individual, given this individual transmits a deviant allele.

So why return to these issues? One reason is that in specific models the rj’s are often computed in a way not consistent with definitions given in elementary treatments (e.g. Grafen, 1985; Maynard Smith, 1998) as well as more mathematically orientated derivations ( Taylor, 1996) of inclusive fitness measures. These treatments refer to a ‘pedigree definition’ stemming from, e.g. Malécot (1939) and Cotterman (1940; reprinted 1974), and found in most textbooks, whereas specific models for subdivided populations often use a ‘mutation definition’ of identity by descent also owing to Malécot (1948). It is important to realize that this measure is in general not a measure of pedigree relationships ( Seger, 1981). A fitness measure based on a single definition of identity would be useful.

Another reason for a reanalysis is that previous attempts do not provide a rigorous and general justification for inclusive fitness measures by a link to expected changes in gene frequency. Different views have been expressed as to when exactly Hamilton’s rule would apply. Some formulations focus on ‘rare’ alleles (e.g. Taylor, 1989, 1996– there, ‘rare’ means a limit result when gene frequency goes to zero). Others extend the results for rare alleles to any allele frequency, using an argument originally owing to Hamilton (1970) (e.g. Maynard Smith, 1998–‘rare’ then means that ‘unrelated’ individuals cannot have identical genes, which seems to imply that gene frequency is null for an infinite population). Hamilton’s argument was based on the assumption that in the absence of selection, there is a regression coefficient r independent of allele frequency, such that

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where q is the allele frequency in some individual related to a focal individual, E(q|p) is the expectation of q conditional on allele frequency p in the population, and X is the allele frequency in the focal individual. In this case, Hamilton showed that the direction of selection would be independent of allele frequency (e.g. Hamilton, 1970; Harpending, 1979; Seger, 1981; Frank, 1998, p. 42; see also below). On the other hand, if this assumption is not correct, selection is frequency dependent. Frequency-dependence is actually known in some kin selection models (e.g. Seger, 1981; Uyenoyama & Feldman, 1982), not because eqn 1 is necessarily wrong but because the resulting fitness measure would be insufficient to predict the direction of selection. These issues are not considered here.

Grafen (1985, pp. 59–60) raised the point that in subdivided populations, there may not be any ‘regression coefficient’r with the above properties. Simulations (not presented here) show that Grafen’s point is valid, particularly when dispersal is localized. There is also ambiguity whether p above is the present allele frequency (which is necessary for the formula to be useful) or is some ‘ancestral’ allele frequency (which is used in an elementary proof of this formula in Cotterman’s framework; see, e.g. Crow & Kimura, 1970, p. 65).

A more coherent and general basis for game-theoretical measures of kin selection is needed. Here we will show how useful measures of kin selection and relatedness may be constructed by a different approach, introduced in the next section. We focus on a class of kin-selection models characterized by spatially homogeneous dispersal. In other words they have an ‘isolation by distance’ type of dispersal, with the island model as a special case. Such models are basic in the development of kin selection theory, including models of dispersal (e.g. Comins, 1982; many later works) and other competitive interactions (e.g. Taylor, 1989, 1992b). Our analysis resolves the above problems for finite populations. It leads to some disagreement with current interpretations of the relationship between inclusive fitness measures and allele frequency dynamics. In particular, eqn 1 is not true in general, but at the same time it is not always required. Implications of these results for the use of measures of kin selection in game-theoretical analyses are also discussed.

Measuring selection

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Models of kin selection generally focus on the direction of selection over one generation, given allele frequency is p in the population. They aim to give a first-order (‘weak selection’) approximation for frequency change in terms of easily calculable relatedness coefficients. However, when selection is frequency-dependent, this is not very informative for the outcome of selection. A formal answer to the latter problem would involve the expected distribution of allele frequency under the joint effects of frequency-dependent selection and genetic drift. However, such an answer is usually beyond our reach.

One useful measure that turns out to be accessible to analysis is the expected frequency of a gene. In the low-mutation limit, it directly reflects differences in the probability of fixation of unique mutants as a result of selection. More specifically, if π[RIGHTWARDS ARROW] A and πA [RIGHTWARDS ARROW] a are the probabilities of fixation of an a mutant in an A population and of an A mutant in an a population, respectively, then in the low-mutation limit the expected frequency of allele A is πa [RIGHTWARDS ARROW] A/(πa [RIGHTWARDS ARROW] A + πA [RIGHTWARDS ARROW] a), and this is also the probability that allele A is fixed in the population. The present framework aims to derive a first-order measure for the effect of selection on expected gene frequency and on probability of fixation, and then to define measures of relatedness and of fitness which bear a simple link with such quantities.

The model

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

The main notations used below are summarized in Table 1.

Table 1.   Summary of main notations. Thumbnail image of

Formulation of the model

Population structure and life cycle

The population is made of a finite number n of demes which are homogeneously distributed on a lattice. For simplicity, we consider a closed one-dimensional lattice on a circle (see Fig. 1), with symmetrical migration from each deme in both directions. The extension to two-dimensional lattices is straightforward (see Gandon & Rousset, 1999, for examples). We consider models where, as shown on Fig. 1:

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Figure 1.  Life cycle and geometry of interactions in the model. See text for details.

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1 individuals affect the fecundity of other individuals, i.e. the relative number of gametes they produce;

2 gamete dispersal occurs;

3 differential gamete death according to parental genotype may occur at this stage; and

4 further competition occurs between all surviving gametes that have dispersed to one deme, independent of genotype, so that only N of them develop into haploid adults.

