A sibling's child, who will carry one-quarter of the individual's genes, will then represent 1/2 offspring equivalent (and so on - see coefficient of relationship for further examples).
It enables us to see how the average trait value of a population is expected to evolve under the assumption of small mutational steps.
[5] Belding's ground squirrel provides an example; it gives an alarm call to warn its local group of the presence of a predator.
[6] Synalpheus regalis, a eusocial shrimp, is an organism whose social traits meet the inclusive fitness criterion.
[7] Inclusive fitness is more generalized than strict kin selection, which requires that the shared genes are identical by descent.
Consequently, to apply Hamilton's rule to biological systems the conditions under which fitness can be approximated to being linear in trait value must first be found.
[10][2] Performing a partial regression requires minimal assumptions, but only provides a statistical relationship as opposed to a mechanistic one, and cannot be extrapolated beyond the dataset that it was generated from.
Linearizing via a Taylor series approximation, however, provides a powerful mechanistic relationship (see also causal model), but requires the assumption that evolution proceeds in sufficiently small mutational steps that the difference in trait value between an individual and its neighbours is close to 0 (in accordance with Fisher's geometric model): although in practice this approximation can often still retain predictive power under larger mutational steps.
As such it cannot be considered sufficient to determine evolutionary stability, even when Hamilton's rule predicts no change in trait value.
This is because disruptive selection terms, and subsequent conditions for evolutionary branching, must instead be obtained from second order approximations (quadratic in trait value) of fitness.
[11] They suggest that one should "use standard population genetics, game theory, or other methodologies to derive a condition for when the social trait of interest is favoured by selection and then use Hamilton's rule as an aid for conceptualizing this result".
[11] It is now becoming increasingly popular to use adaptive dynamics approaches to gain selection conditions which are directly interpretable with respect to Hamilton's rule.
The American evolutionary biologist Paul W. Sherman gives a fuller discussion of Hamilton's latter point:[13] To understand any species' pattern of nepotism, two questions about individuals' behavior must be considered: (1) what is reproductively ideal?, and (2) what is socially possible?
Only when ecological circumstances affecting demography consistently make it socially possible will nepotism be elaborated according to what is reproductively ideal.
For example, if dispersing is advantageous and if it usually separates relatives permanently, as in many birds, on the rare occasions when nestmates or other kin live in proximity, they will not preferentially cooperate.
For example, if reproductives generally die soon after zygotes are formed, as in many temperate zone insects, the unusual individual that survives to interact with its offspring is not expected to behave parentally.
[14][15][16] Only in species that have the appropriate traits in their gene pool, and in which individuals typically interacted with genetic relatives in the natural conditions of their evolutionary history, will social behaviour potentially be elaborated, and consideration of the evolutionarily typical demographic composition of grouping contexts of that species is thus a first step in understanding how selection pressures upon inclusive fitness have shaped the forms of its social behaviour.
Richard Dawkins gives a simplified illustration:[17] If families [genetic relatives] happen to go around in groups, this fact provides a useful rule of thumb for kin selection: 'care for any individual you often see'.
"[17]Evidence from a variety of species[18][19][20] including primates[21] and other social mammals[22] suggests that contextual cues (such as familiarity) are often significant proximate mechanisms mediating the expression of altruistic behaviour, regardless of whether the participants are always in fact genetic relatives or not.
This is nevertheless evolutionarily stable since selection pressure acts on typical conditions, not on the rare occasions where actual genetic relatedness differs from that normally encountered.
(Park 2007, p860)[23]Such misunderstandings of inclusive fitness' implications for the study of altruism, even amongst professional biologists utilizing the theory, are widespread, prompting prominent theorists to regularly attempt to highlight and clarify the mistakes.
[17] An example of attempted clarification is West et al. (2010):[24] In his original papers on inclusive fitness theory, Hamilton pointed out a sufficiently high relatedness to favour altruistic behaviours could accrue in two ways—kin discrimination or limited dispersal.
Furthermore, a large number of authors appear to have implicitly or explicitly assumed that kin discrimination is the only mechanism by which altruistic behaviours can be directed towards relatives... [T]here is a huge industry of papers reinventing limited dispersal as an explanation for cooperation.
However, Wang et al. has shown in one of the species where the effect is common (fire ants), recombination cannot occur due to a large genetic transversion, essentially forming a supergene.
[29] Equally, cheaters may not be able to invade the green-beard population if the mechanism for preferential treatment and the phenotype are intrinsically linked.
In budding yeast (Saccharomyces cerevisiae), the dominant allele FLO1 is responsible for flocculation (self-adherence between cells) which helps protect them against harmful substances such as ethanol.
[35] Gardner in turn was critical of the paper, describing it as "a really terrible article", and along with other co-authors has written a reply, submitted to Nature.
Under such circumstances Hamilton's rule then emerges as the result of taking a first order Taylor series approximation of fitness with regards to phenotype.
[10] This assumption of small mutational steps (otherwise known as δ-weak selection) is often made on the basis of Fisher's geometric model[37] and underpins much of modern evolutionary theory.
The parts that were derived from the genotypes of different individuals were terms to the right of the minus sign in the covariances in the two versions of the formula for