Sibling rivalry in a variable environment

l-HEORETICAL

41, 135-160 (1992)

POPULATION

BIOLOGY

Sibling

Rivalry in a Variable

L. SCOTT FORBES’

AND

Environment

RONALD C. YDENBERG

Behavioural Ecology Research Group, Department of Biological Sciences, Simon Fraser University, Burnaby, British Columbia, Canada WA IS6 Received June 8, 1990 Unpredictable variation in food availability is widely thought to be a key factor in the evolution of avian brood reduction. We develop here two models to examine how interannual variability in parental provisioning, M (a function of the level of food in the environment), and the ability to assess M affect the brood reduction decisions of a senior sib in a brood of two chicks. Obligate execution is always favored by natural selection if M never exceeds mca,r, the minimum level of M that, from the senior sib’s perspective, is required to sustain both chicks. Obligate execution can be favored in a variable environment even if M exceeds mcsir in most years because of the asymmetry in costs and benefits of levels of M below and above mcs,r. Execution will never be favored if A4 always exceeds mcsir. If M varies around III,-s,r, facultative execution will bc favored when M can be assessed accurately, when the cost of assessment is low, and when the cost of an error is small. These costs arise from the possibility, due to sampling effects, that the senior sib arrives at an incorrect decision, killing its sib in good years, or more importantly, allowing it to live in a bad year. A senior sib must allow its junior sib to remain alive in order to derive information about the state of M (e.g., from the pattern of parental provisioning), but has the disadvantage that the senior sib must share food with the possible victim, which yields no payoff if the eventual verdict is execution. The results of a dynamic programming model suggest that early execution is favored, as the senior sib has the most to gain in future provisioning, and uncertainty about M is greatest. Increasing total provisioning through extension 0 1992 Academic Press, Inc. of the nestling period favors clemency.

INTRODUCTION

Success in sibling competition is a strong determinant of subsequent fitness for a nestling bird. Winners gain access to more food and enjoy enhanced prospects for growth and survival, whereas losers are burdened with food shortfalls, often with lethal consequences(Lack, 1947; Ricklefs, 1965; Procter, 1975; Gargett, 1978; O’Connor, 1978; Ryden and Bengtsson, 1980; Mock, 1984; Fujioka, 1985a,b; Greig-Smith, 1985; Roskaft and Slags/old, 1985; Bortolotti, 1986a,b; Cash and Evans, 1986; Hebert and Barclay, 1986; Slagsvold, 1986; Mock et al., 1987; Anderson, 1989; Drummond and Garcia Chavelas, 1989; Lessels and Avery, 1989; Parker et al., ’ Present address: Department

of Zoology,

University

of Oklahoma,

Norman, OK 73019.

135 0040-5809/92 $3.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

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1989; Poole, 1989; Lamey, 1990; Magrath, 1990; Amundsen and Slagsvold, 1991; Lamey and Mock, 1991). Sibling competition for food occurs in an array of forms. Among many passerine birds, competitive begging occurs without the use of overt aggression (Ryden and Bengtsson, 1980; Stamps et al., 1989; Lamey and Mock, 1991), but at the opposite extreme, sibling rivalry is a brutal spectacle. First-hatched black eagle (Aquila oerreaxi) chicks, for example, bludgeon junior nestmates into an early death (Gargett, 1978). Although the form of such sibling competition varies considerably, a common thread exists: the decisions of stronger sibs affect the fitness of their weaker, and usually younger, nestmates. Dominant siblings may deny junior sibs accessto food directly by aggressive means (Ploger and Mock, 1986; Mock et al., 1987) or indirectly simply by consuming more of the delivered food, leaving a smaller residual share for other nestlings (Forbes, 1991a; Lamey and Mock, 1991). Thus stronger sibs may often be able to effect the death of junior sibs if they so choose. Alternatively, stronger sibs may retain weaker sibs, necessitating that they share some of the total food. For the purpose of our discussion, we shall define these tactics as execution and clemency. Three possible strategies arise from these two behavioral tactics: obligate execution, facultative execution, and obligate clemency. The behavioral decisions of the offspring are integral to the brood reduction process, as the strategic decisions of the offspring, particularly the stronger sibs, will determine who lives and who dies. Unpredictable variation in food availability is widely thought to be a key factor in the evolution of brood reduction strategies (O’Connor, 1978; Mock, 1984; Magrath, 1989), but its role has rarely been considered in a formal theoretical structure (but see Temme and Charnov, 1987). Here we consider the role of the offspring in brood reduction decisions. An obvious advantage of facultative execution over the two obligate strategies is that it allows a nestling to incorporate information about the state of the environment into the decision to perform execution. Ideally, an individual would estimate resource availability, and allow a junior sib(s) to survive when food is abundant, but eliminate it when food is short. Obligate execution and clemency both preclude the use of such information. While gathering information about the state of the environment that will determine execution decisions, a senior sib must allow a junior sib to remain alive. During this period of clemency a senior sib invests time and energy in aggression, shares some of the total food, and may risk injury or a future dominance reversal in doing so (Gargett, 1978; Hahn, 1981; Mock, 1984; Fujioka, 1985a,b; Ploger and Mock, 1986). Hence there are costs and benefits to clemency: Forbes (1991b) referred to this tradeoff of future food for enhanced inclusive fitness as the “burgers or brothers principle.”

