Behavioral Ecology Vol. 11 No. 6: 640–647
Reproductive skew and group size: an
N-person staying incentive model
Hudson Kern Reeve and Stephen T. Emlen
Department of Neurobiology and Behavior, Cornell University, Ithaca, NY 14853, USA
Transactional models of social evolution emphasize that dominant breeders may donate parcels of reproduction to subordinates
in return for peaceful cooperation. We develop a general transactional model of reproductive partitioning and group size for
N-person groups when (1) expected group output is a concave (decelerating) function g[N] of the number N of group members,
and (2) the subordinates may receive fractions of total group reproduction (‘‘staying incentives’’) just sufficient to induce them
to stay and help the dominant instead of breeding solitarily. We focus especially on ‘‘saturated’’ groups, that is, groups that have
grown in size just up to the point where subsequent joining by subordinates is no longer beneficial either to them (in parentoffspring groups) or to the dominant (in symmetric-relatedness groups). Decreased expected output for solitary breeding
increases the saturated group size and decreases the staying incentives. Increased relatedness decreases both the saturated group
size and the staying incentives. However, in saturated groups with symmetric relatedness, an individual subordinate’s staying
incentive converges to 1 ⫺ g[N* ⫺ 1]/g[N*]) regardless of relatedness, where N* is the size of a saturated group, provided
that the g[N] function near the saturated group size N* is approximately linear. Thus, staying incentives can be insensitive to
relatedness in saturated groups, although the dominant’s total fraction of reproduction (total skew) will be more sensitive. The
predicted ordering for saturated group size is: Parent-full sibling offspring ⫽ non-relatives ⬎ symmetrically related relatives.
Strikingly, stable groups of non-relatives can form for concave g[N] functions in our model but not in previous models of group
size lacking skew manipulation by the dominant. Finally, symmetrical relatedness groups should tend to break up by threatened
ejections of subordinates by dominants, whereas parent-offspring groups should tend to breakup via unforced departures by
subordinates. Key words: cooperative breeding, dominance, ecological constraints, group size, relatedness, skew, eusociality.
[Behav Ecol 11:640–647 (2000)]
M
odels of ‘‘optimal skew’’ in reproduction attempt to explain the degree of reproductive dominance (skew)
within animal societies by predicting the conditions under
which a dominant breeder should yield just enough reproduction to a subordinate to make it favorable for the subordinate to stay in the group and cooperate peacefully rather
than to leave the group and reproduce independently or fight
for exclusive control of the group’s resources. The potential
applicability, as well as the limitations, of these models has
been extensively discussed in several recent reviews and critical commentaries (Bourke, 1997; Cant, 1997; Clutton-Brock,
1998; Emlen, 1999; Keller and Reeve, 1994; Reeve, 1998). We
refer to these models as ‘‘transactional’’ because group members are envisioned as trading parcels of reproduction for
peaceful cooperation (Reeve et al., 1998). Reproductive payments that prevent subordinates from leaving (Emlen, 1982;
Reeve, 1991; Reeve and Ratnieks, 1993; Vehrencamp, 1979,
1983) are called staying incentives; payments that prevent subordinates from fighting to the death for complete control of
group resources are called peace incentives (Reeve and Ratnieks, 1993). The Reeve and Ratnieks model, which integrated both incentives for two-person groups, predicted that the
skew should tend to increase (i.e., the reproductive sharing
should become less equitable) as (1) the relatedness between
dominants and subordinates increases, (2) the probability of
successful solitary reproduction by the subordinate decreases
(i.e., for stronger ecological constraints), (3) the subordinate’s contribution to colony productivity increases, and (4)
the subordinate’s relative fighting ability decreases. Recently,
Johnstone et al. (1999) have shown that similar relationships
Address correspondence to H. K. Reeve. E-mail: [email protected].
Received 16 September 1999; revised 7 April 2000; accepted 18
April 2000.
2000 International Society for Behavioral Ecology
hold for three-person groups in which a dominant fully controls the reproduction of its two subordinates.
In this article, we shall develop a comprehensive, transactional theory of reproductive skew and group size for arbitrarily sized (N-person) groups in which staying incentives may
be given to subordinates by a dominant. We assume that there
is an increasing, but concave (decelerating) relationship between group output and group size, at least in the neighborhood of the saturated group size (i.e., maximal group size
above which subordinates will not be added). The resulting
theory predicts how both the saturated group size and the
reproductive skew will vary with changes in the genetic and
ecological variables of the transactional models. We also compare the predictions of our model of group size to those of
previous models lacking skew adjustments by a dominant
group member (Giraldeau and Caraco, 1993; Higashi and Yamamura, 1993).
An N-person transactional model of skew and group size
In the dyadic transactional model of skew, the magnitudes of
the staying and peace incentives yielded to subordinates by
dominants, as well as the conditions under which the latter
incentives are given, are obtained by use of the Hamilton’s
rule inequality (Hamilton, 1964). In particular, an action i is
favored over an action j if:
Bi ⫺ Bj ⫹ r(Ki ⫺ Kj) ⬎ 0,
(1)
where Bi (or Bj) is the personal reproductive output associated
with the ith (or jth) action, Ki (or Kj) is the corresponding
partner’s reproductive output and r is the genetic relatedness
between the two partners (here assumed symmetrical, but allowed to become asymmetrical below).
