Behavioral Ecology Vol. 11 No. 6: 640-647
© 2000 International Society for Behavioral Ecology
Reproductive skew and group size: an N-person staying incentive model
Department of Neurobiology and Behavior, Cornell University, Ithaca, NY 14853, USA
Address correspondence to H. K. Reeve. E-mail: hkr1{at}cornell.edu .
Received 16 September 1999; revised 7 April 2000; accepted 18 April 2000.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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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) functiong[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 parent-offspring 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 concaveg[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.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Models 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 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; Higashia 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:
 | (1) |
In the N-person skew model, the dyadic version of Hamilton's rule
is generalized to take the form:
 | (2) |
) 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):
 | (3) |
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]:
 | (4) |
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:
 | (5) |
 | (6) |
 | (7) |
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).
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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:
 | (8) |
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] 
S, it follows that the
staying incentive (Equation 4) is possible (i.e., 
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:
 | (9) |
g[N]) and
relatively large if the positive effect of a single subordinate 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:
 | (10) |
; 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 three-person 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*.
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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 + r2)/(1 - r2)2] as group size 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 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:
 | (11) |
 | (12) |
g[N] -
g[N - 1], that is, at group sizes larger than the maximum
size at which complete skew occurs (S =
r{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 1/2, 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:
 | (13) |
,
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:
 | (14) |
 | (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 parent-offspring 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 = 1/2 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 1/2 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, = 1/2, and 1/2 <
2/N only if N < 4.) The new condition for favored joining
now becomes:
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| (16) |
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
in 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 -
r2)], which is positive.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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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 N-person
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
concaveg(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 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 subordinates 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 same-sized 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 ecologically-determined 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 |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
|---|
Changes in the saturated group size N* as the solitary output S = xg[1] changes
We can use a second-order Taylor expansion to approximate g[N] as g[N - 1] + g'[N - 1] + (1/2)g''[N - 1]. Likewise, g[N - 2] can be approximated as g[N - 1] - g'[N - 1] + (1/2)g''[N - 1]. Substituting these expressions into Equation 6, we obtain:
 | (17) |
 | (18) |
|
ACKNOWLEDGEMENTS |
|---|
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.
|
REFERENCES |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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