Mutation

Mutation occurs at a locus determining the trait under study. A symmetric two-allele (a and A) model with mutation rate u will be considered. Two alleles are identical in state if they are of the same allelic type and identical by descent if no mutation occurred in the lineages leading to their common ancestor (the ‘mutation definitions’ of identity in state and by descent). As the total population size is finite, the probability of identity of any pair of genes goes to 1 when u [RIGHTWARDS ARROW] 0 owing to the random fixation of one of the alleles. Frequency (p) is the frequency of allele A among the Nn adults. It includes all demes connected by migration. It is not an ‘ancestral’ allele frequency.

Selection

The genotype of a haploid individual is a random variable with value a or A, and so is the associated phenotype, with value za or zA. The generic notation Z will be used for the latter random variables. A variable Z(i) may be defined for each individual i (i=1,…,Nn). Fitness is the expected number of adult offspring of an individual. The fitness W(i) of an individual i is defined as this expectation, evaluated conditionally on its phenotype and on the phenotypes of other individuals (this corresponds to neighbour-modulated fitness, Hamilton, 1970). We consider here models where interactions are spatially homogeneous, so that the fitness of each individual may be expressed as a function of its own phenotype, of the phenotype of other individuals in the same deme, of the phenotype of individuals one step further on the lattice, of the phenotype of individuals two steps further on the lattice, and so on. Thus, it is convenient to express fitness as a function of phenotypes of individuals at positions relative to that of the focal individual on the lattice, rather than considering absolute positions (i.e. the i index). All of the individuals will be considered in turn as the focal individual, and given individual i is taken as the focal individual, we write Z for its phenotype Z(i), and z for the realized value of Z(i). Likewise, the phenotypes of its neighbours at different distances form a vector of random variables, Z ≡ Z(i), whose elements will be indexed according to their position relative to the focal individual. Thus Z0 ≡ Z0(i) will be the average phenotype of neighbours in the same deme as individual i taken as the focal individual, Z1 ≡ Z1(i) will be the average phenotype of neighbours in a deme adjacent to the focal individual and Zn−1 ≡ Zn−1(i) will be the average phenotype of neighbours in the other adjacent deme, etc.

We consider models in which fitness takes the following form. Let gr be the relative contribution of an individual to gametes competing after phase (iii) in a deme r lattice steps apart. Migration rates between demes depend only on their relative position, so that the relative contribution of an individual in deme j to gametes competing in deme l may be written as gl−j (where gl−j ≡ gj−l). Let Zj be a vector obtained by circular permutation of the elements of Z and with first element Zj (for example, Z1=(Z1, Z2,…,Z0)). Then the expected number of adult offspring of focal individual i is

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hence we consider the fitness function w (z, z) written as

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where z is the variable describing the phenotype of a focal individual (thus z is equivalent to y in Taylor & Frank, 1996), z ≡ (z0,z1,…,zn−1) where zj is the variable describing the average phenotype of individuals at j steps from this individual on the lattice (here including the focal individual itself for j=0), and zj is a vector obtained by circular permutation of the elements of z and with first element zj (for example, z1=(z1, z2,…,z0)). Examples are given below in eqns 19, 20 and above in eqn 7. This form of the fitness function follows from the assumptions of spatial homogeneity of dispersal and other interactions. The number of gametes produced is a function of (zj, zj) by the assumption of spatial homogeneity of interactions determining fecundity, and the l − j index in glj follows from the assumption of spatial homogeneity of backward dispersal rates: the fraction of gametes produced in deme j which is sent to deme l is a function of the distance l − j but not of l and j separately.

As fitness is the number of adult offspring, and the total number of adult offspring is also the number of adult parents, Nn, the average fitness over all individuals in the population ( below) is necessarily 1, whatever may be the genetic composition of the parental population. Note that this does not appear to be always so for the ‘mean fitness’ () in, e.g. Frank (1997), and that the results below apply only when fitness is defined as here. Another consequence of this definition is that

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where ∑n−1j = • is a sum over all variables of the fitness function. A simple formal argument is given by computing dw(z,…,z)/dz in two ways. First, the expected number of adults offspring of all individuals is 1 when all individuals are identical: w(z,…,z)=1 for any z. Thus the derivative is zero. Second we compute the same quantity as a total derivative dw(f(z), f0(z),…, fn−1(z))/dz where we set set all functions fj(z)=z, i.e. as

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Thus, the latter expression is 0. Sums of higher order partial derivatives are likewise shown to be zero. It will be very useful at several steps in our analysis. In particular, it implies that we can express any partial derivative as the sum of the others, and therefore we can eliminate one derivative from some expressions when convenient.

Measures of genetic identity

Q will be used as a generic notation for genetic identity. Thus Qj:A|p reads as the probability that two genes at j steps from each other on the lattice are identical and of allelic type A, conditional on the frequency of A being p in the population. The qualificative |p is removed when the unconditional probability is considered, and the :A is removed when identity is considered regardless of the allelic type.

Constructing the measures of selection

Consider the change in gene frequency of A when copies of a express phenotype za and copies of A express phenotype zA ≡ za + δ. The argument developed in Appendix 1 yields a compact and intuitive expression for changes in gene frequency owing to selection over one generation, Epp]sel:

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(recall Q•:A|p=p). All derivatives here are evaluated in za for all variables. This expression weighs the effects of neighbours at distance j on the number of offspring of the focal individual, ∂w/∂zj, by the probabilities that this individual shares allele A with a random neighbour at distance j (here conditional on p). Thus the contribution of a neighbour is higher when it is more likely to share the A allele with the focal individual (higher Qj:A|p) and, given both individuals bear the A allele, when its effect on the fitness of the focal individual is higher (higher ∂w/∂zj).