In the N-person skew model, the dyadic version of Hamilton’s rule is generalized to take the form:
Reeve and Emlen • N-person skew models
冘 r (K
N
m ⫽1
m
m,i
⫺ K m,j ) ⬎ 0,
641
(2)
(Bulmer, 1994) where N is the group size, m indexes the
group member, and, if s is the index for self, rs ⫽ 1.0 and Ks,i
⫽ Bi and Ks,j ⫽ Bj from the dyadic model. Note that the lefthand side of Expression 2 can be viewed as the difference of
two inclusive fitnesses (one for strategy i and the other for
strategy j), each inclusive fitness having the form ⌺rmKm. The
use of the additive version of Hamilton’s rule and of these
component inclusive fitnesses can be justified if actions are
conditional upon role (Parker, 1989), in this case dominance
rank.
The variables entering into the dyadic skew model are: x,
the expected reproductive output of a solitary subordinate
(standardized relative to an expected output of 1.0 for a solitary dominant); k, the total output of the dyad relative to
that of a solitary dominant; r, the genetic relatedness; and f,
the probability that a subordinate would win a lethal fight with
the dominant. We make the plausible assumptions (as in prior
models) that k ⬎ 1 (i.e., the dyad does better than a lone
dominant) and that r ⬍ 1.0. Which individual is dominant
and which is subordinate is determined by the relative efficiency of an individual in converting group resources into an
increased share of reproduction (Reeve et al., 1998). We assume that the dominant has a sufficiently high efficiency that
it can push the subordinate’s share of reproduction down to
the latter’s staying or peace incentive.
Using Expression 1, the dyadic model staying incentive ps
can be calculated by finding that fraction of the dyad’s reproduction that just makes staying (versus leaving) favorable for
the subordinate (Reeve and Ratnieks, 1993):
ps ⫽
x ⫺ r(k ⫺ 1)
.
k(1 ⫺ r)
(3)
Alternatively, ps can be obtained by equating ps k ⫹ r(1 ⫺ ps)k
(⫽ the subordinate’s inclusive fitness for staying) and x ⫹ r
(⫽ the subordinate’s inclusive fitness for leaving).
Hamilton’s rule also is used to determine under what conditions the staying incentive will be donated by dominants.
Reeve and Ratnieks (1993) showed that the subordinate will
stay and help with no direct reproduction under strong ecological constraints, which corresponds to the condition x ⬍
r(k ⫺ 1). Dominants will yield the staying incentive (Equation
3) instead of ejecting the subordinate under moderate ecological constraints, which corresponds to the condition r(k ⫺
1) ⬍ x ⬍ k ⫺ 1. No stable group forms (unless peace incentives are given) under weak ecological constraints, that is, x
⬎ k ⫺ 1.
Let us now generalize these results to a model of staying
incentives in groups of arbitrary size. (For simplicity, we ignore the possibility of peace incentives in this article.) We
assume that the expected total reproductive output of a group
consisting of N individuals is given by the function g[N],
which is concave (decelerating), at least in the vicinity of the
saturated group size. Each group consists of a single dominant
and N ⫺ 1 subordinates of roughly equal competitive efficiencies (as defined in Reeve et al., 1998). We shall express a
focal subordinate’s staying incentive if it remains in the group
as ps[N], that is, as a function of the total group size N. Thus,
if a focal subordinate were to leave the group, the staying
incentive of each of the remaining subordinates would become ps[N ⫺ 1].
Symmetric relatedness, positive staying incentives
First we model the case of a symmetric relatedness r among
group members, as occurs when group members are of the
same generation (e.g., when they are full siblings). Each subordinate is free to join the group or to breed solitarily and
will receive positive staying incentives if the dominant provides
them. The expected absolute reproductive output of a solitary
breeder is equal to S ⫽ xg[1]. Decreasing x leads to decreasing S, corresponding to increasingly harsh ecological constraints on solitary reproduction (as in the two-person models). When a group just reaches the size at which joining is
no longer favored over breeding solitarily (because the dominant is favored to eject the subordinate or the subordinate is
favored to leave the group), then we say that the group is
saturated, that is has reached its maximal stable size. Interestingly, it turns out that the staying incentive has special properties in saturated groups.
First, we determine the magnitude of the staying incentive
for a subordinate in a group of size N (i.e., the staying incentive for a focal subordinate that joins a group of size N ⫺ 1).
This staying incentive ps[N] will be such that the inclusive
fitness for joining a group of N ⫺ 1 individuals, that is,
g[N]ps[N] ⫹ rg[N](1 ⫺ ps[N]) will just equal the inclusive
fitness for solitary reproduction, that is, xg[1] ⫹ rg[N ⫺ 1].
After equating these two inclusive fitnesses, solving for ps[N],
and simplifying, we obtain the following as the N-person subordinate staying incentive ps[N]:
ps [N ] ⫽
xg[1] ⫺ r(g[N ] ⫺ g[N ⫺ 1])
.
g[N ](1 ⫺ r)
(4)
The latter staying incentive also applies simultaneously to all
of the other N ⫺ 2 subordinates, not just the focal one, because if any one of the N ⫺ 1 equivalent subordinates does
not benefit from leaving, then none will (thus, Equation 4
incorporates the decisions of all subordinates, not just the focal one). Providing a staying incentive is always possible as
long as the right-hand side of Equation 4 is less than one,
which will occur if S ⬍ g[N] ⫺ rg[N ⫺ 1]. Below we show that
the latter must always be true if it pays the dominant to yield
the staying incentive, so ultimately it is the dominant that will
determine the group size.
