Behavioral Ecology

Behavioral Ecology Advance Access originally published online on June 15, 2005
Behavioral Ecology 2005 16(5):880-888; doi:10.1093/beheco/ari062

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Predicting the temporal dynamics of reproductive skew and group membership in communal breeders

Andrew G. Zink^a and H. Kern Reeve^b

^a Field of Ecology and Evolutionary Biology, and ^b Department of Neurobiology and Behavior, Cornell University, Ithaca NY 14853

Address correspondence to Andrew G. Zink, who is now at the Department of Entomology, University of California, One Shields Avenue, Davis, CA 95616, USA. E-mail: agzink{at}ucdavis.edu.

Received 28 May 2003; revised 25 April 2005; accepted 17 May 2005.

	ABSTRACT

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
Reproductive skew models attempt to predict the fraction of reproduction contributed by each individual that participates in a communal brood. One potential limitation of these models is that individuals make a single, fixed decision about group membership and reproductive allocation at the beginning of the breeding period. While this is appropriate for animals that reproduce in a synchronous bout, many cooperative breeders produce offspring over a prolonged period of time. It is likely that these species adjust reproductive allocation and group membership over time in response to temporal shifts in group productivity and ecological constraints. In this paper we adapt transactional models of reproductive skew to a continuous form, generating time-dependent functions of reproductive allocation. We derive a general method for predicting temporal changes in group membership as well as a general expression for reproductive skew across the regions over which a group is stable. Using a linear approximation for time-dependent reproduction, we derive new expressions for reproductive skew in cases where the subordinate departs during the breeding period. In this case we find that the traditional model always overestimates the subordinate's share of reproduction when dominants are in control of both reproductive shares and group membership (i.e., concessions models). Conversely, we find that the traditional model always underestimates the subordinate's share of reproduction when subordinates are in control of reproductive shares (i.e., constraint models). We discuss the implications of these new calculations in relation to the traditional skew models and more recent empirical tests of reproductive skew in animal societies.

Key words: communal breeding, dynamic model, kin selection, reproductive skew.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
Reproductive allocation among communal breeders can range from complete monopolization of resources by one dominant individual to equal contribution by each member of a breeding group. Models of reproductive skew predict the degree of monopolization of reproductive output by one or more females in response to group productivity, ecological constraints, and dominance hierarchies (see reviews in Johnstone, 2000; Keller and Reeve, 1994). These models have been used to predict a wide variety of social behaviors such as infanticide (Johnstone and Cant, 1999a), lekking (Widemo and Owens, 1995), social queuing (Kokko and Johnstone, 1999), group foraging (Hamilton, 2000), aggression (Reeve, 2000; Reeve and Ratnieks, 1993), manipulation (Crespi and Ragsdale, 2000), and the origin of sterile castes (Jeon and Choe, 2003). This generality and broad applicability of skew models has led some scientists to herald these models as a first step toward a unified theory of social behavior (Keller and Reeve, 1994; Reeve and Keller, 2001; Sherman et al., 1995).

As originally conceived, models of reproductive skew assumed that a dominant individual has complete control over reproductive allocation among group members (Emlen, 1982a,b; Vehrencamp, 1979, 1983a,b). These models considered the minimum amount of reproduction that a dominant should yield to a subordinate (for a given level of group productivity and ecological constraints on solitary breeding) in order to keep her at the nest. Reeve and Ratnieks (1993) formalized these models and extended them to (1) analyze the conditions under which stable groups should form and (2) include the possibility that dominant females will provide peace incentives to subordinates. Reeve and Ratnieks also emphasized that a dominant should be expected to yield the minimum fraction of combined reproduction (i.e., staying incentive) required to keep a subordinate at the nest. These models have been termed "concession" models because the dominant concedes a fraction of the group's reproduction to the subordinate (Clutton-Brock, 1998).

In contrast to the concessions approach, a second class of skew models termed "restraint" models were developed with new assumptions that were thought to be more appropriate for vertebrates rather than social insects (Johnstone, 2000; Johnstone and Cant, 1999b). For example, these models assumed that the subordinate, not the dominant, has full control over reproductive allocation while the dominant retains control over group membership. Therefore a subordinate can be expected to take the maximum fraction of overall reproduction that is possible before it is more beneficial for the dominant to evict (versus tolerate) the subordinate. This approach predicts an upper limit (as opposed to the concession's model's lower limit) to the subordinate's fraction of the group's reproduction. In this case it is the threat of eviction by the dominant, rather than the threat of departure by the subordinate, that sets the fraction of subordinate reproduction (Johnstone, 2000; Johnstone and Cant, 1999b). Taken together, the concessions and restraint models have been termed "transactional" models of reproductive skew (Johnstone, 2000; Reeve, 2000; Reeve et al., 1998a) because the overall level of reproductive skew is determined by an exchange of benefits among the communal breeders. These models are also united because they make common predictions about the qualitative relationships between reproductive skew and relatedness or skew and ecological constraints (if the dominant is the individual controlling reproductive shares; Reeve and Keller, 2001).