We have considered identity in state, no particular population structure or size, no particular mutation process and mutation rate. p is not the allele frequency in some ancestral ‘reference’ population; it is the frequency in the collection of demes we are examining. Nor is identity in state supposed to be a function of identity by descent and of ancestral allele frequency.

To make clear the significance of the different parameter, consider the following elementary example of selection in a panmictic population of N individuals, where A’s fecundity is (1 + sδ) relative to a, irrespective of allele frequency. It is well known that the change in gene frequency owing to selection is δsp(1 − p)/(1 + δsp)=δsp(1 − p) + o(δ) in this model (e.g. Wright, 1969, p. 30). In the above framework we may write za=0, zA=δ, and w(z,z0)=(1 + sz)/(1 + s(z + (N − 1) z0)/N) where z0 is the average phenotype of all other individuals (the focal one being excluded here). Then ∂w/∂z=(N − 1)s/N=−∂w/∂z0 (all derivatives being evaluated in z=z0=za=0). Furthermore, we have only one Q:A|p which is simply the frequency of AA pairs of genes among all pairs of different genes in the population: Q:A|p=p2 + p(p − 1)/(N − 1). Hence

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as expected. Note that in more complex models, different population configurations with the same allele frequency p may have different frequencies of AA pairs of genes for a given type j of pair of genes. Then, Q:A|p is only the expected frequency of AA pairs over the different possible populations where frequency is p, generated by the stochastic process, not the frequency of AA pairs in any particular population. All of the conditional expectations Ep[…] considered here must be interpreted similarly. The main reason for considering such conditional expectations will be to compare results with previous measures of ‘inclusive fitness’ and of ‘relatedness’.

Two measures of inclusive fitness

From eqn 6 we now define two different quantities that have been used as ‘inclusive fitness’ in previous work.

Hamilton’s approach to define ‘inclusive fitness’WIF(p) was to express gene frequency change in the form p(1 − p)WIF(p)δ + o(δ), as in eqn 7. A convenient expression for WIF(p) is obtained noting that property (4) implies that we can add any constant term we wish in all the coefficients of the partial derivatives of the fitness function without changing the whole expression (6). If we subtract p2, the sum in (6) becomes

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This is of the form p(1 − p)WIF(p) for

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If eqn 1 is valid, for every j there is some rj independent of p such that

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(see eqn A.2). Then WIF(p) takes the form

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which is frequency-independent. Equations 9 and 11 have an interpretation similar to eqn 6, with the difference that we are looking at ‘fitness’ rather than at expected change in gene frequency. The first term is the effect of the actor on itself given it bears allele A and is independent of gene frequency. The other terms weigh the effects of an individual at distance j, ∂w/∂zj, by the ‘regression’ coefficient rj measuring again the similarity between an actor and the focal individual. The interpretation as a regression coefficient follows from interpreting Qj:A|p − p2 as a covariance and p − p2 as a variance in genic value (e.g. Taylor, 1996, p. 669).

Another quantity that has been de facto used as ‘inclusive fitness’ in previous works may be obtained from eqn 6 by considering the unconditional expectation of Epp]sel over the distribution of allele frequency p. In this way the unconditional probabilities of identity Qj:A substitute for the conditional ones Qj:A|p, as follows:

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Here we consider a two-allele model with mutation rate u for each allele and hence the expected frequency E[p] of allele A is E[p]0=1/2 is in the absence of selection. Likewise the probability Qj:A that two genes are both A is the same as the probability that two genes are both a, and is half the probability that two genes are identical, Qj/2. Thus, we can write the first-order coefficient in (12) as S/2 where

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This expression may be seen as an extension of fitness measures generated by the direct fitness method ( Taylor & Frank, 1996): derivatives of the fitness function are weighted by measures of genetic similarity. If we substitute identity by descent to identity in state in the above formula, we have the inclusive (or direct) fitness measure used in Taylor (1992b, 1994), which will be denoted here. Comins (1982) similarly considered identity by descent, but his fitness measure seems different from that obtained by application of the direct fitness method; see Gandon & Rousset (1999).

In the literature, ‘inclusive fitness’ refers to both and WIF(p). This is confusing because is of the order of the mutation rate: in the low-mutation limit, the allele frequency distribution concentrates on p=0 and p=1, and there is no gene frequency change owing to selection in these points, so that vanishes. Indeed, by the same argument one not only has Epp]selO(u) + o(δ) but also Epp]sel=O(u), i.e. higher order terms in δ are also O(u). By contrast WIF (p) in general does not vanish in this limit. Thus is not a ‘fitness’ measure as usually understood, and we will refer to S as the mean selection measure.

Although has been used as a measure of inclusive fitness, its relationship to gene frequency changes has not been given previously. We now show why and how a useful measure of kin selection can be constructed from .

Expected gene frequency and probability of fixation

There is a simple result which supports the use of S as a measure of kin selection: it gives the first-order effect of selection on the expected frequency of allele A. To see this, we only need to note that, from eqns 12 and 13, the unconditional expected change in frequency owing to selection is Sδ/2 + o(δ) where the o(δ) term is also O(u) as noted above. Thus, the unconditional change in frequency owing to selection is Sδ/2 + O(u)o(δ).