What are the conditions under which the dominant will be
willing to concede a staying incentive versus eject the subordinate? In this case we need to apply the generalized Hamilton’s rule (Expression 2) to the decision of the dominant to
yield the staying incentive of Equation 4 rather than eject the
subordinate. If the focal subordinate joins, the dominant receives g[N]{1 ⫺ (N ⫺ 1)ps[N]} offspring, and the focal subordinate and the other N ⫺ 2 subordinates each receive
g[N]ps[N] offspring. If the focal subordinate is ejected
(leaves), the dominant receives g[N ⫺ 1](1 ⫺ (N ⫺ 2)ps[N ⫺
1]) offspring, the focal subordinate receives S ⫽ xg[1] offspring, and the other N ⫺ 2 subordinates each receive g[N ⫺
1]ps[N ⫺ 1] offspring. Thus, from Expression 2, the appropriate Hamilton’s rule inequality for concession of the staying
incentive by the dominant to the focal subordinate versus
ejecting the subordinate is:
g[N ]{1 ⫺ (N ⫺ 1)ps [N ]} ⫺ g[N ⫺ 1](1 ⫺ (N ⫺ 2)ps [N ⫺ 1])
⫹ r(N ⫺ 2){g[N ]ps [N ] ⫺ g[N ⫺ 1]ps [N ⫺ 1]}
⫹ r {g[N ]ps [N ] ⫺ S } ⬎ 0.
(5)
The values of both ps[N] and ps[N ⫺ 1] can be obtained from
Equation 4 (the latter by substituting N ⫺ 1 for N). Clearly,
Expression 5 is less likely to be satisfied as the absolute solitary
output S increases (i.e., as x increases). We seek the critical
value of S at which the left-hand side of the inequality becomes negative, because this will tell us when the group is
saturated (i.e., the point at which the dominant should start
ejecting subordinates). Thus, we next solve Expression 5 as an
Behavioral Ecology Vol. 11 No. 6
642
⳵Scrit (N ⫺ 2)(g[N ⫺ 2] ⫺ g[N ⫺ 1] ⫹ g[N ] ⫺ g[N ⫺ 1])
⫽
.
⳵r
(1 ⫹ r) 2
(8)
Figure 1
Group size in the N-person staying incentive model for non-relatives
or parent-offspring groups. The size of a saturated colony is
approximately the value N* for which g[N*] ⫺ g[N* ⫺ 1] ⫽ S,
where S is the solitary output. Thus, the saturated group size
increases as S decreases. In groups of symmetrically-related relatives
with partial skew, the saturated group size decreases below that for
non-relatives or parent-offspring groups as the relatedness r
increases.
equality in terms of S (appropriately substituting for ps[N] and
ps[N ⫺ 1] using Equation 4) and obtain the critical value for
S ⫽ Scrit:
([1 ⫹ r(N ⫺ 1)]{g[N ] ⫺ g[N ⫺ 1]}
⫺ r(N ⫺ 2){g[N ⫺ 1] ⫺ g[N ⫺ 2]})/(1 ⫹ r).
(6)
Note that Scrit is a function of group size, relatedness, and
characteristics of the g[N] curve, whereas S is the actual value
for solitary success. When S is greater than Scrit, then the dominant is favored to eject the subordinate. Thus, the group is
saturated when S ⫽ Scrit. We can now find the staying incentive
for a saturated group by substituting Expression 6 for S in
Equation 4. After simplification, this yields the staying incentive in a saturated group,
{[1 ⫺ r ⫹ (N ⫺ 1)r ⫺ r 2 ]g[N ]
⫺ [1 ⫺ 2r ⫹ 2(N ⫺ 1)r ⫺ r 2 ]g[N ⫺ 1] ⫹ r(N ⫺ 2)g[N ⫺ 2]}
⫼ {(1 ⫺ r)(1 ⫹ r)g[N ]}.
(7)
Expressions 6 and 7 look complicated, but they can be unpacked to make some surprisingly strong predictions about
group size and skew.
Saturated group size
First, we can ask what will happen to the size of saturated
groups when ecological constraints on solitary reproduction
are relaxed, that is, x increases. When x increases, the group
size N* at saturation must change so that Expression 6 increases to match the increased value S ⫽ xg[1]. Expression 6
will increase only if N decreases, owing to the concave shape
of g[N] (see Appendix).
From Expression 6 directly, we can derive the saturated
group size N* for the special case of non-relatives: N* satisfies
g[N*] ⫺ g[N* ⫺ 1] ⫽ S (see Figure 1).
It can also be shown that the saturated group size N* decreases as the relatedness increases. The partial derivative of
Expression 6 with respect to the relatedness r is given by:
If, as assumed, g[N] is decelerating, the numerator in the
right-hand side of Equation 8 must be negative (because g[N]
⫺ g[N ⫺ 1] ⬍ g[N ⫺ 1] ⫺ g[N ⫺ 2]) and so Scrit will decline
with increasing relatedness. This means that colonies will become saturated at a lower value of S as relatedness increases.