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Here we use two simple models to explore the effects of uncertainty about parental provisioning on the behavioral decisions of stronger siblings. The first is a simple analytical model that examines the effects of interannual variability in the abundance of food on the fitness payoffs for clemency and execution. The second model uses the techniques of stochastic dynamic programming to assesswithin the nestling period (i) if and when a senior sib should execute a junior sib, and (ii) whether a senior sib can use information about the current state of the environment to make better brood reduction decisions.

A MODEL OF OBLIGATE STRATEGIES We assume that the fitness (survival to breeding) of an offspring is a function, P, of the resources, m, it consumes.We loosely follow the protocol and terminology of MacNair and Parker (1979) and Parker et al. (1989). It is convenient here to think of m as the total provisioning obtained by an offspring over the entire period of parental care, so the sum of m to all offspring is M. We begin by defining the environmental yield of resources in a given year as a discrete random variable, Y. We assume that within a year, Y is constant. We also assume that the resources gathered by the parents and delivered to the brood over the period of parental care (M) is proportional to Y. The function f(m) is zero up to an arbitrary minimum, mMIN, above which it deceleratesto some asymptote such that f(m) shows diminishing returns above mMIN. Consider two sibs, c(and 8, together in a nest. GLis older and larger than b, is dominant, and can take any proportion, p, of parental resources, M, it desires. Now consider the fate of a rare mutant dominant allele that causes c( to take proportion p’ of M. Assuming that CLand /I are full sibs, the probability that p also carries the mutant allele is a. The replication rate of the mutant allele, A, is thus rl =fqMp’)

+ $tp{ (1 - p’)M}.

(1)

From the perspective of TV,the optimal value p* is found by setting iQ./dp’ = 0. At the optimum, f’(Mp*)

= - @((l

- p*)M}.

(2)

That is, tx should take an additional unit of M (i.e., p* increases) when the benefit it derives is greater than half the benefit B would derive from the same unit of M, in accordance with Hamilton’s (1964) inclusive fitness result.

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We now define an explicit function for P(m), k(m -mMIN) f(m)=

if m >mMIN (3)

m

I

0

otherwise,

where k, a positive constant, is the asymptotic value of f(m). (f(m) converges to k as m -+ co). For simplicity we shall set both k and mMIN equal to unity so that (3) simplifies to if m>l (4) otherwise. Substituting Eq. (4) into Eq. (3) gives

Differentiating with respect to p’ and setting dA/dp’ = 0 gives (6)

This simplifies to 4p* - 2 = (p*)*,

(7)

so p* = 0.59. Thus 1 - p*, the share of A4 to /I, is 0.41. This value is independent of M, except as noted in Eq. (4). When Mp ( =m) is less than 1, f(m)=O. /I’ th erefore has no chance of surviving to breed until M= 2.4 (i.e., m = A4p = 0.41 * 2.4 = 1). However, as shown in Fig. 1(top), u’s inclusive fitness is not improved by permitting /I to live until M= 3.8 ( =m,,,,). Therefore /I does not receive any resources until M exceeds 3.8 at which point its allocation becomes 0.41. Fitness in a Temporally