In the two-allele model considered here, it is well known that the change in gene frequency owing to mutation is of the form u(1 − p) − up=u(1 − 2p) (e.g. Wright, 1969, p. 24). In the presence of selection with maternal control of the trait, this applies to gene frequencies in gamete pools: the conditional change in frequency due to mutation is u(1 − 2(p + Epp]sel)). So the unconditional change in frequency is

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At equilibrium, the selection and mutation pressures balance each other, Sδ/2 + o(δ)O(u) + (1 − 2E[p])u + Ou2)=0, so that

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Equation 15 can then be related to first-order effects on probabilities of fixation: if the probability of fixation of an A mutant in an a population is written πa[RIGHTWARDS ARROW]A=1/(Nn) + δO(1) + o(δ), the probability of the reverse event can be written πA[RIGHTWARDS ARROW]a=1/(Nn) − δO(1) + o(δ) for the same O(1) term. Then

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and πa[RIGHTWARDS ARROW]A=1/(Nn) + δ limu[RIGHTWARDS ARROW]0 S/(2Nnu) + o(δ). Thus, the first-order effect on probabilities of fixation is φ ≡ limu[RIGHTWARDS ARROW]0S/(2Nnu).

Equation 15 is only an asymptotic result for small δ and u: it gives the marginal effect of selection on expected frequency E[p]. We can investigate the quantitative accuracy of this result as an approximation predicting the expected frequency of mutants with a small phenotypic effect δ > 0. The validity of eqn 15 has been checked in simulations (not shown).

However, we are more interested in qualitative results, i.e. whether selection favours an allele or not, because the most common use of measures such as S is to find evolutionarily stable strategies (ESS), or more generally ‘singular strategies’ in the evolutionary dynamics of a character (e.g. Eshel, 1996; Geritz et al., 1998 ). In the present framework, singular strategies z* are those for which limu[RIGHTWARDS ARROW]S/u=0 (mutants near z* are neutral to first-order in δ) and a convergence condition for evolutionary stability of trait value is limu[RIGHTWARDS ARROW]dS/dza|za=z*/u < 0. These computations identify z* such that when individuals in the population express trait values around za < z*, mutants with δ > 0 are favoured in that their probability of fixation is higher than that of a neutral mutant; when individuals in the population express trait value around za > z*, mutants with δ < 0 are favoured. Thus, limu[RIGHTWARDS ARROW]S/u=0 corresponds to condition (6.2) of Eshel, and limu[RIGHTWARDS ARROW]0 dS/dza|za=z*/u < 0 corresponds to his condition (6.4). Note that these measures do not provide a condition analogous to condition (6.3) of Eshel, which determines whether a polymorphism with two different phenotypes can evolve from a population at z*.

Thus, when scaled to mutation rate (as in S/(4u)) or to some measure of variability, S is a useful ‘fitness’ measure. Now we can also define an equivalent measure in terms of some more useful relatedness coefficients. Rather than scaling to mutation rate, we may divide S by some measures of genetic variability, which itself depends on the mutation rate. In particular it will often be easier to evaluate S/(1 − Q0D), where Q0D is the probability of identity within a deme after migration but before competition (i.e. in the pool of juveniles). Of interest is the fact that, using eqn 4 to eliminate ∂w/∂z,

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which involves probabilities of identity only through the ratios (Qj − 1)/(1 − Q0D). The present choice of the denominator is convenient because models of population structure are often given simpler formulae for such ratios than for other possible choices of the denominator. Moreover, as (1 − 0D)=2Nnu + O(u2) (a result obtained in a different form by Slatkin, 1987), the first-order effect on probability of fixation, limu[RIGHTWARDS ARROW]0 S/(2Nnu), is also the low-mutation limit value of the above expression.

The above ratios and similar ones have several useful properties which are analogous to properties of pedigree measures of relatedness. We will first describe some of these properties, then we will give examples showing the relevance of these results and also illustrating some additional techniques.

Computation and properties of ‘relatedness’ coefficients

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Computation of identity measures

Specific techniques for computing probabilities of identity and relatedness measures are summarized in Appendix 2. It may be skipped at first reading.

Properties of relatedness

Several properties of measures of relatedness of the form (Q0 − Qj)/(1 − Q0), and of other ratios of 1 − Q’s, may be read from the mathematical expressions in Appendix 2. We only note two of them here, which were first emphasized by Crow & Aoki (1984) for F-statistics in the island model. First, they are approximately independent of the number of demes, so in general the ESS will be approximately independent of the number of demes. Second, the low-mutation limit formulae hold whether we consider the infinite allele model or other mutation models: see Rousset (1996, 1997) for details, examples and discussion. Thus, we may consider probabilities of identity by descent where we have considered identity in state Q above, because in both cases we get the same ESS conditions, and we may compute these ESS conditions using existing theory for identity by descent in subdivided populations.