In other words, if relatedness increases for a saturated colony
and a fixed value of S, then now Scrit ⬍ S, and some subordinates would be ejected that otherwise would be accepted at
lower values of relatedness, that is, the group size will decrease. Consequently, the saturated group size N* must decline with increasing relatedness if g[N] is decelerating. However, if g[N] is virtually linear in the neighborhood of N* (i.e.,
the concave curvature is negligible), the numerator in the
right-hand side of Expression 8 will essentially be zero, and
the saturated group size will be rather insensitive to the value
of the relatedness. In particular, for concave g[N] functions
of the general form cNa, where c is a constant and 0 ⬍ a ⬍
1, the right-hand side of Equation 8 converges to zero as
group size increases, that is, the sensitivity of group size to
relatedness will decline to zero with increasing group size.
Whenever g[N] is virtually linear from N ⫺ 2 to N, N* will
generally satisfy g[N*] ⫺ g[N* ⫺ 1] ⫽ S (see Figure 1), regardless of relatedness.
The saturation size is determined by the dominant’s threshold S for ejecting potential joiners, not by the other resident
subordinates’ ejection thresholds. The latter is obtained by
solving the equivalent of Expression 5 from a resident subordinate’s point of view, which reduces to just S ⬍ g[N ⫺ 1]
⫺ g[N ⫺ 2] (relatedness cancels out here because the resident
subordinates are equally related to the dominant and to the
joining subordinate). Because g[N] is concave, g[N ⫺ 1] ⫺
g[N ⫺ 2] ⬎ g[N] ⫺ g[N ⫺ 1], so resident subordinates will
always benefit from admitting an additional subordinate if the
dominant does. However, the latter inequality shows that the
dominant may not benefit from admitting a subordinate while
the other subordinates do. Interestingly, this predicts that it
will be the dominant (not subordinates) that first drives out
potential joiners. Also, because g[N*] ⫺ g[N* ⫺ 1] ⫽ S implies that g[N*] ⫺ rg[N* ⫺ 1] ⱖ S, it follows that the staying
incentive (Equation 4) is possible (i.e., ⱕ 1) and thus maximal
group size will be determined entirely by the dominant’s willingness to provide staying incentives and not by voluntary departures of subordinates. (It also should be noted that the
condition g[N*] ⫺ g[N* ⫺ 1] ⫽ S guarantees that a dominant
will always have a greater reproductive share than a subordinate, which occurs when S ⬍ [1 ⫹ r(N* ⫺ 1)]g[N*]/N* ⫺
rg[N* ⫺ 1].) When a group is at saturation size, dominants
should either eject or threaten to eject additional subordinates (threatened subordinates may leave without actual physical conflict).
Staying incentives in saturated groups
Predictions also can be extracted from Expression 7 about the
characteristics of the staying incentive in saturated groups. If
group members are unrelated (r ⫽ 0), the staying incentive
of an individual subordinate takes a simple form:
1⫺
g[N* ⫺ 1]
.
g[N*]
(9)
In this case, the saturated staying incentive will be close to
zero if the expected group output with a subordinate is only
slightly greater than without that subordinate (g[N ⫺ 1] 艐
g[N]) and relatively large if the positive effect of a single sub-
Reeve and Emlen • N-person skew models
643
ordinate on group output is large. Note that Expression 9
together with the condition giving the saturated group size,
g[N*] ⫺ g[N* ⫺ 1] ⫽ S, allows us to trace the effects of
decreasing solitary output S (i.e., decreasing x) simultaneously
on both the group size and the staying incentive. As S decreases, the saturated group size N* must increase since g[N]
⫺ g[N ⫺ 1] declines with increasing N. Now as g[N] ⫺ g[N
⫺ 1] decreases, the staying incentive in Expression 9 necessarily decreases. In sum, when relatedness is zero, one would
expect low solitary outputs to be associated with large saturated group sizes and high skews, whereas high solitary outputs would be associated with small saturated group sizes and
low skews.
How will the staying incentive change with group size and
solitary breeding success in groups that are below saturation
size? By differentiating Equation 4 with respect to N, it is seen
that the staying incentive will increase with increasing N if S
⬍ r{g[N](g⬘[N ⫺ 1]/g⬘[N]) ⫺ g[N ⫺ 1]}. That is, the staying
incentive increases with N if the solitary breeding success S is
sufficiently low, group size N is sufficiently low, and relatedness r is sufficiently high. Otherwise, the staying incentive decreases with increasing N; in particular, the staying incentive
always decreases with increasing group size if the relatedness
is zero and S ⬎ 0. Since ⳵ps/⳵S ⫽ 1/g[N](1 ⫺ r) ⬎ 0, it also
follows that, for a given group size, the staying incentive will
always increase as solitary breeding success increases.
How will the saturated staying incentive vary with relatedness? The partial derivative of Expression 7 with respect to
relatedness is given by:
⳵ps [N ]
⫽ {(N ⫺ 2)(1 ⫹ r 2 )(g[N ⫺ 2] ⫺ g[N ⫺ 1] ⫹ g[N ]
⳵r
⫺ g[N ⫺ 1])}/{(1 ⫺ r 2 ) 2 g[N ]}.
