Hamilton's rule meets the Hamiltonian: kin selection on dynamic characters

Many biological characters of interest are temporal sequences of decisions. The evolution of such characters is often modelled using dynamic optimization methods such as the maximum principle. A quantity central to these analyses is the ''Hamiltonian'' function, named after the mathematician William R. Hamilton. On the other hand, evolutionary models in which individuals interact with relatives are usually based on Hamilton''s rule, named after the evolutionary biologist William D. Hamilton. In this article we present a generalized maximum principle that includes the effects of interactions among relatives and we show that a time-dependent (dynamic) version of Hamilton''s rule holds involving the Hamiltonian. This result brings together the power and generality of both the maximum principle and Hamilton''s rule thereby providing a natural framework for understanding the evolution of ''dynamic'' characters under kin selection.     

Density-dependent habitat selection in muskrats: a test of the ideal free distribution model

Summary Two predictions of the ideal free distribution model, a null hypothesis of habitat selection, were examined using free-ranging muskrats. We rejected the prediction that the proportion of the animals found in each of five habitats was independent of population size. Data on over-winter occupancy of muskrat dwellings tend also to refute the prediction of equal fitness reward among habitats. Habitat type and water-level had a profound effect on the suitability of a site for settlement. We concluded that the observed pattern of muskrat distribution followed more closely an ideal despotic distribution where some individuals benefited from a higher fitness because of resource monopolization. Current theories of density-dependent habitat selection, which assume an ideal free distribution, would not apply to muskrats and possibly to many other mammal species.     

Putting competition strategies into ideal free distribution models: habitat selection as a tug of war

When resources are patchily distributed in an environment, behavioral ecologists frequently turn to ideal free distribution (IFD) models to predict the spatial distribution of organisms. In these models, predictions about distributions depend upon two key factors: the quality of habitat patches and the nature of competition between consumers. Surprisingly, however, no IFD models have explored the possibility that consumers modulate their competitive efforts in an evolutionarily stable manner. Instead, previous models assume that resource acquisition ability and competition are fixed within species or within phenotypes. We explored the consequences of adaptive modulation of competitive effort by incorporating tug-of-war theory into payoff equations from the two main classes of IFD models (continuous input (CI) and interference). In the models we develop, individuals can increase their share of the resources available in a patch, but do so at the costs of increased resource expenditures and increased negative interactions with conspecifics. We show how such models can provide new hypotheses to explain what are thought to be deviations from IFDs (e.g., the frequent observation of fewer animals than predicted in "good" patches of habitat). We also detail straightforward predictions made uniquely by the models we develop, and we outline experimental tests that will distinguish among alternatives.     

Hamilton's rule and the causes of social evolution

Hamilton''s rule is a central theorem of inclusive fitness (kin selection) theory and predicts that social behaviour evolves under specific combinations of relatedness, benefit and cost. This review provides evidence for Hamilton''s rule by presenting novel syntheses of results from two kinds of study in diverse taxa, including cooperatively breeding birds and mammals and eusocial insects. These are, first, studies that empirically parametrize Hamilton''s rule in natural populations and, second, comparative phylogenetic analyses of the genetic, life-history and ecological correlates of sociality. Studies parametrizing Hamilton''s rule are not rare and demonstrate quantitatively that (i) altruism (net loss of direct fitness) occurs even when sociality is facultative, (ii) in most cases, altruism is under positive selection via indirect fitness benefits that exceed direct fitness costs and (iii) social behaviour commonly generates indirect benefits by enhancing the productivity or survivorship of kin. Comparative phylogenetic analyses show that cooperative breeding and eusociality are promoted by (i) high relatedness and monogamy and, potentially, by (ii) life-history factors facilitating family structure and high benefits of helping and (iii) ecological factors generating low costs of social behaviour. Overall, the focal studies strongly confirm the predictions of Hamilton''s rule regarding conditions for social evolution and their causes.     

