- Top of page
- Homogeneous population
- A class-structured population
Two standard mathematical formulations of kin-selection models can be found. Inclusive fitness is an actor-centred approach, which calculates the fitness effect on a number of recipients of the behaviour of a single actor. Direct fitness is a recipient-centred approach, which calculates the fitness effect on the recipient of the behaviour of a number of actors. Inclusive fitness offers us a powerful heuristic, of choosing behaviour to maximize fitness, but direct fitness can be mathematically easier to work with and has recently emerged as the preferred approach of theoreticians. In this paper, we explore the fundamental connection between these two approaches in both homogeneous and class-structured populations, and we show that under simple assumptions (mainly fair meiosis and weak selection) they provide equivalent formulations, which correspond to the predictions of Price's equation for allele frequency change. We use a couple of examples to highlight differences in their conception and formulation, and we briefly discuss a two-species example in which we have a class of ‘actor’ that is never a ‘recipient’, which the standard direct fitness method can handle but the usual inclusive fitness cannot.
- Top of page
- Homogeneous population
- A class-structured population
Kin selection is a deterministic evolutionary force that acts when individual fitness is affected by traits expressed by relatives. The objective of evolutionary modelling is to obtain conditions under which a kin-selected trait will change in frequency or in character. A standard approach is to assume that the trait in question is genetically determined and to find conditions for the change in frequency of the underlying alleles. The classic covariance formula of Price (1970) provides a simple direct equation for allele frequency change, but when fitness is affected by the behaviour of others who might also carry the alleles in question, the calculations required by this approach can be complicated.
Inclusive fitness, a powerful heuristic method introduced by Hamilton (1964), was designed to handle these complications. Hamilton introduced his idea with the observation ‘if we were to follow the usual approach of the progress due to natural selection in a generation, we should attempt to give formulas for the neighbour-modulated fitness…’ and he suggests that the calculations for this might be ‘rather unwieldy’. This prompted him to introduce an ‘alternative approach’, which restricts attention to the fitness effects of a single focal actor, and this is what we call inclusive fitness. The decades following Hamilton's paper saw a wonderful development of his idea, simplifying it, extending it, and relating it to various exact genetic formulations. But the neighbour-modulated approach, now usually called direct fitness (Taylor & Frank, 1996; Frank, 1998), has also been developed and the main point of this article is that these two approaches are computationally equivalent, although in different models one approach or the other often seems more immediate or natural. In fact, in recent years, direct fitness has emerged as the preferred method of modelling kin selection (e.g. Gandon, 1999; Perrin & Mazalov, 2000; Day, 2001; Leturque & Rousset, 2002, 2003; Wild & Taylor, 2005; Pen, 2006). However, it is important to say that inclusive fitness, in tracking the various fitness effects of a single individual's behaviour, mirrors the way evolutionary biologists think, and likely for that reason, it remains the preferred mode of analysis by most biologists.
Here, we describe and compare these two heuristic approaches with particular attention to their formulation in a class-structured population (Taylor, 1990; Taylor & Frank, 1996; Rousset, 2004). Under a wide range of assumptions the two approaches come up with the same conditions for the evolutionary increase or decrease in allele frequency, but there are conceptual and technical differences in their formulation and our purpose here is to make these clear. Basically, we will argue that each approach arises from the other by a simple re-indexing process. Suggestions have been made that direct and inclusive fitness are essentially different (Frank, 1997, 1998), but it seems to us that this arises from (interesting) attempts to extend the methods beyond their normal range of applicability, e.g. to fitness interactions between species. We will present an example to try to clarify this situation.
In our experience, the difficulties that often arise in the formulation of inclusive and direct fitness arguments, and in their comparison, have to do with the wide range of notations and assumptions found in the literature, and it seems to us that there is a real need for a simple general scheme and a self-contained exposition of it. A key ingredient to a clear understanding and easy implementation of any method is a good notational scheme and the literature has not consistently provided one (not even in our own papers). We have tried to remedy that here and to employ a common scheme to facilitate comparison between the two approaches. We will apply our formulation to three models of altruistic behaviour, a model of altruistic interaction between offspring, a class-structured model in which offspring provide a benefit to parents, and a host–parasite model with interaction between species.