Varying Environment

In temporally varying environments, variation in reproductive sucess should be considered in the measure of fitness. If variation in reproductive successresults from stochastic temporal fluctuations in the environment on the same scale as the,generation length of the organism and such variation is population-wide, the strategy favored by natural selection is that

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FREQ. 004

M FIG. 1. Top. Fitness in the senior sib, I, when following a policy of either execution of clemency at differing levels of parental provisioning, M. If M> mCRIT then clemency is favored. If M
maximizing geometric mean fitness (Dempster, 1955; Cohen, 1966; Gillespie, 1977; Seger and Brockmann, 1987). In such stochastic environments, more variable strategies are more vulnerable to extinction all else being equal, and hence bet-hedging (sensu Seger and Brockmann, 1987) may be favored by natural selection. That is, the arithmetic mean success of a strategy may be sacrificed for the sake of reduced reproductive variance (Slatkin, 1974; Gillespie, 1977; Real, 1980; Rubenstein, 1982; Boyce and Perrins, 1987; Seger and Brockmann, 1987; Boyce, 1988). As noted above, the geometric mean measure of fitness makes restrictive assumptions about the nature of temporal and spatial variation. We suspect these will not strictly hold. For example, year-to-year variation in reproductive successwill be less important to long-lived organisms since lifetime reproductive success will be the product of many reproductive attempts--e.g., a failure to produce offspring in one year is not

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catastrophic. For such animals, longer term processes, such as slowly changing climatic or marine conditions, will be more relevant. Similarly, the importance of temporal variation in reproductive success also is diminished if the effectsare not population-wide. We expect, therefore, that the arithmetic and geometric means will bracket the true best measure of fitness (see also Yoshimura and Shields, 1987). As such, we shall consider both measures of fitness. The arithmetic mean fitness of obligate execution, I,, and clemency, L, are

EC&] = f y WW > v,tlN( E[n,]= 5 M-L;1 I Y-p)-1 2 Ml-P) ( mMlN

>

Pr(M),

(9)

where E denotes mathematical expectation, and Pr(M) is the probability density distribution of M (M> m MIN). Similarly, the geometric mean fitness of obligate execution, E[ln A,], and clemency, E[ln J.,] are E[ln A,] = f In mMlN E[lni,]

(10)

= f In VdlN

Pr(M).

(11)

Obligate execution will be favored over obligate clemency when

EC&l ‘EC&l,

(12)

when the arithmetic mean is the best measure of fitness, and E[ln lE] > E[ln &.],

(13)

when the geometric mean is the best measure of fitness. Consider the situation now where 0: possessesperfect information about the current state of M. Execution will be favored in those years when M < mCRITywhereas c1should allow /I to live when M > mCRIT(Fig. 1, top). It is clear that if M never exceedsmCRIT,obligate execution will be favored since E[&] will always exceed E[&] (and E[ln A,] will always exceed E[ln A,]). Conversely, if M never falls below mCRITexecution will never be favored. It seems unlikely though that either case will ever strictly hold. Rather, variability in A4 will result in M> mcRIT in some years and in others. The threshold for a policy of obligate execution vs MC mCRIT clemency, MT., is a function not only of the mean level of M, but of the

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variance in M as well, MT will be greater than mCRIT when M varies from year to year (i.e., the senior sib is more disposed to execution). The reason for this is simple. At M > mCRIT, there are diminishing inclusive fitness returns to the senior sib for retaining its junior sib (Fig. 2). As M falls from mCRIT to 1.7, however, there is an accelerating cost to the senior sib for retaining its junior sib (Fig. 2). As M falls from 1.7 to mMIN, the cost of clemency falls to zero (Fig. 2). With the proviso that if M never falls below 1.7, MT will increase as variability in M never falls below 1.7, MT will increase as variability in M increases (this follows from Jensen’sinequality (Feller, 1966; 151)). This analogous to the concept of risk aversion in foraging theory (Stephens and Krebs, 1986). We explored the effect of increasing variability in M on the arithmetic mean payoffs for obligate execution and clemency using a simple simulation model. The mean level of M was set at 3.8 ( =mCRIT), and M was allowed to vary above and below M according to a Gaussian distribution simulated on a computer; examples of the frequency distribution of M at different levels of variation are illustrated in Fig. 1 (bottom). When there is no variability in M (C = 0, Fig. 3), the arithmetic mean payoffs for execution and clemency are identical (Fig. 3). As D increases,however, the payoff for execution initially exceedsthat for clemency, but at very high levels of (r, the payoffs for the two strategies once again converge. This convergence occurs as more extreme values of M (particularly M < 1.7) become more frequent; in this region the cost of clemency diminishes, but the benefit of clemency at very high levels of M( > 5.9) continues to increase. That is,