Quantitative genetic versions of Hamilton's rule with empirical applications

Hamilton''s theory of inclusive fitness revolutionized our understanding of the evolution of social interactions. Surprisingly, an incorporation of Hamilton''s perspective into the quantitative genetic theory of phenotypic evolution has been slow, despite the popularity of quantitative genetics in evolutionary studies. Here, we discuss several versions of Hamilton''s rule for social evolution from a quantitative genetic perspective, emphasizing its utility in empirical applications. Although evolutionary quantitative genetics offers methods to measure each of the critical parameters of Hamilton''s rule, empirical work has lagged behind theory. In particular, we lack studies of selection on altruistic traits in the wild. Fitness costs and benefits of altruism can be estimated using a simple extension of phenotypic selection analysis that incorporates the traits of social interactants. We also discuss the importance of considering the genetic influence of the social environment, or indirect genetic effects (IGEs), in the context of Hamilton''s rule. Research in social evolution has generated an extensive body of empirical work focusing—with good reason—almost solely on relatedness. We argue that quantifying the roles of social and non-social components of selection and IGEs, in addition to relatedness, is now timely and should provide unique additional insights into social evolution.     

Hamilton's rule and kin competition: the Kipsigis case

Evolutionary studies of human behavior have emphasized the importance of kin selection in explaining social institutions and fitness outcomes. Our relatives can nevertheless be competitors as well as sources of altruism. This is particularly likely when there is local competition over resources, where conflict can lead to strife among nondispersing relatives, reducing or even negating the effects of relatedness on promoting altruism. Here, I present demographic data on a land-limited human population, utilizing large within-population variation in land ownership to determine the interactions between local resource competition and the benefits of kin in enhancing child survival, a key component of fitness in this population. As predicted, wealth affects the extent of kin altruism, in that paternal relatives (specifically father's brothers) appear to buffer young children from mortality much more effectively in rich than in poor households. This interaction effect is interpreted as evidence that the extent of nepotism among humans depends critically on resource availability. Further unanticipated evidence that maternal kin play a role in buffering children from mortality in situations where paternal kin control few resources speaks to the important role that specific local circumstance plays in shaping kin contributions to child welfare.     

Aggregations from using inadvertent social information: a form of ideal habitat selection

Social information in breeding site selection has received extensive study; however, few attempts have been made to link this process to pre‐existing models. We examine the importance of social information to three pertinent models of habitat selection that describe breeding aggregations and spatial patterns: 1) the ideal despotic distribution (IDD) which considers conspecific competition and habitat availability, 2) the perceptual constraints model which accounts for patch selection when animals experience a threshold of undetectable difference in quality, and 3) the “neighbourhood model” which predicts concordance between resources and settlers can be disrupted by conspecific attraction when resources are patchy. These models all predict initial settlers will select a high quality patch first. However, their predictions of subsequent settlement behaviour in remaining patches differ: the IDD predicts subsequent settlers will be distributed regularly, the perceptual constraints model predicts a random distribution, and the neighbourhood model predicts clustering from conspecific attraction. We examined which model best described settlement patterns of bobolink Dolichonyx oryzivorus and savannah sparrow Passerculus sandwichensis, in the context of social information. We observed settlement timing, quantified available resources, and determined where they occurred in the highest (local population “core”) and lowest densities (local population “periphery”). We then assessed whether individuals in the periphery settled in greater concordance with resources or conspecific presence. Core territories were clustered strongly on relevant resources, and these territory holders were older than in the periphery. Peripheral territories were likewise clustered but did not always co‐occur with the best available resources, matching the neighbourhood model prediction that social information may not always direct them to the best sites available. This suggests older individuals used their own experience to locate ideal habitat, whereas younger individuals attempted to aggregate on seemingly ideal habitat by using conspecific location; such information asymmetry due to age can be viewed as an “ideal aggregative distribution”.     

Dispersal among habitats varying in fitness: reciprocating migration through ideal habitat selection

Current evolutionary models of dispersal set the ends of a continuum where the number of individuals emigrating from a habitat either equals the number of individuals immigrating (balanced dispersal) or where emigrants flow from a source habitat to a corresponding sink. Theories of habitat selection suggest a more sophisticated conditional strategy where individuals disperse from habitats where they have the greatest impact on fitness to habitats where their per capita impact is lower. Asymmetries between periods of population growth and decline result in a reciprocating dispersal strategy where the direction of migration is reversed as populations wax and wane. Thus, for example, if net migration of individuals flows from high- to low-density habitats during periods of population growth, net migration will flow in the opposite direction during population decline. Stochastic simulations and analytical models of reciprocating dispersal demonstrate that fitness, carrying capacity, stochastic dynamics, and interference from dominants interact to determine whether dispersal is balanced between habitats, or whether one habitat or the other acts as a net donor of dispersing individuals. While the pattern of dispersal may vary, each is consistent with an underlying strategy of density-dependent habitat selection.     