We consider a population of individuals. This might be homogeneous consisting of individuals all of the same type or heterogeneous with individuals of different classes (male, female, or adult, juvenile or large, small, etc.) or even different species. Our objective will be to track the changing frequency of an allele found in a number of members of the population. This allele will affect the behaviour of a number of individuals who carry it and this in turn will affect the fitness of this individual and of other interactants and this will cause a change in the frequency of the allele. This is a complex situation – many individuals in different situations affecting the fitness of different configurations of other individuals. We will handle this by classifying the interactions and then adding the effects of the different types of interactions that affect a random individual. This is essentially a differential approximation, justified by an assumption of small effects (Grafen, 1985a).
We use the term ‘interaction’ to refer to a basic ‘cause and effect’ unit. Each interaction will involve a single actor and a single recipient and it is crucial to be aware of the distinction between these. An actor is characterized by its phenotype; a recipient is characterized by its genotype. Some individuals in the population will be both actors and recipients; as an actor their phenotype is what counts; as a recipient their genotype is what counts. Other individuals might be only recipients and will have only genotypes; still others might be only actors and have only phenotypes. Each type of interaction will have an effect on allele frequency and the overall population-wide change will be obtained by adding up these effects.
An example of an individual which is a recipient but not an actor is found in a class-structured population in which adult females care for juveniles. The adults are both actor (in providing care) and recipient (in paying the cost of care), but the juveniles are only recipients (in receiving the care). An example of an individual which is an actor but not a recipient is found in a two species interaction in which a parasite inhabits a host. If we are interested only in a parasite trait, the parasite is both actor and recipient, but the host who responds to the parasite is only an actor. For this last example, there will usually be alternative or more general modelling approaches which would work better. Our notation is found in Table 1.
Table 1. A summary of the main notation used in the text.
|cj||ujvj, the reproductive value of class j|
|dwk||Effect of deviant behaviour on the fitness of kth recipient|
|dwjk||Effect of deviant behaviour on the fitness of the kth recipient belonging to class j|
|G||Genotypic value of the focal individual|
|Gj||Genotypic value of a focal individual belonging to class j|
|Gk||Genotypic value of the kth recipient belonging to the social group of the focal individual|
|Population-wide genotypic value, also the frequency of the target allele|
|Average genotypic value among individuals belonging to class j|
|nj||The number of class j recipients in the social group of the focal actor (inclusive fitness) or equivalently the number of class j actors in the social group of the focal recipient (direct fitness)|
|P||Phenotypic value of the focal individual|
|Pk||Phenotypic value of the kth actor belonging to the social group of the focal individual|
|Normal phenotypic value|
|Rk||cov(G,Pk)/cov(G,P), the relatedness of the kth actor to the focal recipient in his social group (direct fitness arguments), or cov(Gk,P)/cov(G,P), the relatedness of the focal actor to the kth recipient in his social group (inclusive fitness arguments). Note that both formulations of relatedness are equivalent|
|Rjk||cov(Gjk,P)/cov(G,P), the relatedness of the focal actor to the kth recipient of class j (inclusive fitness arguments). This is equivalent to the coefficient, Rkj used in direct fitness arguments|
|Rkj||cov(Gj,Pk)/cov(G,P), the relatedness of the kth actor to the class j focal recipient (direct fitness arguments). This is equivalent to the coefficient, Rjk used in inclusive fitness arguments.|
|uj||The steady-state frequency of class j|
|vj||The reproductive value of an individual belonging to class j|
|W||Fitness of the focal individual|
|Wj||Fitness of a focal individual belonging to class j|
|Population-wide average fitness|
|Fitness of a normal class j individual|
|Wdir||Direct fitness effect|
|Winc||Inclusive fitness effect|
|wi||The i-fitness of a random individual defined as her contribution to class i in the next generation|
|wij||= E[wi| individual belongs to class j], the average i-fitness of a random individual in class j|
|w||= [wij], a square matrix that maps the frequency distribution of classes from one time step to the next|
- Top of page
- Homogeneous population
- A class-structured population
In a homogeneous population, interactions are symmetric and each individual is both actor and recipient. The fundamental result describing allele frequency change is Price's (1970) covariance formula. If we let the genotypic value G of an individual be the frequency of the target allele in its genotype and let its fitness be W, then Price's formula tells us that the population-wide allele frequency will increase exactly when G and W are positively correlated. Precisely:
where is population-wide average fitness. Some simplifying assumptions are needed for this, notably that meiosis is ‘fair’, that on average there is no change in allele frequency in transmission from parent to offspring.
where the derivatives are evaluated in the uniform population. Then eqn (1) becomes:
At this point it is worth mentioning Queller's (1992) paper which shows that if fitness W is regressed directly on genotypic value (of an individual and neighbours), we get a simple general form of inclusive or direct fitness without the many assumptions needed to make phenotypic arguments work. This elegant result is mainly of theoretical interest, as we typically get hold of the partial regressions by using phenotype as an intermediate variable. (Queller's, 1992 discussion of the role of phenotype was to some extent foreshadowed by Cheverud, 1984.)