(10)
If g[N] is decelerating (as assumed), the numerator in the
right-hand side of Equation 10 must be negative (because
g[N] ⫺ g[N ⫺ 1] ⬍ g[N ⫺ 1] ⫺ g[N ⫺ 2]) and so the saturated staying incentive will decline with increasing relatedness,
as generally occurs in the two- and three-person models
( Johnstone et al., 1999; Reeve and Ratnieks, 1993). However,
if g[N] is nearly linear in the neighborhood of N* (i.e., the
concave curvature is very slight), the right-hand side of Equation 10 will essentially be zero, and the saturated staying incentive will essentially equal Expression 9 and be quite insensitive to the value of the relatedness. In particular, for concave
g[N] functions of the general form cNa, the right-hand side
of Equation 10 converges to zero as group size increases, that
is, the sensitivity of the staying incentive to relatedness will
decline with increasing group size (see Figure 2). In sum, if
the g[N] function is nearly linear in the neighborhood of the
saturated group size, then the condition for the saturated
group size converges to g[N*] ⫺ g[N*] ⫽ S, and the staying
incentive converges to Expression 9, regardless of relatedness.
These predictions show that the generally positive relationship
between skew and relatedness predicted by two- and threeperson models may not always hold for saturated N-person
groups, particularly if skew is ascertained solely from the magnitude of an individual subordinate’s staying incentive and the
g[N] function is nearly linear in the neighborhood of N*.
Although the individual staying incentive may be quite insensitive to relatedness in saturated groups, the dominant’s
total fraction of reproduction (i.e., the total skew ⫽ 1 ⫺ sum
of individual subordinate staying incentives) may remain quite
sensitive (see Figure 2). The partial derivative of the total skew
with respect to relatedness is equal to the right hand expression in Equation 10 multiplied by ⫺(N ⫺ 1). For concave
g[N] functions of the form cNa, this derivative converges not
to zero but to ca(a ⫺ 1)[(1 ⫹ r 2)/(1 ⫺ r 2)2] as group size
Figure 2
Reproductive skew versus relatedness and number of subordinates
(N ⫺ 1) in the N-person model; g[N] ⫽ N½. (Top) The critical
value of relative solitary success x below which joining the group
with the given relatedness and size is favored. (Middle) An
individual subordinate’s staying incentive in a saturated colony as a
function of the relatedness r and the number of subordinates n ⫽
N ⫺ 1. Note that the staying incentive is sensitive to relatedness
only for smaller group sizes, because g[N] becomes more nearly
linear at high N. (Bottom) The dominant’s total fraction of
reproduction as a function of the relatedness r and the number of
subordinates n ⫽ N ⫺ 1. Note that this total fraction, unlike an
individual subordinate’s staying incentive, is sensitive to relatedness
even at large group sizes.
increases, that is, the sensitivity of the staying incentive to relatedness will not vanish with increasing group size (see Figure
2). Note that the latter derivative becomes small for large saturated group size only if a is close to 0 or 1, that is, the g[N]
function is nearly linear.
Symmetric relatedness, no staying incentives (complete
skew)
In two-person groups of relatives, complete skew (zero staying
incentive) can result if the ecological constraints are strong
Behavioral Ecology Vol. 11 No. 6
644
enough (Reeve and Ratnieks, 1993). From (4) we can easily
derive the condition for complete skew in the N-person model. The staying incentive will be zero when:
S ⬍ r(g[N] ⫺ g[N ⫺ 1]).
(11)
(Note that if the staying incentive is zero for a group size of
N, it must also be zero for all smaller group sizes, owing to
the concave curvature of g[N].) Thus, complete skew will be
expected only when r ⬎ 0. From Expression 5, the dominant
will be favored to retain subordinates receiving no staying incentive if:
S⬍
g[N ] ⫺ g[N ⫺ 1]
,
r
(12)
which will always hold true if Inequality 11 is true. Note that
groups with complete skew are likely to be unsaturated with
subordinates, because the dominant will yield staying incentives to subordinates whenever S ⬍ Scrit 艐 g[N] ⫺ g[N ⫺ 1],
that is, at group sizes larger than the maximum size at which
complete skew occurs (S ⫽ r{g[N] ⫺ g[N ⫺ 1]}).
Asymmetrical relatedness (parent-offspring groups)
For simplicity, we assume that the subordinates are all offspring of the dominant parent(s) (i.e., they are genetic full
siblings). The subordinate is related to itself by 1.0, to its siblings by ½, and to the dominant effectively by 1.0 (because its
future siblings are as reproductively valuable as offspring
[Reeve and Keller, 1995; Reeve et al., 1998]). First, we apply
Hamilton’s rule (2) to the decision of a subordinate to stay
in versus leave from the group. We assume that, as in the
above model, when a new subordinate attempts to join a
group of size N ⫺ 1 (forming a group of size N), all N ⫺ 1
subordinates become equivalent in their required staying incentives:
g[N ]ps [N ] ⫺ S ⫹ (1/2)(N ⫺ 2)
⫻ (ps [N ]g[N ] ⫺ ps [N ⫺ 1]g[N ⫺ 1])
⫹ {1 ⫺ (N ⫺ 1)ps [N ]}g[N ] ⫺ {1 ⫺ (N ⫺ 2)ps [N ⫺ 1]}
⫻ g[N ⫺ 1] ⬎ 0.
(13)
We first note that the left-hand side of Inequality 13 strictly
declines as ps[N] increases, that is, the derivative of this expression with respect to ps[N] is equal to ⫺(½)(N ⫺ 2)g[N].
