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How to Measure Inclusive Fitness, Revisited Anim. Behav., 1996, 51, 225–228 COMMENTARIES How to measure inclusive fitness, revisited JEFFREY R. LUCAS*, SCOTT R. CREEL† & PETER M. WASER* *Department of Biological Sciences, Purdue University †Rockefeller University Field Research Center (Received 14 September 1994; initial acceptance 31 October 1994; final acceptance 13 March 1995; MS. number: -1136) An individual’s inclusive fitness is derived by verbal definition) is equal to its Darwinian fitness augmenting its traditional Darwinian fitness by stripped of the effects of helpers. This is equival- certain components, and stripping it of others ent to the reproductive success of a breeder in (Hamilton 1964). The component to be added is an unaided pair. But in obligately cooperative the sum of all effects of the individual on his species, unaided pairs cannot rear young (e.g. neighbours, weighted by the coefficient of genetic dwarf mongooses, Helogale parvula: Creel 1990a; relationship (r) between them. In the original African wild dogs, Lycaon pictus: Malcolm & derivation of inclusive fitness, the component to Marten 1982). Under the original verbal definition be subtracted was verbally defined as ‘all compo- of e0, the inclusive fitness of breeders in groups of nents which can be considered as due to the any size would therefore be zero. Bringing the individual’s social environment’ (Hamilton 1964). verbal definition of e0 into line with its original It is not widely appreciated that this verbal defi- algebraic definition resolves the bizarre problem nition of the component to be subtracted does not of zero inclusive fitness for breeders. But as we agree with its algebraic definition. Creel (1990a) discuss here, it raises two new and conceptually used Hamilton’s algebraic definition of inclusive thorny issues. fitness to show that the component to be sub- (1) Calculation of e0 is not always irrelevant tracted, e0, is actually equal to the average effect when using Hamilton’s rule. Creel (1990a) sug- of one individual on others’ fitness (dT0,in gested that Hamilton’s rule (that help is favoured Hamilton’s 1964 formulation). Thus, the calcu- when rb>c, where b is the effect of a helper lation of e0 requires that we know the fraction of on breeder reproductive success, r is the genetic individuals in the population that provide help relatedness between helper and breeder, and c is and the mean amount of help provided per helper. the cost of helping to the helper) is unaffected by e0 is simply the product of these two numbers (see the definition of e0, because e0 is stripped from the below). Subtracting e0 resolves what has become inclusive fitness of both breeders and helpers. known as the ‘double accounting’ problem Thus ‘e0 drops out of an inequality comparing (Grafen 1984; Brown 1987). inclusive fitness for alternative strategies’ (Creel The verbal definition of e0 is important, because 1990a). The implication is that the correct defi- it is primarily the verbal definition of inclusive nition of e0 is not relevant to a correct calculation fitness that has been put to empirical use (Grafen of Hamilton’s rule. However, this statement does 1982, 1984). The flaw in the original verbal defi- not hold if inclusive fitness effects are summed nition of e0 can be seen by applying it to the case over more than a single reproductive season and if of a breeder in an obligately cooperative species alternative decisions differ in their risk of mortal- (Creel 1990a). In a group with a single breeding ity. This is true because e0 is subtracted from each pair, the breeder’s inclusive fitness (using the old individual’s fitness each time during its lifespan that direct or indirect fitness components are Correspondence: J. R. Lucas, Department of Biological measured, that is, once every breeding season. We 0 Sciences, Purdue University, West Lafayette, IN 47907- can illustrate why e will not cancel out using a 1392, U.S.A. (email: [email protected]). simple example. 0003–3472/96/010225+04 $12.00/0 ? 1996 The Association for the Study of Animal Behaviour 225 226 Animal Behaviour, 51, 1 Assume that if an individual dispersed it would Therefore the helper’s inclusive fitness is produce, on average, 0.5 offspring as a breeder in (r"p)#b. This will be negative whenever the a new group the current year but would have zero coefficient of relationship is less than the pro- probability of surviving another year. The inclu- portion of individuals in the population that are sive fitness of this individual would be 0.5"e0. non-breeding helpers. Alternatively, assume that the individual could From published demographic data (e.g. Stacey remain in its current group, and if it did so it & Koenig 1990) it appears that negative inclusive would survive the current year as a helper, helping fitness for helpers will be common, unless helpers its sibling produce 0.5 more offspring than it often obtain hidden direct reproduction. For 12 otherwise