Examples

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Island model

We first consider an island model to show how our expressions reduce to previous ones in this case. Here the fitness of an individual depends on its own phenotype z, on the average phenotype of a group of neighbours in a deme (excluding the focal individual), z0, and on the average phenotype of all individuals in other demes, z1. Let w ≡ w(z,z0,z1) be the fitness function, then S=∂w/∂z + ∂w/∂z0Q0 + ∂w/∂z1Q1, which, using eqn 4, may be written (1 − Q1)(∂w/∂z + ∂w/∂z0(Q0 − Q1)/(1 − Q1)). The factor ∂w/∂z + ∂w/∂z0(Q0 − Q1)/(1 − Q1) is in the form of Hamilton’s rule, −c + Rb, for R ≡ (Q0 − Q1)/(1 − Q1). This R reduces to usual formulae for relatedness in the infinite island, infinite allele model because limu[RIGHTWARDS ARROW](Q0 − Q1)/(1 − Q1)=limu[RIGHTWARDS ARROW]0 (0 − 1)/(1 − 1), and because some techniques for computing R actually give the latter quantity as 0. These techniques, which consider that 1=0, implicitly compute probabilities of identity in the limit case u [RIGHTWARDS ARROW] 0, n [RIGHTWARDS ARROW] ∞, nu [RIGHTWARDS ARROW] ∞: 1=O((nu)−1[RIGHTWARDS ARROW] 0 in this limit. However, this result is specific to the island model.

The more general utility of the direct fitness method as applied here is that the coefficients that come in factor with the partial derivatives of the fitness function in eqn 13 are always probabilities of identity, rather than probabilities of identity in some cases and implicit or explicit ratios of such probabilities in other cases.

Our formalism also makes clear another point. Queller (1994) noted that in subdivided populations with localized dispersal the concept of ‘relatedness’ should be local. The relatednesses in Queller’s eqn 1 may indeed be interpreted as functions of the identity by descent within demes, 0, as his equation is for Taylor’s (1992a) island model. But it should also be noted that estimates of such a parameter using molecular markers are actually based on the estimation of (Q0 − Q1)/(1 − Q1), i.e. on the comparison of genes both within and between demes ( Cockerham & Weir, 1993) so comparisons of genes between islands are necessary. Ultimately, however, the real question is which genes should be compared when dispersal is really localized. Equation 17 says that probabilities of identity only up to some distance j, given by the model, need to be considered, not an average Q for the whole population.

Sex-ratio in a stepping stone

From the above results, a practical way to find potential evolutionarily stable trait values z* is, first, to express as a function of a trait value z (in particular all derivatives of the fitness function are evaluated in z for all its variables); then to evaluate

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where probabilities of identities are also expressed as a function of z if they are affected by the trait value; to solve φ(z)=0 for the trait value; and to check that dφ/dz < 0 in z*. As one of the simplest illustrations of this method under localized dispersal, we consider a model of evolution of the sex ratio in a one-dimensional stepping stone population with sex-specific dispersal ( Taylor, 1994). In the model we consider hermaphrodites who produce male and female gametes. These gametes disperse with sex-specific rates (2α for males, 2β for females) and compete for fertilization. The z phenotypes are here the sex ratios (proportion of male gametes among gametes produced by each individual). The fitness function for male gametes is

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and for female gametes it is

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Here z1 stands for neighbours in each of two adjacent demes. We do not need to distinguish them as they interact in the same way with the focal individuals and are characterized by the same probability of identity. For w=(wm + wf)/2, by the direct fitness method the mean selection measure is

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which obviously decomposes in terms with (1 − Q) factors as in eqn 17. To evaluate φ we use the values of the relatedness measures for a one-dimensional stepping stone model on an infinite lattice (eqn A.11, where here m=α + β). Then, if there is an evolutionarily stable sex ratio z*, it is given by

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which reduces to Taylor’s equation 5 when N=1, the only case he considered. If one sex disperses at the rate optimal in the model without sex-specific dispersal, and if the other disperses at some other rate, we expect the sex ratio to be biased towards the sex with the optimal dispersal rate ( Hamilton, 1967; Taylor, 1994; references therein). In this one-dimensional stepping stone model, the optimal dispersal rate is 3/4 for all N ( Gandon & Rousset, 1999). It can be verified from the above equation that the sex ratio is biased toward the sex with dispersal rate 3/4.

Likewise, we can check whether there is convergence to this point in populations starting from another value z, i.e. whether mutants closer to the value z* will be favoured, by checking that dφ/dz < 0 near z*. For example, when N=1, one finds that dφ/dz is of the sign of

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which is always negative because the coefficients of (1 − z)2 and z2 are both negative for any α and β.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

The above theory provides a method to find evolutionarily stable strategies, or more generally ‘singular points’, in the class of models considered here, as follows:

1 write a fitness function w. This function must give the expected number of adult offspring of an individual as a function of the phenotypes of other individuals in the population;

2 compute the measure , made of partial derivatives of w and of probabilities of identity by descent;

3 divide by 1 − 0D or some other measure of genetic variability (1 − 0D is often most convenient as noted above);

4 compute the low-mutation limit of this ratio, using the theory of identity by descent in subdivided populations. Let φ be this limit: φ may be viewed as a measure of ‘inclusive fitness’; and

5 solve φ=0.

The total size of the population generally has only weak effects on ESSs (see, e.g. Gandon & Rousset, 1999, for examples). Thus, it is often convenient to approximate φ by its value in an infinite population model (n [RIGHTWARDS ARROW] ∞).

We have seen that this method is equivalent to the previously described one ( Taylor & Frank, 1996) in the infinite island model. In other cases, such as the analysis of evolutionarily stable dispersal rates under localized dispersal ( Gandon & Rousset, 1999), results are different from previous ones obtained. We have also reanalysed and generalized another model, giving a coherent argument for, and extending the conclusions of, previous analyses.