This means that subordinates will never benefit from staying
incentives, unlike in the symmetrical relatedness case. Thus,
when subordinates are equivalent in their required staying incentives, complete skew is always expected in parent-offspring
groups. (Reeve et al., 1998, and Emlen, 1999 leave open the
possibility that staying incentives can be given in parent-offspring groups in other situations, e.g., when subordinates are
not equivalent. Such models are beyond the scope of the present paper.) From Inequality 13 and ps[N] ⫽ 0, we can easily
derive the condition for favored joining when there is complete skew in the N-person parent-offspring model. Joining
will be favored over voluntary leaving if:
S ⬍ g[N] ⫺ g[N ⫺ 1].
(14)
(Note that if the required staying incentive is zero for a group
size of N, it must also be zero for all smaller group sizes, owing
to the concave curvature of g[N].) From Hamilton’s rule (Inequality 2), it is also easy to show that the dominant will be
favored to retain subordinates receiving no staying incentive
if:
S ⬍ 2(g[N] ⫺ g[N ⫺ 1]),
which will always hold true when Inequality 14 is true.
(15)
Thus, the saturated group size is determined solely by the
condition (derived from Inequality 14): g[N*] ⫺ g[N* ⫺ 1]
⫽ S. This condition applies because if N is below N*, the
group is unsaturated (subordinates are still favored to join),
whereas if it is above N*, subordinates will leave without coercion. Thus, as in the symmetric relatedness model, the saturated group size increases as S decreases since g[N] is concave (see Figure 1). However, in the case of asymmetric relatedness, exceeding the saturated group size results in voluntary
departure, not eviction, of subordinates.
The saturated group size for parent-offspring groups will
tend to be higher than the size of saturated groups containing
symmetrically related relatives. This is because the maximum
possible saturated group size for symmetrical relatedness
groups is given by the same condition that applies to parentoffspring groups, g[N*] ⫺ g[N* ⫺ 1] ⫽ S. The group size
associated with the latter condition is closely approached for
symmetrical relatedness groups only for groups of non-relatives or when g[N] is virtually linear in the range N ⫺ 2 to N,
for example, for sufficiently large groups near an upper asymptote for g[N]. For all other situations, the saturated group
size in symmetrical relatedness groups will be less than that
for saturated parent-offspring groups.
Of course, offspring may not always be full siblings. For
example, because of extra-pair paternity, the relatedness of a
sibling to a subordinate (relative to the value of the subordinate’s own offspring) may be less than 1, with r ⫽ ½ in the
extreme case of all subordinates being half-siblings to each
other. In this case, Inequality 13 would be modified so that
the subordinate’s average relatedness to the other subordinates’ offspring is r/2 instead of ½ and its relatedness to the
dominant parent’s offspring (relative to its relatedness to its
own offspring) is r instead of 1. In the latter case, it can be
shown with the logic above that subordinates still would require no staying incentive unless r is less than 2/N (the derivative of the subordinate’s inclusive fitness with respect to its
staying incentive is negative if the latter inequality is not true).
For example, even if all other subordinates are half-siblings
to the focal subordinate, no staying incentive will be required
unless there is only one other subordinate; for groups of more
than two subordinates, complete skew is always expected, regardless of the frequency of extra-pair paternity (in this case,
⫽ ½, and ½ ⬍ 2/N only if N ⬍ 4.) The new condition for
favored joining now becomes:
S ⬍ r(g[N] ⫺ g[N ⫺ 1]).
(16)
Thus, as the frequency of half-sibs increases (decreasing average r), the saturated group size decreases. (The dominant’s
condition for retaining subordinates is not affected by the
presence of half-sibs provided that all subordinates are its genetic offspring.)
Cost of ejecting subordinates
If ejection of a subordinate incurs a cost e for the dominant
in the symmetric relatedness case, e should be added to the
dominant’s direct fitness difference in the left-hand side of
Inequality 5 since this condition corresponds to acceptance
of the focal subordinate, which avoids this cost. Thus, e is added to the numerator of Scrit in Expression 6, with the result
that increasing ejection cost should increase the saturation
size N* (as is intuitive). Increasing ejection cost would also
increase the staying incentive in a saturated group: The derivative of the saturated staying incentive with respect to the
ejection cost is 1/[N*(1 ⫺ r 2)], which is positive.
Reeve and Emlen • N-person skew models
DISCUSSION
Our N-person concessions model of the joint evolution of reproductive skew and group size substantially increases the
number of ways in which transactional models of skew can be
empirically tested. We summarize the most striking predictions below, most of which have not been systematically investigated.
Predictions concerning saturated group size
1. The predicted ordering for saturated group size in our Nperson model (assuming identical values for the ecological
parameters) is: parent-offspring groups in which all offspring
are full-sibs (complete skew) ⫽ groups of non-relatives (partial
skew) ⬎ groups of symmetrically related relatives (partial or
complete skew) or parent-offspring groups with a high frequency of half-siblings among the offspring (complete or partial skew).
These predictions contrast markedly with those earlier models of group size without skew manipulation by a dominant
(Giraldeau and Caraco, 1993; Higashi and Yamamura, 1993).