would have produced. Also, assume cooperatively breeding species ranging from lions, that this helper then bred the next year, pro- Panthera leo, to splendid fairy wrens, Malurus ducing 1.0 offspring and dying thereafter. The splendens, mean helper–breeder relatedness falls lifetime inclusive fitness of this option is between 0.23 and 0.47 (Macdonald 1980; (0.5#r"e0)+(1.0"e0). In this case, the fitness Woolfenden & Fitzpatrick 1984; Austad & components to be subtracted from dispersers’ and Rabenold 1985; Emlen & Wrege 1988; Mills 1990; non-dispersers’ fitnesses will not cancel out. More Packer et al. 1991; Creel & Waser 1994; also see generally, differences in mortality risks between Stacey & Koenig 1990). If the dichotomy between alternatives will cause differences in expected helping and breeding were absolute, this range of longevity, which will in turn cause differences in r-values would yield negative fitness for helpers in the relative weighting of e0 on lifetime inclusive groups larger than 2.6 (for r=0.23) to 3.8 (for fitness. We emphasize that since e0 relates to the r=0.47). Many cooperatively breeding mammals average ability of individuals to give (or equiva- live in group sizes large enough to yield negative lently to receive) help, it should be calculated on a fitness for helpers (e.g. African wild dogs with per-breeding-season basis, as we use it here. mean adult pack size of 11: Fuller et al. 1992). A number of recent analyses have shown that Modal group size for most cooperatively breeding future effects are important components in the birds is less than four (two, in most cases: Stacey estimation of inclusive fitness (e.g. Brown 1987; & Koenig 1990), but even for these species Mumme et al. 1989; Creel 1990b; Lucas et al., in many helpers will have a negative fitness unless press). In addition, differences in mortality risks relatedness is high. between alternative behaviour patterns (e.g. dis- Of course, the calculation of inclusive fitness is persal versus non-dispersal, or helping versus non- affected by direct fitness effects. Direct production helping) are likely to be nearly ubiquitous (Brown of offspring by helpers has recently been dis- 1987; Waser et al. 1994). Together, these data covered in several species (e.g. dwarf mongooses: suggest that an accurate evaluation of e0 is Keane et al. 1994; stripe-backed wrens: Rabenold warranted. et al. 1990). This direct reproduction will obvi- (2) Helpers often will have negative inclusive ously increase the inclusive fitness of helpers, but fitness. Applying the definition of e0 in a dynamic at least in dwarf mongooses, the number of young optimization analysis of dispersal decisions in produced by helpers is insufficient to give them cooperatively breeding dwarf mongooses, Lucas positive inclusive fitness (Lucas et al., in press). et al. (in press) found that subordinate helpers The role of the indirect component of inclusive of many age–sex classes had negative inclusive fitness on the evolution of helping behaviour has fitness. To illustrate this, assume that helpers do been controversial. Some authors argue that in- not breed (and breeders do not help), so that a direct effects are relatively unimportant (Zahavi helper’s fitness is entirely indirect (Brown 1987). 1974; Ligon 1981; Walters et al. 1992); others Indirect fitness is the product of a helper’s effect suggest that indirect effects cannot be overlooked on breeder reproductive success (b) multiplied by (Mumme et al. 1989; Emlen & Wrege 1991; Creel its relatedness to the breeder (r). Inclusive fitness is & Waser 1994). Although we have shown here obtained by stripping e0 from this indirect fitness, that the inclusive fitness of helpers can actually be where e0 is the product of an average helper’s negative, we want to emphasize that this does not effect on breeder reproductive success (b) multi- mean that indirect fitness effects will be unimport- plied by the proportion of individuals in the ant in the evolution of helping behaviour in these population that are helpers (p) (Creel 1990a). species. For example, we show elsewhere that, Commentaries 227 despite negative indirect fitness for subordinate ness to other group members, and so on. The dwarf mongooses, indirect effects can play a alternative genotypes that compete in the real major role in the evolution of social behaviour in world are likely to be more complicated, this species (Lucas et al., in press). and might for example influence their bearers’ Non-breeders will likely suffer negative fitness, tendencies to disperse conditional on some com- irrespective of their choice of behaviour, if the bination of, for example, dominance and related- frequency of helpers in a population is large. ness. An approach that explicitly allows the Indeed, the inclusive fitness of non-helping non- modelling of such conditional traits is dynamic breeders will always be negative whenever helping optimization; we have presented a model of exists.
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