The significance of inclusive fitness techniques

Measures of kin selection are often used in contexts where a limit result when gene frequency goes to zero is sought, such as applications of game theory where one considers in principle whether a ‘rare’ mutant ‘invades’ or not. The assumption inherent in the ‘regression definition’ (1) essentially holds in some models, such as models of family interactions in a panmictic population, and in infinite island models. Hamilton (1964, 1970) noticed that it should not work for ‘rare’ alleles, and assumed this would have little impact, which is indeed correct in such models.

But more generally the assumption inherent in the ‘regression definition’ is not valid. It has not been proved even as a limit result in models of isolation by distance on an infinite lattice, and it should be clear that it has no general mathematical basis. Simulation results (not detailed here) suggest that a better and weaker conjecture would be that ‘inclusive fitness’WIF(p) is of constant sign for all 0 < p < 1 in the infinite population limit (n [RIGHTWARDS ARROW] ∞), its sign being that of S. This conjecture would imply that at the ESS (such that φ=0), WIF(p) may approach 0 as total population size increases. The condition 0 < p < 1 excludes ‘rare’ alleles, i.e. when there is a finite number of copies of the gene in the population.

If eqn 1 was really important, then the fact that it is not exact for rare alleles would be important. What this paper shows is that a coherent argument for the inclusive fitness measures considered here does not depend on eqn 1 being true or not. Our analysis leads us to conclude that ‘inclusive fitness’ does not measure selection specifically on rare alleles. A more general interpretation of measures such as φ is that they give the effect of selection on the probability of fixation of a mutant. In the low-mutation limit a positive value of φ means that the probability of fixation of an A mutant in an a population is higher than the probability of fixation of an a mutant in an A population. At the ESS these probabilities are equal.

If the concept of ESS is defined from consideration of selection on rare alleles, one may question whether a solution of φ=0 may be called an ESS. However, the frequent emphasis on what happens to rare alleles may be sometimes misleading. For example it gives little clue about the probability of fixation of a mutant. Ultimately, the notion of ‘rare’ allele may need clarification because for example a limit as p [RIGHTWARDS ARROW] 0 is not uniquely defined in an infinite population. Any finite number of copies of the A allele will satisfy ‘p=0’ in an infinite population, yet in subdivided populations inclusive fitness will differ depending on these different numbers of A copies. When p > 0, either WIF(p) is frequency dependent, or it is pointless to consider WIF(p) specifically for some low value of p.

Our framework may be applied to other models. For example, theoretical analyses of the concept of reproductive value are often based on limit results when gene frequency goes to zero (e.g. Taylor, 1989; Charlesworth, 1994). If reproductive value is to be used in selection measures analogous to S, it should follow from results not based on this assumption, and interestingly such results may be attained by arguments similar to the argument used here (H. Leturque and F. Rousset, unpublished results). Such results may be used to analyse kin selection under more general migration models than the ones considered here.

Concepts of relatedness in subdivided populations

We have emphasized measures of relatedness which have the same properties as the measured effects on probability of fixation of mutants: weak dependence on total population size and no dependence on mutation rate. In these respects they are similar to pedigree measures of relatedness.

In the case of localized dispersal, the present approach gives accurate expressions for inclusive fitness in terms of local relatedness coefficients. Here, we have converted measures of the form ∑bjQj into measures of inclusive fitness of the form ∑bjaj, where the aj ≡ (1 − Qj)/(1 − Q0D) are local measures of relatedness if the competitive interactions (including migration) are local (small values of j). There, the ‘gene diversity’ between neighbours at distance j, 1 − Qj, is compared with a measure of ‘gene diversity’ within a deme, 1 − Q0D. In other words, such ratios measure how different are two gametes at distance j relative to two gametes competing for the same deme. As noted above, other measures of gene diversity could be considered in the denominator, so there are different possible definitions of ‘relatedness’. In general the measure ∑bjaj may not simplify to a measure of the form bR − c with R independent of selection parameters.

The classical pedigree measures of relatedness are measures of genealogical relationships only and do not depend on the mutation rate and similar genetical features. In the present models the measures of genetic identity are determined by the genealogical relationships and by mutation, and therefore to qualify as measures of genealogical relationship they should have a nontrivial low-mutation limit depending only on the expected genealogical relationships between the genes considered. The coefficients of the mean selection measure S, being always probabilities of identity, generally do not have this property. The (1 − Qj)/(1 − Q0D) coefficients have this property, as would any ratio of sums of (1 − Q)’s.