Equation (3) arises directly from Price's formula and for that reason is called a direct fitness analysis of the fitness interaction (Taylor & Frank, 1996), also referred to as ‘neighbour-modulated’ fitness (Hamilton, 1964). More precisely, we define the direct fitness increment to be:
with everything evaluated at . Here P is the phenotypic value of the focal individual and Rk is the relatedness of the kth actor (or the type k actor depending on the setup) to the focal recipient. Provided cov(G,P) > 0 (which can always be arranged), Wdir will have the same sign as the change in average allele frequency.
The direct fitness approach fastens attention on a random individual recipient and adds up the effects on its fitness of the behaviour of all actors. An alternative formulation, inclusive fitness, introduced by Hamilton (1964), again takes a random individual but adds up the effects of its behaviour on the fitness of all recipients. This is essentially a re-allocation of the direct fitness effects, each effect being credited to the actor rather than to the recipient. But with this different formulation comes a different conceptualization. Instead of a fitness function W(P1,P2,P3,…) which expresses the fitness of an individual in terms of the behaviour of others, the inclusive fitness version starts with a random actor, with phenotypic value P, and tabulates the effects of its behaviour on the fitness of a number of recipients, with genotypic values Gk. This actually already requires the ‘allocation of effects’ found in eqn (2) and for this reason, inclusive fitness can be more difficult to formulate; that is, inclusive fitness ‘begins’ with the differential effects of behaviour on fitness. Let dwk be the effect of a deviant act on the fitness of the kth recipient. Then the inclusive fitness effect is:
where Rk is the relatedness of the focal actor to the kth recipient (or to the type k recipient depending on the setup).
Notice that although Rk appears to be differently defined in eqns (4) and (5) the difference between formulae is purely technical and derives from the re-indexing. If we take these effects to be the fitness derivatives from eqn (2): dwk = ∂W/∂Pk, then eqns (4) and (5) are equivalent. However, the formulae are thought of in different ways. The terms of the direct fitness effect (eqn 4) are effects of the behaviour of others on the fitness of the focal individual; the terms of the inclusive fitness effect (5) are the effects of the behaviour of the focal individual on the fitness of others.
An example of cooperation between patchmates
As an example we take a patch with n asexual haploid breeders and consider a cooperative interaction between random offspring. An actor will have three categories of fitness effects: on itself (the cost c), on its partner (the benefit b) and on others who experience increased competitive effects due to the extra products of the cooperation. Note that the list of ‘recipients’ of an item of behaviour must include all those whose fitness is affected by the behaviour, not only the ‘direct’ effects, in this case the cost and benefit, but also ‘indirect’ effects of altered competition for limited resources, first identified by Hamilton (1971) and attributed to what he called ‘population viscosity’ and incorporated into inclusive fitness arguments by Wilson et al. (1992) and Taylor (1992a,b). Continuing with the argument, we take dw0 = −c (the cost to the actor), dw1 = b (the benefit to the partner) and dw2 = −(b − c) (the loss of fitness by competitors). The inclusive fitness effect is then:
where R0 = 1 is the relatedness of the actor to itself, R1 is the relatedness between offspring native to the same patch and R2 is the relatedness of the actor to the competitively displaced individuals.
To identify these displaced individuals in the last term of eqn (6), we need further details of the population structure. As an example, suppose that the population has Wright's (1943) island structure and the islands are breeding patches. Suppose that after the fitness interaction there is random dispersal of offspring at rate d to an infinitely distant patch, followed of course by the arrival of an equal number of immigrants. Finally, we have competition for the n breeding spots. In that case, of the b − c additional individuals created, a proportion d will disperse and will compete with those who are unrelated to the actor. The remaining 1 − d will stay on the patch and will compete with the post-dispersal inhabitants. Now a proportion d of these will be immigrants and will also be unrelated to the actor, and the remaining 1 − d will be native to the patch. That is two factors of 1 − d and the relatedness of the actor to the competitively displaced offspring will be R2 = (1 − d)2R1, where R1 as above, is relatedness between offspring native to the same patch. These verbal arguments can in fact be tricky and the direct fitness calculation will give us a chance to check our reasoning. This analysis allows us to write the inclusive fitness effect as:
Notice that our original description of the model used an inclusive fitness format – specifying the effects of the behaviour. This is typically the most natural way to present a model, and it also guides our intuition and our reasoning. This accounts in large part for the popularity of the inclusive fitness paradigm.