Despite the promise of these transactional models, empirical studies have yielded mixed results. While some studies of insects such as ants, wasps, and bees reveal support for these transactional skew models (e.g., Bourke et al., 1997; Hannonen and Sundström, 2003; Paxton et al., 2002; Reeve et al., 2000), other insect studies do not support the models and suggest that alternative approaches such as the "tug-of-war" model may be more appropriate (Field et al., 1998; Reeve and Keller, 2001; Reeve et al., 1998a; Seppä et al., 2002). Interpreting data on reproductive skew can be complicated because in many insect societies the degree of skew changes over time (Eggert and Müller, 2000; Field et al., 1998; Reeve et al., 2000). In addition, many insects exhibit changes in group membership (e.g., the arrival or departure of subordinates) over the course of a breeding period (Dunn and Richards, 2003; Reeve et al., 1998b). Temporal changes in reproductive skew and group membership are also relevant for vertebrates, where skew measured in a single season may not be representative of the total duration of a cooperative association. Like in insects, empirical tests of the transactional model of reproductive skew theory yielded mixed results in mammals (Cant, 2000; Clutton-Brock et al., 2001; Packer et al., 2001) and birds (Haydock and Koenig, 2002; Magrath and Heinsohn, 2000; Öst et al., 2003). Many of these studies find low degrees of overall reproductive skew over longer periods of time, with different individuals monopolizing reproduction (i.e., high skew) within a given breeding bout (Alberts et al., 2003; Haydock and Koenig, 2002; Macedo et al., 2004; Packer et al., 2001). This suggests that, similar to insect societies, vertebrate societies experience temporal fluctuations in the degree and direction of reproductive skew among group members.

It is likely that some of this disconnect between empirical studies and transactional models of reproductive skew can be attributed the fact that, as originally conceived, the models may be overly simplistic for a particular species. Kokko (2002) has shown, for example, that the decisions predicted by these models may not be evolutionarily stable if subordinates have imperfect knowledge about the benefits provided to them if they remain in the group. It is likely, however, that selection will favor mechanisms that enable individuals to monitor these benefits with some degree of accuracy. Theoreticians have also pointed out that because cooperative associations may be maintained across several breeding seasons, the probability of inheritance can be more important than relatedness or group productivity in its influence on reproductive skew (Kokko and Johnstone, 1999; Kokko et al., 2002; Ragsdale, 1999). Crespi and Ragsdale (2000) argue that there may also be additional mechanisms that dominants employ to keep subordinates at the nest, such as reducing the benefits of solitary breeding through manipulation or punishment (see also Hamilton, 2004); these effects can be incorporated into existing models. For cases in which reproductive control is diffuse, as in large societies, Reeve and Jeanne (2003) have developed a "majority rules" transactional model that relaxes the assumption of complete control by a dominant female.

In this paper we add to these recent extensions of reproductive skew theory by allowing individuals to adjust their decisions about reproductive allocation and group membership over time. This modification is particularly relevant for cooperatively breeding species that reproduce over a prolonged period of time, rather than in one discrete bout. It should be recognized that models of reproductive skew already incorporate temporal changes in reproduction by predicting the summed total of reproduction across an entire breeding period. However, this approach also assumes that this overall future reproductive allocation is fixed at the beginning of the breeding period such that individuals cannot adjust their decisions at a later time when conditions have changed. Even though reproduction is a fixed quantity, however, these same skew models require that females continually assess reproductive allocation in order to prevent cheating. If this is true, then it is likely that females can also adjust their decisions about reproductive allocation and group membership.

To address this issue we develop an extended and more general form of the continuous skew model introduced by Reeve et al. (2000) to explain temporal changes in reproductive allocation in the social paper wasp, Polistes fuscatus. In particular, we focus on how changes in group and solitary productivity across a breeding season will affect the temporal allocation of reproduction and the stability of groups. We find that the conditions for group stability may be violated over portions of a breeding period, causing apparent premature departure by the subordinate individual (or a mid-season stability of a previously unstable group). This result is in sharp contrast to traditional skew models, which assume that a group remains together throughout the entire breeding period. We derive new, general solutions for reproductive skew that take into account changes in group membership, revealing that traditional (static) models may be overestimating or underestimating the magnitude of reproductive skew.