The latter models assumed symmetrical relatedness r (and
thus cannot apply to parent-offspring groups). Our symmetric
relatedness N-person skew model is most like the ‘‘group-controlled entry’’ versions of these earlier models, since group
size in our model is determined by the dominant’s ejection
behavior. Intriguingly, the earlier models predict that groups
of non-relatives should never form in the case of concave g(N)
functions, because addition of a non-relative would reduce
every group member’s mean (per capita) reproductive output. For example, Scrit in the Higashi and Yamamura (1993)
model [‘‘w(1)’’ in their Equation 1] is equal to negative infinity when group members are unrelated, that is, such groups
should never form. By contrast, such groups can occur according to our staying incentive model because the dominant
steals some reproduction from joining non-relatives, leaving
just enough so as to secure their cooperation and thereby
reap some of the benefits of grouping. Thus, evidence of the
existence of cooperative associations of non-relatives when the
group output function is concave will by itself support the
staying incentive (transactional) model.
2. Increasing ecological constraints on independent reproduction will result in larger saturated group sizes.
3. Less strongly decelerating group output functions (g[N])
will result in larger saturated group sizes.
4. For symmetrical relatedness groups, increasing relatedness will result in smaller saturated group sizes. However,
group size will become insensitive to relatedness as the group
output function approaches linearity in the neighborhood of
the saturated group size.
This prediction contrasts strongly with those of the earlier,
non-incentive models of group size (Giraldeau and Caraco,
1993; Higashi and Yamamura, 1993). As mentioned above,
our symmetric relatedness N-person skew model is most like
the ‘‘group-controlled entry’’ versions of these earlier models,
because saturated group size in our model is determined by
the dominant’s ejections. In the non-incentive models, group
size increases with increasing relatedness r when group size is
determined by one or more group members, a prediction that
is exactly opposite to ours. This difference arises because, in
the non-incentive models, when addition of joiners reduces
per capita personal fitness (as occurs for a concave group output function), group members are more likely to tolerate joiners of higher relatedness as a kind of altruism. In contrast, in
our staying incentive model, the more potent effect of relatedness is the following: As relatedness r increases, joiners become less attractive to the dominant because such joiners
645
would yield increasing indirect benefits for the dominant if
they were forced to breed solitarily.
5. For asymmetrical relatedness (parent-offspring) groups
where all offspring are genetic offspring of one parent, an
increase in the relative frequency of half-sibling dyads among
subordinates (e.g., as might occur if a breeding female reproduced with several males) will result in smaller saturated
group sizes.
Predictions concerning the magnitude of saturation staying
incentives
6. The saturated staying incentive will decrease with increasing
constraints on independent reproduction (as in two- and
three-person models; Johnstone et al. 1999; Reeve and Ratnieks, 1993).
7. For symmetrical relatedness groups, increasing relatedness will usually result in smaller saturated staying incentives
(as in two- and three-person models; Johnstone et al., 1999;
Reeve and Ratnieks, 1993). However, the magnitude of the
staying incentive (but not the total skew) will become quite
insensitive to relatedness as the group output function approaches linearity in the neighborhood of the saturated group
size. This prediction has important implications for how skew
should be measured, a subject of much current debate (Tsuji
and Tsuji, 1998). If relatedness effects on reproductive partitioning are to be tested, the total skew, that is, the alpha’s
total fraction of reproduction, should be more sensitive to
relatedness than will be an individual subordinate’s fraction
of reproduction (see Figure 2). An important consequence of
Predictions 6 and 7 is that, in general, the predicted relationship between skew and ecological factors should be more robust than will the predicted positive correlation between skew
and relatedness.
8. The individual staying incentive in unsaturated groups
will increase as the group size approaches the saturation size
if relatedness is sufficiently high, group size is sufficiently
small, and solitary breeding success is sufficiently small. On
the other hand, if relatedness is sufficiently low, group size is
sufficiently high, and solitary breeding success is sufficiently
high, then the staying incentive will decline with increasing
group size in unsaturated groups. The latter negative effect
of group size on the staying incentive would appear less likely
to occur than the former positive effect, because large group
size is inconsistent with high solitary breeding success (groups
cannot be larger than the saturation size). However, a robust
prediction is that the staying incentive should always decline
with increasing group size for groups of non-relatives (r ⫽ 0).
At a fixed group size below the saturation size, the staying
incentive should always increase as solitary breeding success
increases.
9. Symmetrical relatedness groups with complete skew are
likely to be unsaturated with subordinates.
Predictions concerning behavioral mechanics of group
breakups
10. Parent-offspring associations will tend to breakup by voluntary (unforced) departure of subordinate offspring, whereas associations of symmetrically related group members (or
groups of non-relatives) will tend to breakup via ejection, or
at least behavioral threats of ejection, of subordinates by dominants (and less often by other subordinates).
Valid testing of Predictions 1–8 requires knowledge of
whether groups are or are not saturated. Predictions 9 and 10
offer ways of identifying whether groups are saturated or unsaturated. According to Prediction 9, symmetrical relatedness
groups with complete skew are likely unsaturated—if subor-
Behavioral Ecology Vol. 11 No. 6
646
dinates join or can be added to such groups, Prediction 8 (for
example) predicts that a subordinate’s fraction of reproduction will increase in the enlarged group. Prediction 10 also
offers a possible way of assessing saturation: Ejection by a dominant (symmetrical relatedness groups) or voluntary departure by a subordinate (parent-offspring groups) will indicate
that the group is near saturation. In such cases, for symmetrical relatedness groups, the special staying incentive represented by Expression 9 is predicted and can be tested (it is
predicted to be exactly true for groups of non-relatives and
approximately true for relatives if the group output function
is nearly linear in the vicinity of the saturated group size).