The direct fitness approach requires us to write the fitness of an actor as a function of the behaviour of all potential interactants. Suppose that phenotypic value P is the probability of cooperating, and let P0 and P1 be the phenotypic values of the actor and of a random offspring native to the same patch. Then if there are N individuals on a patch, the number after dispersal will be
If neutral fitness is normalized at , then normalized fitness of an actor can be written as:
The first expression partitions the fitness according to whether the actor disperses (probability d) or does not disperse (probability 1 − d), and the second expression provides a first-order approximation which assumes that c and b are small. This approximation gives us a linear expression for W, and the direct fitness effect is clearly
and we see that this is the same as the inclusive fitness effect.
However, note that the terms of eqns (7) and (9), are interpreted in a different way. The terms of the inclusive fitness (eqn 7) are the fitness effects on all individuals (weighted by relatedness) of a single interaction. The terms of the direct fitness (eqn 9) are the fitness effects on a single individual of all interactions (again weighted by relatedness). This leads to a different analysis of the two last terms, the competitive effects. In eqn (7) the dispersal effect (1 − d)2, is counted as a component of the relatedness term (although there are other ways to do the analysis) whereas in eqn (9) this is counted as a component of the fitness effect (see also Gardner & West, 2006).
- Top of page
- Homogeneous population
- A class-structured population
We have examined two classic formulations of kin-selection, direct and inclusive fitness. Direct fitness is a reformulation of Price's covariance formula, and therefore, under suitable assumptions, principally fair meiosis and small genetic effects, it provides a true measure of allele frequency change. Inclusive fitness is essentially a reorganization of the direct fitness calculation and therefore also measures allele frequency change. In particular, direct and inclusive fitness always give the same answer.
The two formulations use slightly different forms of the relatedness coefficient, cov(G,Pk)/cov(G,P) in direct fitness (eqn 4) and cov(Gk,P)/cov(G,P) in inclusive fitness (eqn 6), and in their generalizations in (eqn 17) and (eqn 20). The first was introduced by Orlove & Wood (1978), the second by Michod & Hamilton (1980) and both forms have been compared and their merits discussed by many others, notably Seger (1981), Grafen (1985b) and Queller (1985, 1992). There are two things to mention here. First, these two forms emerge naturally from the two methodologies, direct fitness which is centred on a focal recipient (i.e. eqn 4) and inclusive fitness which is centred on a focal actor (i.e. eqn 6). The many discussions in the literature of the relationship between these two forms often fail to draw attention to this critical connection. Secondly, the two forms are in fact equivalent (Queller, 1992). This is clear if the two approaches are regarded as sums over fitness interactions, in each case the same sum, but organized differently, with exactly the same relatedness weight applied to each fitness effect. To emphasize this we write both expressions as cov(GR,PA)/cov(G,P). The numerator is the covariance between recipient genotype and actor phenotype, and the denominator is the covariance between genotype and phenotype in the actor class.
We have assumed small selective effects (weak selection) and it is important to note where this is used. In fact, the only place we need this is to obtain the (approximate) linearity of individual fitness as a function of the phenotypes of others (eqn 2), but if fitness happens to be linear in phenotype, we do not require weak selection at this point. If W is linear in the Pk, then all our equations are valid, even if fitness effects are large. The other place where inclusive and direct fitness arguments typically use weak selection is in the calculation of relatedness. The covariances in eqns (4) and (6) are typically calculated by assuming phenotype is linearly related to genotype and then using the notion of identity by descent, and this last argument is valid only when the alleles are neutral. When selection is weak, these ‘neutral’ covariances provide a good approximation. Work is ongoing on a more complete examination of the effects of strong selection on inclusive and direct fitness.
The popularity of direct fitness in the theoretical literature in recent years reflects the fact that it is often mathematically more natural to formulate. But the inclusive fitness paradigm continues to have a powerful presence, no doubt because it positions the modeller as an agent choosing behaviour to maximize fitness. This ‘individual as maximizing agent’ analogy (Grafen, 1999) allows us to put ourselves in the position of an individual organism and ask: how can I maximize my inclusive fitness? Not only does this constitute a powerful theoretical construct, but it is also a natural question for us humans to ask, as in our day-to-day lives our behavioural decisions are typically optimal, albeit with regard to complex payoff functions.