	The continuous model of reproductive skew

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
In the concessions model of reproductive transactions (Reeve and Ratnieks, 1993), dominants are assumed to have complete control over reproduction and group membership. Dominants then yield the minimum fraction of reproduction to a subordinate in order to keep the latter in the group (versus breeding solitarily). We reformulate the minimum subordinate share (required to keep a subordinate in the group) by assuming that the traditional static parameter values are now continuous functions of time:

(1)

The left-hand side of Equation 1 represents the future inclusive fitness payoff to a subordinate for breeding solitarily, and the right-hand side represents the inclusive fitness payoff for joining the nest of a solitary dominant. Here, S(t) is a continuous function describing future productivity (i.e., offspring number) of a solitary subordinate from time t forward. L(t) is a continuous function describing the future productivity (i.e., offspring number) of a lone dominant over the same time period. G(t) is a function describing the total future productivity (i.e., offspring number) of the subordinate and dominant. Finally, P_min(t) is a function describing the overall proportion of reproduction yielded to the subordinate over future time, beginning at time t.

In contrast to previous models of reproductive skew, we have used absolute measures of reproductive output rather than values relative to the output of a lone dominant female. One may easily translate back and forth with the old notation, however, by recognizing that dividing S(t), L(t), and G(t) by L(t) reveals the original X(t) = x, 1, and K(t) = k, respectively, of the original skew models (Reeve and Ratnieks, 1993). If we solve for P_min(t) in Equation 1, we get the expression

(2)

(for S(t) > r[G(t) – L(t)]), which is simply a continuous form of the traditional staying incentive (Reeve and Ratnieks, 1993; henceforth referred to as the "minimum subordinate share").

In contrast to the concessions approach, the restraint model (Johnstone and Cant, 1999b) considers the maximum amount of reproduction that the subordinate can take from the dominant before the dominant will eject the subordinate from the nest (i.e., dominant controls membership but not reproduction). We can use the continuous approach to set up the following condition under which a dominant should tolerate a subordinate:

(3)

The left-hand side of Equation 3 describes the inclusive fitness of the dominant if it ejects the subordinate from its nest. The right-hand side of the inequality describes the dominant's inclusive fitness if the subordinate remains in the nest. We can solve for P_max(t), which is the maximum proportion of reproduction that the subordinate can take from the dominant before being ejected:

(4)

(for G(t) – L(t) – rS(t) > 0). This is simply a continuous form of the traditional expression for the maximum subordinate share (Johnstone and Cant, 1999b).

The functions G(t), L(t), and S(t) are forward-looking in that they describe the expected values of present plus future reproduction at some time t. We assume that the departure of a subordinate female does not directly impact the survival of past brood (which depends on the value of t), such that females are making reproductive decisions based solely on future reproductive potential. Our attempts to relax this assumption resulted in a series of equations that are quite complex and beyond the scope of this paper. However, we have provided the general forms for these equations in Appendix A. Under these conditions, S(t) takes the form

(5)

where s(x) represents the expected instantaneous accumulation of brood at time x for a solitary subordinate. Likewise, we define G(t) as

(6)

where g(x) represents the expected instantaneous accumulation of brood at time x for the paired dominate plus subordinate. We define L(t) as

(7)

where l(x) represents the expected instantaneous accumulation of brood at time x for a solitary dominant. In Equations 5–7, t is standardized to range from 0 to 1 (proportion of total season). The productivity curves s(x), g(x), and l(x) represent the number of offspring produced at each time t multiplied by the survival probabilities of females and offspring at each time t (Figure 1A).

View larger version (24K): [in this window] [in a new window]   Figure 1 Dynamic productivity curves (a) for linear functions, where a = 1, b = 4, c = 1.5, m = 0, n = –1, and q = 1. (b) Depicts the corresponding maximum and minimum instantaneous subordinate share where r = .2. (c) Shows the functions for the future integrals of these curves (S(t), G(t), and L(t)) as well as the function for L(t) + S(t). The condition for stability, G(t) > L(t) + S(t), is violated at t* = 0.5 when we expect the subordinate to leave the nest. This corresponds with (d), where Pmax(t) < Pmin(t) after t = 0.5.