When they involve pairs of genes compared only at a local geographical scale, such measures of relatedness are estimable using molecular markers with their own mutation rates and process, as their values are also robust to such features (e.g. Rousset, 1997). Here estimation may be done simply by estimating each probability Qj by the frequency of identical pairs of genes at distance j in a sample. Thus, the relationship between the present theoretical measures and statistical estimates of relatedness is straightforward. This would not be necessarily so if some overall Q for the total population was considered in the definition of the relatedness measures.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Two anonymous reviewers expressed various frustrations, some of which led to improvements. We thank M. Raymond, S. Frank, M. Kirkpatrick, and the Editor for comments on various versions of this paper, P. Taylor for more comments and some other matters of concern, and S. Gandon and O. Ronce for many useful discussions. This work was supported by the Service Commun de Biosystématique de Montpellier, and Grants LR963223 from the Région Languedoc-Roussillon, ACCSV- 39503077 and GDR 1105 from Centre National de la Recherche Scientifique. This is paper ISEM 2000-065.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices
  • 1
    Charlesworth, B. 1994. Evolution in Age-Structured Populations, 2nd edn. Cambridge University Press, Cambridge.
  • 2
    Cockerham, C.C. & Weir, B.S. 1993. Estimation of gene flow from F-statistics . Evolution 47: 855 863.
  • 3
    Comins, H.N. 1982. Evolutionarily stable strategies for localized dispersal in two dimensions. J. Theor. Biol. 94: 579 606.
  • 4
    Cotterman, C.W. 1940, reprinted 1974. A calculus for statistico-genetics. PhD Thesis, Ohio State University, Columbus. In: Genetics and Social Structure (P. Ballonoff, ed.), pp. 157–272. Dowden, Hutchinson & Ross, Stroudsburg, Pennsylvania.
  • 5
    Crow, J.F. & Aoki, K. 1984. Group selection for a polygenic behavioural trait: estimating the degree of population subdivision. Proc. Natl. Acad. Sci. USA 81: 6073 6077.
  • 6
    Crow, J.F. & Kimura, M. 1970. An Introduction to Population Genetics Theory. Harper & Row, New York.
  • 7
    Eshel, I. 1996. On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution. J. Math. Biol. 34: 485 510.
  • 8
    Frank, S.A. 1997. The Price equation, Fisher’s fundamental theorem, kin selection, and causal analysis. Evolution 51: 1712 1729.
  • 9
    Frank, S.A. 1998. Foundations of Social Evolution. Princeton University Press,.
  • 10
    Gandon, S. & Rousset, F. 1999. Evolution of stepping stone dispersal rates. Proc. Roy. Soc. (London) B 266: 2507 2513.
  • 11
    Geritz, S.A.H., Kisdi, É., Meszéna, G., Metz, J.A.J. 1998. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12: 35 57.
  • 12
    Grafen, A. 1985. A geometric view of relatedness. Oxford Surv. Evol. Biol. 2: 28 89.
  • 13
    Hamilton, W.D. 1964. The genetical evolution of social behavior. I. J. Theor. Biol. 7: 1 16.
  • 14
    Hamilton, W.D. 1967. Extraordinary sex ratios. Science 156: 477 488.
  • 15
    Hamilton, W.D. 1970. Selfish and spiteful behaviour in an evolutionary model. Nature 228: 1218 1220.
  • 16
    Harpending, H.C. 1979. The population genetics of interactions. Am. Nat. 113: 622 630.
  • 17
    Malécot, G. 1939. Théorie mathématique de l’hérédité mendelienne généralisée. PhD Thesis, Université de Paris.
  • 18
    Malécot, G. 1948. Les Mathématiques de L’hérédité. Masson, Paris.
  • 19
    Malécot, G. 1950. Quelques schémas probabilistes sur la variabilité des populations naturelles. Annales Université Lyon A 13: 37 60.
  • 20
    Malécot, G. 1951. Un traitement stochastique des problèmes linéaires (mutation, linkage, migration) en génétique de population. Annales Université Lyon A 14: 79 117.
  • 21
    Malécot, G. 1975. Heterozygosity and relationship in regularly subdivided populations. Theor. Popul. Biol. 8: 212 241.
  • 22
    Maruyama, T. 1970. Effective number of alleles in a subdivided population. Theor. Popul. Biol. 1: 273 306.
  • 23
    Maynard Smith, J. 1998. Evolutionary Genetics, 2nd edn. Oxford University Press, Oxford.
  • 24
    Price, G. 1970. Selection and covariance. Nature 227: 520 521.
  • 25
    Queller, D.C. 1994. Genetic relatedness in viscous populations. Evol. Ecol. 8: 70 73.
  • 26
    Rousset, F. 1996. Equilibrium values of measures of population subdivision for stepwise mutation processes. Genetics 142: 1357 1362.
  • 27
    Rousset, F. 1997. Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance . Genetics 145: 1219 1228.
  • 28
    Sawyer, S. 1977. Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob. 9: 268 282.
  • 29
    Seger, J. 1981. Kinship and covariance. J. Theor. Biol. 91: 191 213.
  • 30
    Slatkin, M. 1987. The average number of sites separating DNA sequences drawn from a subdivided population. Theor. Popul. Biol. 32: 42 49.
  • 31
    Tachida, H. 1985. Joint frequencies of alleles determined by separate formulations for the mating and mutation systems. Genetics 111: 963 974.
  • 32
    Taylor, P.D. 1989. Evolutionary stability in one-parameter models under weak selection. Theor. Popul. Biol. 36: 125 143.
  • 33
    Taylor, P.D. 1992a. Altruism in viscous populations – an inclusive fitness model. Evol. Ecol. 6: 352 356.
  • 34
    Taylor, P.D. 1992b. Inclusive fitness in a homogeneous environment. Proc. Roy. Soc. (London) B 249: 299 302.
  • 35
    Taylor, P.D. 1994. Sex ratio in a stepping stone population with sex-specific dispersal. Theor. Popul. Biol. 45: 203 218.
  • 36
    Taylor, P.D. 1996. Inclusive fitness arguments in genetic models of behaviour. J. Math. Biol. 34: 654 674.
  • 37
    Taylor, P.D. & Frank, S.A. 1996. How to make a kin selection model. J. Theor. Biol. 180: 27 37.
  • 38
    Uyenoyama, M.K. & Feldman, M. 1982. Population genetic theory of kin selection. II. The multiplicative model. Am. Nat. 120: 614 627.
  • 39
    Wright, S. 1969. Evolution and the Genetics of Populations. II. The Theory of Gene Frequencies. University of Chicago Press, Chicago.