Hamilton's forces of natural selection after forty years

In 1966, William D. Hamilton published a landmark paper in evolutionary biology: "The Moulding of Senescence by Natural Selection." It is now apparent that this article is as important as his better-known 1964 articles on kin selection. Not only did the 1966 article explain aging, it also supplied the basic scaling forces for natural selection over the entire life history. Like the Lorentz transformations of relativistic physics, Hamilton's Forces of Natural Selection provide an overarching framework for understanding the power of natural selection at early ages, the existence of aging, the timing of aging, the cessation of aging, and the timing of the cessation of aging. His twin Forces show that natural selection shapes survival and fecundity in different ways, so their evolution can be somewhat distinct. Hamilton's Forces also define the context in which genetic variation is shaped. The Forces of Natural Selection are readily manipulable using experimental evolution, allowing the deceleration or acceleration of aging, and the shifting of the transition ages between development, aging, and late life. For these reasons, evolutionary research on the demographic features of life history should be referred to as "Hamiltonian."     

The evolution of altruism between siblings: Hamilton's rule revisited

This paper explores the validity of Hamilton's rule in the case of other-only altruism in which the benefits are shared by other members of the sibling group excluding the donor. It presents a model of competition between two alleles which code for different kinds of altruism. It derives a simple replicator equation for allele frequencies under conditions of strong selection. This equation does not depend on the size of the sibling group. In mathematical form, the equation is similar to Hamilton's original rule in the case of inbreeding, although the causal mechanism is different. The paper derives a simple criterion to determine whether there will be a polymorphism in which both alleles coexist permanently. Such an event is rare and victory will normally go to the allele with the higher value of 1/2b-c, where b is the total benefit which an offspring confers on its siblings and c is the cost to the donor. The paper also considers how an offspring will behave in particular circumstances. Using a specialized version of the basic model, it shows how, in the absence of polymorphism, natural selection should take the system towards the point of 50% marginal altruism. With this type of altruism, an offspring will perform any act for which the expected cost to the donor is at most half the expected benefit to its siblings. Acts which do not satisfy this criterion are not performed. This accords with Haldane's quip that he would sacrifice his own life for two of his brothers, but not for less. Numerical simulation is used to explore these issues in greater depth. The paper also examines briefly the implications of heterozygote advantage for Hamilton's rule. It concludes with a brief discussion of the connection between other-only altruism and whole-group altruism, in which the donor gains some benefit from its actions.     

The robustness of Hamilton's rule with inbreeding and dominance: kin selection and fixation probabilities under partial sib mating

Assessing the validity of Hamilton's rule when there is both inbreeding and dominance remains difficult. In this article, we provide a general method based on the direct fitness formalism to address this question. We then apply it to the question of the evolution of altruism among diploid full sibs and among haplodiploid sisters under inbreeding resulting from partial sib mating. In both cases, we find that the allele coding for altruism always increases in frequency if a condition of the form rb>c holds, where r depends on the rate of sib mating alpha but not on the frequency of the allele, its phenotypic effects, or the dominance of these effects. In both examples, we derive expressions for the probability of fixation of an allele coding for altruism; comparing these expressions with simulation results allows us to test various approximations often made in kin selection models (weak selection, large population size, large fecundity). Increasing alpha increases the probability of fixation of recessive altruistic alleles (h<1/2), while it can increase or decrease the probability of fixation of dominant altruistic alleles (h>1/2).     

Not my brother's keeper: A thought experiment for Hamilton's rule

The now popular ‘selfish gene’ view defines evolutionary fitness at the gene level - in terms of the number of gene copies residing in future generations (or propelled from previous generations). Yet, most current biology textbooks still apply the concept of fitness to the individual, where it is defined more traditionally as the number of descendants residing in future generations. The existing literature remains ambiguous regarding whether one of these concepts is more meaningful than the other, or whether they both represent legitimate, functional definitions of fitness. In support of the latter view, I present a composite perspective that recognizes the gene as evolutionarily ‘selfish’, but also the individual as a ‘selfish vehicle’ for resident genes. Hamilton's rule explains, based on genetic relatedness, why natural selection has favoured behaviours that compel individuals (as ‘donors’ of help) to act for the good of copies of their genes residing in close kin (‘recipients’). I propose however, that natural selection should particularly favour helping behaviours directed at those recipient kin who have the highest relative probability of being the vehicle for a remarkably adaptive newly mutant gene, weighted by the proportion of genes shared with the donor. According to this ‘adaptive-genetic-novelty-rescue’ (AGNR) hypothesis, these favoured vehicles for shared gene copies are more likely to involve direct descendants (e.g. offspring) than other close kin from one's collateral lineage (e.g. siblings), even when the donor (e.g. a father) shares fewer genes with an offspring (e.g. a son) than with a sibling (e.g. a brother).     