 
With these values of S(t), L(t), and G(t) we can calculate the cumulative future subordinate share P(t) at time t using Equations 2 and 4 (Figure 1D). Because P(t) describes the subordinate share over all future time (from t to 1) rather than at a particular instant, we term this the "cumulative" subordinate share. Because the functions s(x), l(x), and g(x) are necessarily greater than 0 (i.e., no negative reproductive output), the integrals S(t), L(t), and G(t) will always decrease as time increases (Figure 1C).

While traditional skew models are only applicable to t = 0 (i.e., the beginning point of the breeding period), our continuous skew model provides an extension of these models to any point in time. However, these values for cumulative skew do not necessarily predict the level of skew for a given point in time (actual value measured empirically). To get the instantaneous subordinate share, which we will call p(t), we can solve the following equation based on the idea that the present plus future subordinate share P(t) is just the instantaneous subordinate share averaged over all future times:

(8)

which is equivalent to

(9)

Note that in Equation 8 the instantaneous subordinate share (p(x)) is multiplied by the actual group productivity at that same time period (g(x)), standardized across all future group productivities (denominator). Equation 9 gives us a general equation for the subordinate's share of reproduction at any point in time during the breeding season. In Equations 8 and 9 we assume that p(t) does not take on values greater than 1 or less than 0 because the dominant can at most yield all of the reproduction to the subordinate and the subordinate can at most steal all of the reproduction. If p(t) exceeds 1 or goes below 0, we would be forced to invoke future reciprocity in order to restore the reproduction required to exceed these physical limits.

Applying Equation 9 to Equations 2 and 4 describing the cumulative subordinate share results in the following expressions for the instantaneous subordinate share:

(10)

(11)

The derivatives of Equations 5–7 giving us the equalities S'(t) = –s(t), L'(t) = –l(t), and G'(t) = –g(t) can be substituted into Equations 10 and 11 to get the following expressions:

(12)

(13)

Importantly, these equations for the instantaneous subordinate share Equations 12 and 13 are identical in form to the equations for the cumulative subordinate share Equations 2 and 4. The significance of this result is that females are expected to adjust instantaneous skew at time t in a way that closely tracks the productivity curves at time t (Figure 1B). In addition, the corresponding forms of the instantaneous and cumulative subordinate shares suggest that transactional models of reproductive skew are applicable to any timescale ranging from a point in time to a fraction of the breeding period to an entire breeding period. Appendix B describes the general solutions for this relationship between the cumulative and instantaneous subordinate share.

	Temporal changes in group membership

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
Traditional models of reproductive skew assume that if a group is stable at the beginning of the breeding period, it will remain stable throughout the breeding period. Previous models of reproductive skew have recognized that associations will form only if P_max(t) > P_min(t), where t = 0 (Johnstone, 2000; Reeve and Ratnieks, 1993). In other words, the association will be stable when the minimum subordinate's share of reproduction needed to keep the subordinate at the nest is less than the maximum subordinate's share that is tolerated by the dominant. Combining Equations 2 and 4 reveals the following condition for group stability (P_max(t) > P_min(t)) for time t:

(14)

Note that dividing each side by L(t) gives us the traditional condition for group stability according to both the concessions and restraint models (k > x + 1 or x < k – 1; Reeve and Keller, 2001; Reeve and Ratnieks, 1993).

This continuous skew model reveals that, under certain conditions, groups can become unstable for portions of the breeding period. As a result the model can predict a premature departure by the subordinate female. The actual timing of a subordinate female's departure will depend on the values of S(t), L(t), and G(t). However, it is clear from Equation 14 that increasing S(t) or L(t) (solitary productivity) relative to G(t) (group productivity) will tend to cause a higher likelihood of separation and an earlier time of departure by the subordinate. It is also to find solutions where a subordinate female is expected to join a group later in the breeding season. These switches in group membership can occur at several points during a breeding period depending on the shapes of the productivity curves and their integrals S(t), L(t), and G(t).

The general procedure for determining the timing of group formation and group breakup is the following: let G(t,u) be the cumulative group productivity function from time t until time u and similarly for S(t,u) and L(t,u). In addition, define = d{G(t,u) – [S(t,u) + L(t,u)]}/dt. To find the optimal subordinate strategy at time t, we work from the end of the breeding season backwards, much as in dynamic programming models (Mangel and Clark, 1988). First we solve for t in the equation

(15)