FIG. 2. Costs and benefits of a policy of clemency at varying levels of M. At M> mCRLT, the senior sib derives benefits in inclusive fitness (& - 1, > 0) by retaining its junior sib. At there is a fitness cost (I,-& ~0) for retaining the junior sib. At mMIN < M < IIICRITT no chicks survive whether or not execution (1, - 1, = 0) occurs. MimMIN,

653!41/2-3

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FORBESAND YDENBERG

0 65

0.60 00

05

10

15

20

25

FIG. 3. Results of a Monte Carlo simulation comparing the arithmetic mean successof execution and clemency when M varies. In all simulations the mean value of M was 3.8 ( =~+a,~). The frequency distribution of M for c = 0.5 and (r = 1.5 is shown in Fig. 1, bottom. When there was no variance in M (u = 0), the mean litness, J.?,of execution and clemency was the same. As variability in h4 increases (0 < rr < 1.5), the mean fitness of execution initially exceeds that of clemency, but at very high levels of variability in M (e.g., o > LS), the fitness of the two strategies converges.

obligate execution is favored when variation in M is modest, but clemency is increasingly favored at very high levels of variation in M. How variable M is in nature remains an open and important question. How Frequent Must Bad Years be to Favor Obligate Execution? Obligate siblicide occurs in a variety of eagles, boobies, and pelicans (e.g., Gargett, 1978; Cash and Evans, 1986; Drummond 1989). All lay two eggs, hatching is asynchronous, and usually the elder sib kills its younger sib soon after hatching. Paradoxically, obligate siblicide seemsto be independent of prevailing food at the time of the junior sibling’s demise. This is graphically illustrated in Gargett’s (1978) description of a four-day-old black eagle chick, weighing 163 g, pummelling its newly hatched (and doomed) sib while more than 5 kg of prey remained uneaten in the nest. Stinson (1979) proposed that “pending competition” can explain this behavior. Simply, food might not be limiting at the time of the junior sib’s demise, but may be likely to become so (see also Forbes, 1991b). Here we shall show that obligate brood reduction may even be favored when sutficient food exists in most years to favor clemency. As we have shown above, the frequency distribution of year types affects the fitness payoffs for execution and clemency. We can now ask how frequent good years need be for obligate clemency to be favored. For simplicity, we assume’that there are only two types of years, good and bad, and that the amount of food parents deliver in good and bad years is Mo

SIBLING RIVALRY IN VARIABLE ENVIRONMENT

143

and MB, respectively, subject to the constraint that (Mo + MB)/2 = 3.8 = mcR,,-i.e., the average level of food across years is exactly that where the fitness payoffs for clemency and execution are identical. In a good year, there is sufficient food, from CI’Sperspective, to favor clemency; conversely, in a bad year u favors execution. Let p and 1 - p be the proportion of good and bad years, respectively. At what value of p are & and AE (and ln[&] and ln[n,]) equivalent? The fitness payoffs for execution and clemency in good and bad years, &o, &, &-o, and lcB, respectively, will of course depend upon M, and MB. We define here the deviation from mcRIT, AM. That is, if M in a good year is one unit above mCRIT(M, = MCRIT+ 1= 4.8; AM = l), then M in a bad year will be one unit below mCRIT (MFI = ~CR1-r- 1 = 2.8). If we define fitness as an arithmetic mean, the proportion of good years, p*, where the payoffs for execution and clemency are equivalent will be P*l,,+(l-P*)~EB=P*~CG+(l-P*)~CB.