Appendices

  1. Top of page
  2. Abstract
  3. Introduction
  4. Measuring selection
  5. The model
  6. Computation and properties of ‘relatedness’ coefficients
  7. Examples
  8. Discussion
  9. Acknowledgments
  10. References
  11. Appendices

Appendix 1: constructing the measures of selection

Consider a population with Nn genes (haploid adults) labelled i=1,…,Nn, each of allelic type a or A. Then consider the change in gene frequency of A when copies of a express phenotype za and copies of A express phenotype zA ≡ za + δ (as before, the values za and zA are defined as the value of phenotype given genotype is a or A). As δ [RIGHTWARDS ARROW] 0, we obtain the infinitesimal effect on change in gene frequency of an infinitesimal change in A’s phenotype.

Consider the indicator variables X:A(i), defined so that X:A(i)=1 if the gene born by individual i (i=1,…,Nn) is of allelic type A and X:A(i)=0 otherwise. We will also write Xj:A (j=•,0,…,n − 1) the indicator variable for the allelic type of the focal individual or of a random individual at j steps from the focal individual on the lattice. These indicator variables are related to probabilities of identity as follows:

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If eqn 1 is correct, then Pr(Xj:A=1|p,X•:A = 1)=rj + (1 − rj)p for some rj. Therefore, it implies

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Conditional changes in allele frequency

The change in frequency of A owing to selection obeys a form of Price’s equation ( Price, 1970),

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where Ep[…] denotes conditional expectation given allele frequency p in the population. The last equality is correct only with the definition of the fitness function given above. From this result, we will first derive an expression for the first-order effect of phenotypic change δ on expected change in allele frequency over one generation, Ep[dδp/dzA]=Ep[dδp/dδ]=Ep[X:A(i)dW(i)/dδ], in terms of the derivatives of the fitness function and of probabilities of identity of pairs of genes. We note that this expectation depends only on the effect of selection on the fitness of individuals bearing the A allele, as otherwise X:A(i)=0. We may thus express it in terms of conditional probabilities given X•:A=1:

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Throughout this paper, all derivatives are evaluated in δ=0, and therefore the same is observed in the ‘no selection limit’. We now evaluate dEp[Zj|X•:A=1]/dδ in terms of the probability Qj:A|p of identity in state to allelic type A, conditional on present allele frequency p, of a pair of genes, one in the focal individual and the other in an individual at distance j:

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where we have only used eqn A.1, and Q•:A|p ≡ p. Hence for all j, pdEp[Zj|X•:A=1]/dδ=Qj:A|p, and (A.4) becomes

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which yields eqn 6 in the main text. (Recall that a function g(x) is o(f(x)) when x [RIGHTWARDS ARROW] x0 if limx[RIGHTWARDS ARROW]x0 g/f=0, and that g(x) is O(f(x)) if limx[RIGHTWARDS ARROW]x0 g/f is finite.)

Appendix 2: computation of identity measures

Our model assumes spatially homogeneous migration on a lattice. The lattice model is well documented (e.g. Malécot, 1950, 1951, 1975; Maruyama, 1970; Sawyer, 1977) and exact formulae give the unconditional probabilities of identity in this model, but these facts seem to have been shadowed by the emphasis on some approximate treatments and difficulties with some continuous approximations. We apply these formulae to the probabilities of identity of genes among gametes or juveniles after migration but before competition (i.e. in gamete pools), QjD (which are also the probabilities of identity of genes among adults QjA, except that Q0A is undefined when N=1). Note that the probability Q0 to be considered in the fitness measure S (eqn 13) is Q0D + (1 − Q0D)/N if z0 is defined as the average phenotype of all neighbours in the deme of a focal individual, including this individual: this is so for all N.

Let γ ≡ (1 − u)2 for u the mutation rate. Let ψ(z) be the characteristic function of dispersal distance, ψ(x) ≡ ∑j mj eιjx, for

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and mj the probability that an adult originates from a gamete produced j steps apart (backward dispersal rate). We will use dotted symbols, and , to denote the identity by descent equivalent of S and Q. An exact recursion for the infinite allele mutation model and a lattice population structure is

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( Malécot, 1951; Malécot, 1975, eqn 2), here written in terms of the D’s, so that it is valid also for N=1.

Standard formulae (e.g. Malécot, 1950, 1975) show that

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where for a function f, Lj(f) is the inverse Fourier transform of f. For example on a one-dimensional lattice

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Equation A.8 is also valid for the QA’s when N > 1.

We can use these results to compute inclusive fitness expressions by the following techniques: we express them as a sum of terms involving factors of the form (1 − jD) (which we can always do because of the property (4) of the fitness function) and divide by 1 − 0D. Thus we consider the inclusive fitness measure /(1 − 0D) which is a sum of ratios of the form (1 − jD)/(1 − 0D). These ratios are (L0 − Lj)/N when j ≠ 0. This is convenient because we can compute the low-mutation limits limu[RIGHTWARDS ARROW]0 L0 − Lj. For example in the stepping stone model with migration rate m on an infinite one-dimensional lattice,

inline image

and

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In particular,

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and

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as u [RIGHTWARDS ARROW] 0.

In the symmetrical two-allele model, the exact values of the probabilities of identity are (1 + j(2u))/2 where j(2u) is the probability of identity as given by the formulae mutation rate of the two-allele model (e.g. Tachida, 1985). It also implies that S=(2u)/2 where (2u) is the value of S for an infinite allele model with twice the mutation rate of the two-allele model. This can be used to check that there is generally little difference between the sign of S and that of its identity-by-descent equivalent.