If there is such a t = t* between 0 and 1, then this t* indicates a switch point for the strategy (i.e., the time at which the subordinate joins or leaves the group). We can take the highest such t* and if > 0, at t = t*, then the group is expected to form at time t*. If < 0 at t = t*, then the group is expected to dissolve at time t*. If the group is expected to break up at t*, then we solve for t in the equation G(t,t*) + L(t*,1) + S(t*,1) = S(t,t*) + L(t,t*) + L(t*,1) + S(t*,1) and proceed as before. If the group is expected to form at t*, we solve for t in the equation G(t,t*) + G(t*,1) = S(t,t*) + L(t,t*) + G(t*,1) and proceed as before. Note that both of the latter two equations reduce to just

(16)

Moreover, any solution to G(t,t*) = S(t,t*) + L(t,t*) must also have been a solution to the initial equation G(t,1) = S(t,1) + L(t,1) because the initial equation is obtained by adding a constant (i.e., G(t*,1) = L(t*,1) + S(t*,1)) to both sides of the former. Thus, all of the switching times will be obtained when the initial Equation 15 is solved.

It is important to consider the effects that switch points have on the initial decisions of a subordinate to join a dominant or the decision of a dominant to accept a subordinate. If we consider the condition for group stability at any point in time, Equation 14, we find that future switch points do not change the decisions of females to form a group. Consider the situation in which Equation 14 is satisfied, but the group is expected to separate for a portion of the future breeding period. This means that the group is actually more likely to be stable at time t to t* because it is not constrained to stay together under unfavorable conditions (the implicit assumption of Equation 14 and the static model at t = 0). Therefore, the general solutions for switch points are independent of the number or presence of switch points during the breeding period. In mathematical terms, a group is stable from t to a switch point t* as long as G(t,t*) > L(t,t*) + S(t,t*), regardless of what happens at future switch points. Moreover, the subordinate's overall fraction of reproduction during this time depends only on r, G(t,t*), L(t,t*), and S(t,t*).

For example, let us consider the situation in which there is one switch point during the breeding period. If G(0) > S(0) + L(0), then we expect t* to represent the breaking up of a group that formed at the beginning of the breeding period. In this case the continuous model calculates the subordinate's share (P) from time 0 to t* only. In contrast, the static skew model is constrained to calculate P(0) from time 0 to 1, encompassing the time during which the group is unstable (t* to 1). It can be shown that the subordinate's overall share P_min from 0 to 1 will always equal the quantity P_min from 0 to t* multiplied by the fraction of total group productivity over this same period, G(0,t*)/G(0,1), plus the quantity P_min from t* to 1 multiplied by the group productivity over this same period G(t*,1)/G(0,1). As a result, P_min calculated from 0 to 1 will incorrectly inflate the subordinate's true share of reproduction (from 0 to t*) if P_min from t* to 1 is greater than P_min from 0 to t*. The latter will often be the case because the formula for P_min can be rearranged to equal (A – r)/(1 – r), where A = (S(t) + rL(t))/G(t). If A increases with time, then P_min from t* to 1 will indeed will be higher than P_min from 0 to t*, and the erroneous inflation of P_min will occur. Because the group breaks up at t*, it must be that G is decreasing relative to S + L as time advances, entailing that A is increasing as time advances (provided S/L is constant or decreasing with time, as seems likely for most real situations). As a result, the traditional concessions model incorrectly inflates the subordinate's share (P_min), and when 0 < t* < 1 the traditional restraint model correspondingly incorrectly deflates the subordinate's share (P_max).

To illustrate this general result within a specific context, we focus on the situation in which all three productivity curves are linear functions. Specifically, we define s(t) = a + mt and g(t) = b + nt and l(t) = c + qt. This is appropriate for species that experience constant values of s(t), g(t), and l(t) (i.e., the slopes n, q, and m equal 0) or for species that experience linearly increasing or diminishing productivity. We can solve for the specific conditions of group formation by integrating these linear curves and solving for G(t*,1) = S(t*,1) + L(t*,1), resulting in the expression

(17)

To summarize this outcome in a more compact form we let v = a + c – b and z = m + q – n. Then t* = –2v/z – 1 and occurs between 0 and 1 only if 1/2 < –v/z < 1. If v > 0 and z > 0, a group is never formed, and if v < 0 and z < 0, the group never dissolves. If v > 0 and z < 0, and 1/2 < –v/z < 1, the subordinate is solitary until t*, when it joins the dominant. If v < 0 and z > 0, and 1/2 < –v/z < 1, the subordinate is in the group from the beginning of the season, but leaves the group at t*.

