Predicting Group Size

Stable Group Size Often Exceeds Rate-Maximizing Group Size

Many animals find themselves in a so-called aggregation economy, in which individuals in groups experience higher foraging payoffs than solitary individuals (e.g., Baird and Dill 1996; review by Beauchamp 1998). Peaked fitness functions are the hallmark of such economies (fig. 10.1; Clark and Mangel 1986; Giraldeau and Caraco 2000). By contrast, animals in a dispersion economy experience maximal foraging payoffs when solitary and strictly diminishing payoffs with increasing group size (e.g., Belisle 1998). In an aggregation economy, the per capita rate ofintake increases initially with increasing group size G. However, because competition also increases with group size G, intake rate peaks (at G*) and then falls with further increases in group size. Clearly, this situation favors group foraging, but can we predict group size?

It might seem that the observed group size G should match the intake-maximizing group size G*, at which each group member maximizes its fitness.

Figure 10.1. Hypothetical relationship between group size G and an unspecified surrogate forfitness (e.g., net rate ofenergy intake). This general peaked function is characteristic of an aggregation economy, in which individuals gain fitness with increasing G, at least initially. G* (= 3) is the intake-maximizing group size. G may exceed G* because a solitary individual would receive a fitness gain by joining the group. G may continue to grow until it reaches G (= 6), the largest size at which each individual would do betterto be in the group than to be solitary. G is not expected to exceed G because a joiner that increases G to G + 1 would achieve greater fitness by remaining solitary.

Figure 10.1. Hypothetical relationship between group size G and an unspecified surrogate forfitness (e.g., net rate ofenergy intake). This general peaked function is characteristic of an aggregation economy, in which individuals gain fitness with increasing G, at least initially. G* (= 3) is the intake-maximizing group size. G may exceed G* because a solitary individual would receive a fitness gain by joining the group. G may continue to grow until it reaches G (= 6), the largest size at which each individual would do betterto be in the group than to be solitary. G is not expected to exceed G because a joiner that increases G to G + 1 would achieve greater fitness by remaining solitary.

Many studies, however, have found that G often exceeds G* (Giraldeau 1988). This mismatch is not unexpected. With a peak in the fitness function at G* (see fig. 10.1), the intake-maximizing group is unstable because a solitary forager can benefit from joining the group. A group of size G* will grow as long as foragers do better in that group than on their own, but it should not exceed the largest possible equilibrium group size G. At that point, solitary individuals do better to continue foraging alone than to join such a large group. Equilibrium group size may be as small as the intake-maximizing group size G* and as large as the largest possible equilibrium size G, depending on whether the individual or the group controls entry and on the degree ofgenetic relatedness between individuals (box 10.2).

Thanks to the development of this theory, it is no longer paradoxical to find animals in groups larger than the intake-maximizing group size G*. Yet the role of foraging payoffs in the maintenance of groups of large carnivores remains contentious (see Packer et al. 1990 for a fascinating case study). The wolves discussed in the prologue present a paradox, because pack size routinely exceeds the apparently largest possible equilibrium size G. Why would a wolf belong to a pack when it could forage more profitably on its own? Here we attempt to resolve this paradox while reviewing the theory on group membership.

BOX 10.2 Genetic Relatedness and Group Size

Giraldeau and Caraco (1993) analyzed the effects of genetic relatedness on group membership decisions. Consider a situation in which individuals benefit from increasing group size, and in which all individuals are related by a coefficient r. According to Hamilton's rule, kin selection favors an altruistic act (e.g., allowing an individual tojoin the group) when rB-C > 0, where B is the net benefit for all relatives at which the act is directed and C is the cost of the act to the performer. In the context of group membership decisions, both effects on others (Er) and effects on self (Es) can be either positive or negative, so we rewrite Hamilton's rule as rER + Es > 0. (10.1.1) Group-Controlled Entry