(14)

Using a geometric mean definition of fitness similarly gives

P* lnCh1 + (1 -p*) WEBI =P* W&G1 + (1 -p*) ~nC&J. (15) We solved for p* numerically and the results are presented in Fig. 4. Three patterns emerge. First, p* increases as AM increases.The reason for this is straightforward. As deviations above and below mCRITincrease, the cost of clemency in a bad year exceedsthe benefit of clemency in a good year by an increasing margin (see Fig. 2).

09

p”

07

05 00

05

10

15

2.0

AM FIG. 4. Relationship between the proportion of good years, p*, at which the arithmetic (open squares) and geometric mean (solid squares) payoffs for execution and clemency are equivalent at different values of dM (deviation in M above mcuT in a good year and deviation below ntca,r in a bad year). Obligate execution may be favored even when sufftcient food, from a’s perspective, exists to rear both chicks in most years.

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Second, p* is higher when fitness is defined as a geometric mean than as an arithmetic mean. Once again, the reason is simple. A geometric mean definition of fitness discounts variance, and variance in reproductive success for clemency rises more rapidly than for execution. Thus, as we have argued above, a temporally varying environment should favor execution over clemency. Third, obligate execution can be favored over clemency even when the proportion of good years is very high. Stinson (1979) suggestedthat early, preemptive siblicide might be favored if food was likely to become limiting later in the nestling period. Our results suggest that food limitation need not even be a likely event: ail that is necessary is that the long-run benefit of execution exceed the long-run benefit of clemency, a condition that can be satisfied even if good years are frequent.

FACULTATIVE EXECUTION

Estimating Resource Availability So far we have considered policies of obligate execution and clemency. What about facultative execution? If the senior sib has perfect information about M, it would seem that facultative execution is the best policy. Information, however, will generally be imperfect, and the problem is to estimate whether M is greater or smaller than mCRIT.Four outcomes are possible: (i) execution occurs when there is sufficient food for only one chick; (ii) execution does not occur when there is sufficient food for two chicks; (iii) execution does not occur when there is sufficient food for only one; (iv) execution occurs when there is sufficient food for two. Outcomes (i) and (ii) are correct decisions, whereas (iii) and (iv) are not. Presumably a chick assessesthe state of the environment through the pattern of food deliveries it receives at the nest. Consider the situation where prey arrive at the nest according to a binomial process with mean Y and probability distribution Pr(n 1t, Y) = t Y( 1 - Y)r-n, 0n

(16)

where n is the number of prey delivered in t sampling intervals. We have assumed that within a season Y is constant, but between years it varies as a discrete random variable. That is, when food is scarce (Y is low), the mean rate of food delivery (M) is also low, such that the probability distribution of Y is equivalent to the probability distribution of M (i.e., Pr[ Y= Yj] = Pr[M= Ml]).

SIBLING RIVALRY

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Now consider the situation where c( has some prior expectation of the rate of prey delivery with mean Y’ and probability distribution yi = Pr( Y’ = !P;) (the prior probability distribution). Bayes’ theorem provides a simple and logical method for updating prior probabilities with information derived from sampling (Mange1 and Clark, 1988), stating that for any i, the conditional probability of Uj given V is defined by Pr(UiI V)=

Pr( Ui) Pr( VI Ui) Pr(V) ’

Now let Uj be the event Y’ = Y,! and V be the event n prey arrivals over t sampling intervals. Then,

Pr( Vi) = yi

(18)

Pr( V) = 1 Pr(n, t, ‘Iv/) Pr( Y” = Y;)

(19)

Pr( 1/l Vi) = Pr(n, t, Y/).