It is clear that increasing the intercepts (a and c) of the solitary productivities s(t) and l(t) will increase t* while increasing the intercept (b) of the group productivity g(t) will decrease t*. Similarly, increasing the slopes (m and q) of s(t) and l(t) will increase t* while increasing the slope (n) of g(t) will decrease t*. When v < 0 and z > 0, the timing of subordinate departure is earlier when the initial group productivity decreases or the rate of change in group productivity (over time) decreases (Figure 2a). The timing of separation is also earlier as the initial solitary productivity increases or the rate of change in solitary productivity (over time) increases (Figure 2b). These predictions are similar to those of previous models that favor intraspecific brood parasitism (higher parental care asymmetry) over cooperative breeding as solitary productivity increases and/or group productivity decreases (Zink, 2000, 2001).

View larger version (37K): [in this window] [in a new window]   Figure 2 Timing of separation (a) as a function of initial group productivity (b) and rate of increase in group productivity (n), where a = 1, c = 1, m = 1, and q = 1. When t* = 0 the association will never form, and when t* = 1 the association will remain stable throughout the breeding period. (b) Plots t* versus initial solitary productivity (a) and rate of increase in solitary productivity (m) where b = 2, c = 1, n = 1, and q = 1.

 
When G(0) < S(0) + L(0), we expect t* to represent the time of formation by a group that was not stable at the beginning of the breeding period. In this case the continuous model calculates P from time t* to 1 only while the static skew model does not even attempt to calculate skew because the group is thought to never form. For the linear case we can derive general solutions for converting the traditional, static model skew predictions into a continuous context. When a group that is originally unstable becomes stable at 0 < t* < 1, we know that P_min(t*) = P_max(t*) as defined by the switch point. Using Equation 17 for t* we can find the general solution for the subordinate share across this region of stability (from time t* to 1):

(18)

Likewise, when a group is stable at time t = 0 but becomes unstable at t* < 1, we can find the general solution for the subordinate's share across this region of stability (0 to t*):

(19)

(20)

In total, Equations 18–20 give us the actual equations for calculating skew in the linear case when a group forms or separates during the breeding period. If the productivity curves are clearly nonlinear, one will need to solve for the switch points and then integrate Equation 8 over the time that the group is stable (as we have done here for the linear case). In the linear case, however, the traditional model will always overestimate P_min and underestimate P_max for a group that is stable at t = 0 but separates at t*. Importantly, the degree to which the subordinate's share is underestimated or overestimated increases with relatedness (Figure 3).

View larger version (11K): [in this window] [in a new window]   Figure 3 The predictions for the minimum subordinate share (Pmin) and the maximum subordinate share (Pmax) for the static versus dynamic models. Here, a = 1, m = 0, b = 4, n = –1, c = 1.5, q = 1. Under these conditions the group forms at t = 0 and breaks up at t* = 0.5. This figure and its associated parameter values correspond with Figure 1. The traditional, static model overestimates Pmin and underestimates Pmax. Note that the difference between the two models is exaggerated as relatedness increases.

 

	DISCUSSION

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
Our general solutions for continuous reproductive skew provide confidence in the ability of researchers to apply the transactional models during any portion of the breeding period because of the similarity in form between the instantaneous and cumulative subordinate share. In practice, researchers are often confronted by constraints (biological or logistical), which dictate that data are collected during a restricted portion of the entire breeding period. On a practical level, however, problems will arise if scientists measure reproductive skew and solitary versus group productivity at different timescales. Our model suggests that it is essential to measure these values at consistent time intervals across a time period that is the same for all parameters. Otherwise the data collected will not correspond with the predictions of the models. The most prudent approach for measuring continuous skew would be to take productivity data at regular time intervals for a specific population or species in order to construct the proper curves of l(t), s(t), and g(t). These data could be collected with offspring for subsequent genetic analysis of reproductive skew to test if theoretical dynamics match empirical measures of skew across these same time intervals.

This model also revealed that, in some cases, the transactional models of reproductive skew may provide incorrect estimates because they integrate skew over regions where the group is unstable. We find that, when a subordinate is expected to leave at 0 < t* < 1, the traditional models of reproductive skew tend to overestimate P_min and underestimate P_max. In insects, there are many examples of species where individuals leave the group in the middle of a prolonged course of communal breeding (Dunn and Richards, 2003; Eggert and Müller, 2000; Reeve et al., 1998b). In situations such as these we suggest that it may be more accurate to use the continuous model developed in this paper. In vertebrates for which subordinate females may spend a significant portion of the breeding period away from the group, such as lions (Packer et al., 2001) and meerkats (Clutton-Brock et al., 2001), a similar application of our continuous skew model may be warranted. It is possible that, when matched with our continuous model of skew theory, these insect and vertebrate systems may show more concordance between theory and data.