In some social foragers, group members decide whether to permit solitaries to join the group. Such groups should collectively repel a potential group member (i.e., keep the group at size G) when Hamilton's rule is satisfied. Here Er is the effect of repelling the intruder on the intruder:

and Es is the effect of repelling the intruder on the group:

where ^(1) is the direct fitness of the solitary intruder, ^(G) is the direct fitness of each of G individuals in the current group, and ^(G + 1) is the direct fitness of each individual if the group decides not to repel the intruder. (As we highlight below, the group-level decision is based on the selfish interests of the individual group members.) Substituting these expressions for the effects of repelling the intruder on the intruder [Er; eq. (10.1.2)] and on the group [ES; eq. (10.1.3)] into equation (10.1.1) and dividing all terms by G, we see that selection favors repelling a prospective joiner when

[^(1) - ^(G + 1)] + [^(G) - ^(G + 1)] > 0, (10.1.4)

where we express both the indirect fitness (first term on the left-hand side) and the direct fitness (second term) ofgroup members on a per capita basis.

By extension, group members should evict an individual from the group when rER + Es > 0. Here the effect on the evicted individual Er is

(Box 10.2 continued)

^(1) — ^(G), and the effect on the remaining group members Es is (G — 1)[ft(G — 1) — ft(G)].

Equation(10.1.4)indicatesthat repelling is never favored when 1 < G < G*, where G* is the group size at which individual fitness is maximized, but repelling is always favored when G >G, where G is the largest group size at which the individual fitness of group members exceeds that of a solitary. Thus, equilibrium (stable) group size must fall within the interval G* < G<G. Under group-controlled entry, the effect of increasing genetic relatedness is to increase the equilibrium group size. By contrast, if potential joiners can freely enter the group, genetic relatedness has the opposite effect.

Free Entry

Under free entry, group members do not repel potential joiners; thus, potential joiners make group membership decisions. Any such individual should join a group when Hamilton's rule is satisfied, where Er is the combined effect ofjoining on all the joiner's relatives:

and Es is the effect ofjoining on the joiner:

Substituting, we see that joining a group of size (G — 1) is favored when r(G — 1)[^(G) — ^(G — 1)] + [^(G) — ^(1)] > 0. (10.1.7)

An analysis of equation (10.1.7) reveals that, under free entry, the effect of increasing genetic relatedness is to decrease equilibrium group size. (For derivation of the expressions for equilibrium group size under both entry rules, see Giraldeau and Caraco 2000.)

Rate-Maximizing Foraging and Group Size

In wolf packs, group members control entry. Thus, pack size should fall somewhere between the intake-maximizing group size G* and the largest possible equilibrium size G (see box 10.2). The data show that a group size of two maximizes net per capita intake rate and that individuals would do worse in a larger group than alone (i.e., G* = G = 2; see fig. 3 in Vucetich et al. 2004). Thus, this initial analysis cannot explain pack living.

Variance-Sensitive Foraging and Group Size

Our initial attempt might have failed for lack of biological realism. We assumed that each individual would obtain the mean payoff for its group size. However, in nature, the realized intake rate of an individual might deviate widely from the average rate. In principle, a reduction in intake rate variation with increasing group size could translate into a reduced risk of energetic shortfall. However, a variance-sensitive analysis indicates that an individual will have the best chance to meet its minimum requirement if it forages with just one other wolf (see fig. 4 in Vucetich et al. 2004). Its risk of shortfall will be higher in a group of three or more than alone. Thus, once again, foraging models fail to explain pack living.

Genetic Relatedness and Group Size

So far, foraging-based explanations seem unable to account for the mismatch between group size predictions and observations. Kin selection would seem to provide a satisfactory explanation (e.g., Schmidt and Mech 1997). After all, wolf packs form, in part, through the retention of offspring. However, kin-directed altruism (parental nepotism) does not account for the observation that pack size routinely exceeds the largest possible equilibrium group size G. Although we expect group size to increase with genetic relatedness when groups control entry (see box 10.2), theory predicts that equilibrium group size cannot exceed G, even in all-kin groups (Giraldeau and Caraco 1993).