Saturated groups should predominate when there is a sufficient number of subordinates per group in the population,
the necessary number increasing with increasing ecological
constraints on solitary breeding.
A particularly revealing test of the N-person skew model
would be the comparison of staying incentives between intact
groups and groups from which some but not all subordinates
are experimentally removed. In groups of non-relatives, such
removals will increase the distance from the saturated group
size and should result in an increase in the magnitude of the
individual staying incentives for the remaining subordinates
(i.e., lower skew; Prediction 8). The reverse prediction is
made for groups of close relatives if solitary breeding success
is low and initial group size is low. Another test of the theory
would be a comparison of skews in normal groups and samesized groups for which the solitary breeding success has been
manipulated (e.g., increased by providing nearby available
nests or increasing resource availability).
Does the distribution of group sizes and staying incentives
in nature accord at least roughly with the predictions of our
N-person model? It is well known that the largest groups in
the social insects are parent-offspring (mother queen-daughter worker ⫽ eusocial) groups of ants, bees, wasps, and termites (Bourke, 1997; Michener, 1974; Wilson, 1971), as predicted by the model. Even though many of these societies
contain multiple, symmetrically related queens, the large
group size results almost entirely from the size of the worker
(offspring) population. In vertebrates, the largest group sizes
also are exhibited by parent-offspring groups such as naked
mole-rats (Sherman et al., 1991).
Among groups with symmetrical relatedness, associations of
female non-relatives (e.g., communal wasps and bees, unrelated foundresses in ants; Danforth et al., 1996; Kukuk, 1992;
Kukuk and Sage, 1994; Paxton et al., 1996; Strassmann, 1989)
typically are larger (sometimes much larger) than associations
of symmetrically and positively related females (wasp foundress associations; Reeve, 1991; Reeve and Ratnieks, 1993).
The major exception to this trend is the co-occurrence of
many related queens in tropical, swarm-founding wasps; however, exceptionally strong ecological constraints on solitary
founding caused by intense tropical ant and vertebrate predation probably accounts for the often large numbers of conesting queens in these cases ( Jeanne, 1991), as would be
consistent with Prediction 2.
The above patterns in group size support the N-person theory, but a possible confounding factor is that there usually will
be greater numbers of non-relatives than relatives with which
to form a group, so that groups of non-relatives are more
likely to reach saturation. However, the latter effect does not
prevent parent-offspring groups from exhibiting the largest
societies, in apparent support of Prediction 1. The most sensitive tests of the N-person theory will involve comparisons of
group size among populations of closely-related species or
among or within populations of a single species when social
groups vary in genetic composition. The most discriminating
tests will take into account not only variation in the genetic
structure of the group but also variation in the ecologicallydetermined solitary output S and g[N] function.
There is abundant evidence from cooperatively breeding
birds that group size decreases when ecological constraints are
relaxed (Emlen, 1997), consistent with Prediction 2. This has
been confirmed experimentally in some woodpeckers (Walters et al., 1992), wrens (Zack and Rabenold, 1989), fairywrens (Pruett-Jones and Lewis, 1990) and warblers (Komdeur,
1992). Some support for Prediction 2 also exists in social insects (Bourke, 1997; Bourke and Heinze, 1994; Jeanne, 1991;
Reeve, 1991). Interestingly, in some communal bees, the g[N]
function is approximately linear (in contrast to the concave
relationship characteristic of other bee societies: Michener,
1974), and group sizes are quite large as predicted by the
model (Kukuk and Sage, 1994).
Predictions 3–10 are sufficiently novel that few studies have
assembled all of the data needed to test them. Their tests will
greatly facilitate assessment of the applicability of transactional versus alternative (e.g., Reeve et al., 1998) evolutionary
models of reproductive skew.
APPENDIX
Changes in the saturated group size N* as the solitary
output S ⴝ xg[1] changes
We can use a second-order Taylor expansion to approximate
g[N] as g[N ⫺ 1] ⫹ g⬘[N ⫺ 1] ⫹ (½)g⬙[N ⫺ 1]. Likewise, g[N
⫺ 2] can be approximated as g[N ⫺ 1] ⫺ g⬘[N ⫺ 1] ⫹
(½)g⬙[N ⫺ 1]. Substituting these expressions into Equation 6,
we obtain:
Scrit 艐 g⬘[N ⫺ 1] ⫹
[1 ⫹ r(2N ⫺ 3)g ⬙[N ⫺ 1]
.
2(1 ⫹ r)
(17)
The partial derivative of Equation 17 with respect to the group
size N gives (to a second-order approximation, ignoring g⵮[N
⫺ 1] terms)
⳵Scrit
(1 ⫹ 2r)g ⬙[N ⫺ 1]
艐
.
⳵N
1⫹r
(18)
If g[N] is concave, then g⬙[N ⫺ 1] ⬍ 0, and Scrit increases as
the group size decreases. Thus, as the solitary output S increases, the saturated group size N* must decrease if S is to
become equal to Scrit. The rate at which this decrease occurs
increases the greater the negative curvature of g[N] (because
the absolute value of g⬙[N ⫺ 1] increases).
We thank four anonymous reviewers for comments on a previous version of the manuscript. H.K.R. and S.T.E. were supported by grants
from the U.S. National Science Foundation.
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