Because our continuous skew model revealed these situations in which subordinates depart and/or join a dominant within the same breeding season, we outlined a general procedure for determining the timing of all such behavioral switches. In this case predictions about reproductive skew would necessarily involve a summation across only the time periods in which the group is stable. The decision to remain in an association (by a subordinate) or to eject a subordinate (by a dominant) depends on the integrals of the instantaneous productivity curves over these regions of future group stability. Other work has also suggested that decisions about cooperation must take into account future cumulative direct and indirect fitness effects of the decision to cooperate or disperse (Kokko and Johnstone, 1999). In termites, for example, aspects of colony growth and dissolution make adaptive sense when such a forward-looking fitness approach is adopted (Korb and Schmidinger, 2004; Shellman-Reeve, 1997).

The analysis of linear productivity curves showed that the timing of subordinate departure is delayed with greater future group output or smaller future solitary output. This matches the predictions of previous models of brood parasitism and parental care asymmetry (Zink, 2000, 2001). For example, subordinates should terminate cooperation and pursue selfish options earlier with lower expected present plus future group output. This hypothesis can be tested in social wasps by analyzing the decisions of early-emerging females to stay at the natal nest and act as workers versus disperse to enter early diapause (or pursue other selfish options), as recently documented in P. fuscatus (Reeve et al., 1998b). Early females are most likely to stay on the nest as workers when they are among the first to emerge, which supports the model's theoretical prediction because worker contributions to colony growth likely decline with an increased number of staying workers (Reeve, 1991). Moreover, the mean nest tenure of an early P. fuscatus female (i.e., the duration of time the female stays on the nest before disappearing) is significantly less for smaller nests and less mature nests (Reeve et al., 1998b), again as predicted by the continuous model.

The actual shapes of the solitary and group productivity curves may depend on other variables such as internal states (age, experience, survival) and external states (seasonality, competition, parasitism, and predation), all of which can vary over time. This leaves room for further expansion of the model. For example, it would be interesting to incorporate learning into the model, such that a subordinate's probability of solitary success increases the longer she is with and gains experience by helping a dominant. The explicit incorporation of survival probabilities of adults and brood and inheritance (via Equations 5–7), as suggested by Ragsdale (1999) and Kokko and Johnstone (1999), would also add realism to the model for many biological systems. It is very likely that our continuous model will also apply to species with multiple breeding bouts that occur over discrete time intervals (rather than over continuous time). For example, it would be possible to combine our model with a "binning" method that averages the instantaneous values of productivity (and predicted skew) over short time intervals while retaining cumulative, future measures of productivity (which ultimately determine group membership). This would extend the model to an even greater array of animals, allowing for even broader tests of dynamic skew.

Johnstone (2000), Reeve and Emlen (2000), and Reeve and Keller (2001) point out that it is important to recognize that reproductive skew models are powerful tools for making quantitative predictions about social phenomena other than skew itself. Examples include infanticide (Johnstone and Cant, 1999a) and group size (Reeve and Emlen, 2000), and in this paper we add the duration of cooperation. One can imagine extending this dynamic approach to other models of reproductive skew. We have focused exclusively on two-person transactional models, but this dynamic framework could be used to look at other types of skew models. For example, this model could be extended to three individuals (Johnstone et al., 1999) or an N-person model if the assumption of equivalence of subordinates can be made (Reeve and Emlen, 2000). Hopefully, the ideas presented here will stimulate and refine future research on time-dependent changes in reproductive skew, as well as lead to new predictions about the duration of cooperative associations and the frequency of aggression.

	APPENDIX A

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
General forms of productivity curves
When adapting the traditional skew models for continuous time, we should view the functions G(t), L(t), and S(t) as forward-looking, that is as describing expected values of present plus future quantities at some time t. The most general forms of these parameters will take into account the survival probabilities for the subordinate, the dominant, and their brood. For example, the time-dependent parameter describing the subordinate's expected absolute solitary success S(t) if the subordinate leaves at time t will be equal to

(A1)

where p(x) is the proportion of brood that the subordinate was allocated at time x in the group, g(x) is the total brood produced by the group at time x, (x,t) is the probability that an offspring produced in the group at time x and living at time t will live until the end of the breeding period (t = 1), (x) is the probability that brood produced solitarily at time x will live until the end of the breeding period (t = 1), h(x) is the number of brood produced solitarily at time x, and s(x) is the probability that a solitary subordinate will live until time x. This expression accounts for the effect of the subordinate's leaving decision on both past and future brood produced (because leaving the group may affect the survival of past brood produced while in the group).