Recalling that for wolves, the largest possible equilibrium group size G = 2, kin selection cannot explain pack living. This does not mean, however, that group size should never exceed two. Consider immature wolves, which cannot forage independently. If evicted, they would presumably achieve an intake rate of virtually zero. Under this assumption, Hamilton's (1964) rule (see box 10.2) predicts group membership for nutritionally dependent firstorder relatives (i.e., offspring or full siblings). However, individuals that can achieve the average intake rate ofa solitary adult should not belong to groups, even all-kin groups (fig. 10.2). Thus, while kin selection offers an adequate explanation for packs comprising parents and their immature offspring, we still have not provided a general explanation for wolf sociality. How do we account for packs that include unrelated immigrants and mature individuals? Is there an alternative foraging-based explanation that has evaded us?

Kleptoparasitism and Group Size

Inclusion of a conspicuous feature of wolf foraging ecology, loss of food to ravens (Corvuscorax), increases the predicted group size dramatically (fig. 10.3). Both rate-maximizing (fig. 10.3) and variance-sensitive currencies predict large pack sizes, even for small amounts ofraven kleptoparasitism. Why does

Figure 10.2. The application of Hamilton's rule to predict whether mature and immature solitary wolves should be allowed in packs of various sizes when the pack controls group entry (see also fig. 5 in Vucetich et al. 2004). The pack should repel any individual that attempts to increase the pack size from G to G + 1 when rER + ES > 0 (i.e., above dotted line), where r is the coefficient of relatedness, ER is the fitness effect on a repelled intruder, and ES is the fitness effect of repelling the intruder on the current group members (see box 10.2). The points corresponding to G > 2 are based on the reciprocal exponential function for net rate of food intake (see fig. 10.1). Mature solitaries, assumed to have developed hunting skills, are assumed to achieve the average net intake rate of a solitary adult. Immature solitaries, with undeveloped hunting skills, are assumed to obtain no prey and to expend energy at 3 * BMR (=(3 * 3,724 kJ/d)/(6,800 kJ/kg) = -1.6 kg/d). A group comprising first-order relatives (r = 0.5) should accept an immature solitary with undeveloped hunting skills, but repel any mature solitary even if it is close kin.

Pack size, G

Figure 10.2. The application of Hamilton's rule to predict whether mature and immature solitary wolves should be allowed in packs of various sizes when the pack controls group entry (see also fig. 5 in Vucetich et al. 2004). The pack should repel any individual that attempts to increase the pack size from G to G + 1 when rER + ES > 0 (i.e., above dotted line), where r is the coefficient of relatedness, ER is the fitness effect on a repelled intruder, and ES is the fitness effect of repelling the intruder on the current group members (see box 10.2). The points corresponding to G > 2 are based on the reciprocal exponential function for net rate of food intake (see fig. 10.1). Mature solitaries, assumed to have developed hunting skills, are assumed to achieve the average net intake rate of a solitary adult. Immature solitaries, with undeveloped hunting skills, are assumed to obtain no prey and to expend energy at 3 * BMR (=(3 * 3,724 kJ/d)/(6,800 kJ/kg) = -1.6 kg/d). A group comprising first-order relatives (r = 0.5) should accept an immature solitary with undeveloped hunting skills, but repel any mature solitary even if it is close kin.

including this cost shift the economic picture so dramatically? The key insight here is that individual wolves in larger packs must pay a greater cost in terms of food sharing with other wolves, but this cost is offset by the reduced loss of food to scavenging ravens. Such economic realities may commonly favor sociality in carnivores that hunt large prey and thus are vulnerable to kleptoparasitism (see Carbone et al. 1997; Gorman et al. 1998).

This case study highlights the value of applying formal theory. The failure of kin selection to explain wolf sociality prompted us to continue the search for a foraging-based explanation. Without modern theory on group membership decisions, we might have been satisfied to attribute large pack size in wolves to kin selection and unknown factors. Instead, our conclusions now lead us to ask why group members would prevent entry into the pack and why observed pack size is smaller than predicted (see fig. 10.3). The next subsection offers some perspective.

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