A quantitative test of Hamilton's rule for the evolution of altruism

The evolution of altruism is a fundamental and enduring puzzle in biology. In a seminal paper Hamilton showed that altruism can be selected for when rb - c > 0, where c is the fitness cost to the altruist, b is the fitness benefit to the beneficiary, and r is their genetic relatedness. While many studies have provided qualitative support for Hamilton's rule, quantitative tests have not yet been possible due to the difficulty of quantifying the costs and benefits of helping acts. Here we use a simulated system of foraging robots to experimentally manipulate the costs and benefits of helping and determine the conditions under which altruism evolves. By conducting experimental evolution over hundreds of generations of selection in populations with different c/b ratios, we show that Hamilton's rule always accurately predicts the minimum relatedness necessary for altruism to evolve. This high accuracy is remarkable given the presence of pleiotropic and epistatic effects as well as mutations with strong effects on behavior and fitness (effects not directly taken into account in Hamilton's original 1964 rule). In addition to providing the first quantitative test of Hamilton's rule in a system with a complex mapping between genotype and phenotype, these experiments demonstrate the wide applicability of kin selection theory.     

A generalized habitat matching rule

Summary When fitness of a resource-limited animal depends only on that individual's share of the total resource in a habitat patch and individuals are free to move to the patch where their gains are highest, population density matches resource availability under the simple assumption that individual fitness increases with resource use. Previous theory on habitat matching required the stronger assumption that individual fitness was directly proportional to (rather than monotonically increasing with) resource use. The basic theory suggests conditions under which population density empirically indicates habitat quality. Extensions of this basic theory apply when individuals that are free to move among resource patches interact by interfering with each other's resource extraction or by competing unequally. Analysis of existing models of such ideal free competition yields conditions for a single general matching rule in which the logarithm of crowding is a linear function of the logarithm of resource abundance. Double logarithmic plots of empirical data on habitat use and habitat quality based on this rule furnish possible graphical indicators of the occurrence and intensity of competition in nature.     

State-dependent ideal free distributions

Summary The standard ideal free distribution (IFD) states how animals should distribute themselves at a stable competitive equilibrium. The equilibrium is stable because no animal can increase its fitness by changing its location. In applying the IFD to choice between patches of food, fitness has been identified with the net rate of energetic gain. In this paper we assess fitness in terms of survival during a non-reproductive period, where the animal may die as a result of starvation or predation. We find the IFD when there is a large population that can distribute itself between two patches of food. The IFD in this case is state-dependent, so that an animal's choice of patch depends on its energy reserves. Animals switch between patches as their reserves change and so the resulting IFD is a dynamic equilibrium. We look at two cases. In one there is no predation and the patches differ in their variability. In the other, patches differ in their predation risk. In contrast to previous IFDs, it is not necessarily true that anything is equalized over the two patches.     

Helping in cooperatively breeding long-tailed tits: a test of Hamilton's rule

Inclusive fitness theory provides the conceptual framework for our current understanding of social evolution, and empirical studies suggest that kin selection is a critical process in the evolution of animal sociality. A key prediction of inclusive fitness theory is that altruistic behaviour evolves when the costs incurred by an altruist (c) are outweighed by the benefit to the recipient (b), weighted by the relatedness of altruist to recipient (r), i.e. Hamilton''s rule rb > c. Despite its central importance in social evolution theory, there have been relatively few empirical tests of Hamilton''s rule, and hardly any among cooperatively breeding vertebrates, leading some authors to question its utility. Here, we use data from a long-term study of cooperatively breeding long-tailed tits Aegithalos caudatus to examine whether helping behaviour satisfies Hamilton''s condition for the evolution of altruism. We show that helpers are altruistic because they incur survival costs through the provision of alloparental care for offspring. However, they also accrue substantial benefits through increased survival of related breeders and offspring, and despite the low average relatedness of helpers to recipients, these benefits of helping outweigh the costs incurred. We conclude that Hamilton''s rule for the evolution of altruistic helping behaviour is satisfied in this species.     

The Allee-type ideal free distribution

The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space.