The expected absolute group success if the subordinate stays will be equal to

(A2)

where g(x) is the total brood produced by the group at time x, (x,t) is the probability that an offspring produced in the group at time x and living at time t will live until the end of the breeding (nesting) period (t = 1), and the integrand in the second integral is the expected value of product of the future survival of brood produced at time x and the number of brood produced at time x in the group (which takes into account mortality of the dominant and subordinate).

Finally, the expected output of a lone dominant if the subordinate leaves at time t equals

(A3)

where p(x) is the proportion of brood that the subordinate was allocated at time x in the group, g(x) is the total brood produced by the group at time x, (x,t) is the probability that an offspring produced in the group at time x and living at time t will live until the end of the breeding period (t = 1), '(x) is the probability that brood produced by a lone dominant will live until the end of the breeding period (t = 1), h'(x) is the number of brood produced by a lone dominant at time x, and s'(x) is the probability that a lone dominant will live until time x.

	APPENDIX B

TOP ABSTRACT INTRODUCTION The continuous model of... Temporal changes in group... DISCUSSION APPENDIX A APPENDIX B REFERENCES

 
General relationships between the cumulative and instantaneous subordinate's share
The expressions for p_min(t) and p_max(t) from Equations 12 and 13 can be differentiated with respect to time t to reveal the general conditions under which they will increase or decrease over time. p_min(t) will decrease over time if the proportional rate of change in instantaneous group productivity g'(t)/g(t) is greater than the ratio [rl'(t) + s'(t)]/[rl(t) + s(t)], with the latter seen as the proportional rate of change in the combined solitary instantaneous productivities weighted by the subordinate's relatedness. On the other hand, p_max(t) will increase over time if the proportional rate of change in instantaneous group productivity g'(t)/g(t) is greater than the ratio [l'(t) + rs'(t)]/[l(t) + rs(t)], with the latter seen as the proportional rate of change in the combined instantaneous solitary productivities weighted by the dominant's relatedness. Thus, relatively high rates of increase in instantaneous group productivity will tend to decrease p_min(t) but increase p_max(t).

We can use Equations 8 and 9 to obtain some general properties of instantaneous subordinate shares. Differentiating both sides of Equation 8 with respect to the time t, we obtain

(B1)

Thus, in periods during which the cumulative (present plus future) subordinate share P(t) is decreasing (i.e., the left-hand side of Equation B1 is negative), the instantaneous subordinate share must always exceed the cumulative incentive. Conversely, in periods during which the cumulative subordinate share P(t) is increasing (i.e., the left-hand side of Equation B1 is positive), then the instantaneous subordinate share must always be less than the cumulative subordinate share. Both incentives are equal if there is no time-dependent change in the cumulative subordinate share. Equation B1 can be rearranged to yield an informative expression for the gap between the instantaneous and cumulative subordinate shares:

(B2)

According to Equation B2, the gap between the instantaneous subordinate share and the cumulative subordinate share is directly proportional to the rate at which the cumulative subordinate share changes. In addition, the gap will tend to narrow as the time t increases (for a fixed P(t)/t), and it must close up completely at t = 1.

Differentiating both sides of Equation 9 with respect to the time t, we obtain a useful expression for the rate p(t)/t at which the instantaneous subordinate share changes over small time intervals:

(B3)

From Equation A3 we see that the instantaneous subordinate share must increase with time if g'(t) 0, P(t)/t > 0, and ²P(t)/t² 0, and it must decrease with time if g'(t) > 0, P(t)/t < 0, and ²P(t)/t² 0. For other situations, whether or not the instantaneous subordinate share increases or decreases over time will depend on the time and on the numerical values of the partial derivatives on the right-hand side of Equation A3. For example, the instantaneous subordinate share can increase with time even if the cumulative subordinate share decreases with time if ²P(t)/t² is negative and of sufficiently large magnitude. The instantaneous subordinate share will decrease with time even if the cumulative subordinate share increases with time if ²P(t)/t² is positive and of sufficiently large magnitude. The latter "counter-current" effects will be most likely in early time periods (i.e., t far from 1).

	ACKNOWLEDGEMENTS

 
We would like to thank M. Elgar, S. Emlen, R. Johnstone, H. Kokko, J. Strassmann, S. West, and three anonymous reviewers for comments on earlier versions of this manuscript. A.G.Z. was supported by a National Science Foundation Graduate Research Fellowship during the preparation of this manuscript. H.K.R. was supported by a Presidential Early Career Award.

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