(20)

Substituting these into Eq. (17) gives Yj Prh t, yvl) Pr(y’I ’ n’ t, = xi pr(n, t, y;) pr( y’ = y;)’

(21)

Equation (21) describes the probability distribution of the updated arrival rate in terms of the sampling data n and t, and the prior information, Y’ and yi. This posterior distribution has an important property: as sampling continues (i.e., n increases and the variance of the sample declines), the estimate becomesweighted more heavily toward the sampling information. A simple example illustrates this point. Consider the situation where there are only two types of years, good and bad, denoted by the subscripts G and B, respectively. In a good year sufficient food is delivered to the nest (from CI’Sperspective) for two chicks; in a bad year, there is only sufficient food for one chick. Good and bad years occur with equal frequency (Pr( yG) = Pr( yB) = i), and in a good year the probability, Y&, of a food item being delivered during any sampling period, t, is 0.7; in a bad year the corresponding probability YA is 0.3. u must determine from the rate of food delivery whether it is a good or bad food year. At t = 0 its expectation of a good food year is 4, but as sampling periods pass and prey deliveries occur, a’s expectation of a good year is continually updated (Table I). If for example a prey delivery occurs in the first period, CI’Sexpectation of a good year changes from 0.5 to 0.7; the corresponding expectation of a bad year changes from 0.5 to 0.3. If two prey deliveries occur in 2 days, the

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I

Posterior Probabilities that Year i is a Good Year after Sampling for t Periods with n deliveries (Pr( yGln, t))

j

0

1

2

3

4

5

0

0.500

0.300 0.700

0.155 0.500 0.845 -

0.073 0.300 0.700 0.927

0.033 0.155 0.500 0.845 0.967 -

0.014 0.073 0.300 0.700 0.927 0.986

1 2 3 4 5

-

Note. Probability that year i is a bad year is simply I-(probability of a good year). Prior probabilities of good and bad years are 0.5.

corresponding probability is 0.845. Conversely, if no deliveries occur in the first 2 days, the estimated probability of a good year is 0.155. The Costs and Benefits of Clemency Using an algorithm like this, the senior sib may continually evaluate the state of the environment (in this example, good year or bad year). In deciding whether to eliminate the junior sib, or to extend the period of clemency, the senior sib must take into account its estimate of the type of year, the fitness payoff of each of the possible actions, and the costs should the chosen action prove to be the wrong one. In allowing the junior sib to live, the senior sib must trade off the additional costs of possibly having to eliminate a larger and stronger victim against the benefits of more information about the environment. However, there is also a further consideration. Execution forecloses future options (since the victim cannot be revived if the year turns out to be good after all) but clemency leaves open the option of later execution. A DYNAMIC MODEL

A full analysis of the duration of the clemency period requires a dynamic approach, and here we develop a simple version of such a model using the techniques of stochastic dynamic programming (Mange1 and Clark, 1988). We present an outline of the model here: further details are found in Appendix 1. The general premises of the model are the same as for the analytical model described above. Two sibs, LYand /?, occur together in a nest where they feed and grow over a finite nestling period of T days. The

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ENVIRONMENT

fitness of a chick is a function of its mass at fledging. How much food a chick eats, and hence whether or not execution is favored, depends in part upon whether it is a good or bad food year. Good and bad years occur with equal frequency, and a chick’s prior expectation of a good, y,, or bad, y,, food year is 0.5. Food is delivered to the chicks according to a

0.8

04

00

Pr(N=Ni)

0 00

010

0.05

4

0 00 0

10

20

30

40

50

FIG. 5. Top. Fitness payoff for a (A) from execution and clemency at different levels of N (the total number of food items delivered over the nestling period). Clemency is favored at N > 38; execution is favored at 10 < N < 38. Middle. The binomial probability distribution [Pr(N= Ni)] of N in a bad year, where the nesting period is 50 days in length, and the probability of a food item being delivered on any given day is 0.3. Bottom. The binomial probability distribution [Pr(N= Ni)] of N in a good year, where the nestling period is 50 days in length, and the probability of a food item being delivered on any given day is 0.7.

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binomial random process, with no more than one food item, which the sibs can share, being delivered each day. If clemency is favored, the chicks share food items with 59% going to c1and 41% going to p in accordance with Eq. (7). The probabilities of a food item being delivered on a given day in good, Y,, and bad years. Yy,, are 0.7 and 0.3, respectively. Thus the pattern of food deliveries to the nest can provide the senior sib, CI,with information about whether it is a good or bad year. Such information is directly relevant to b’s decision of whether to kill its junior sib, /I, as in a good year there is a much higher probability of sufficient food being delivered to the nest to favor clemency than in a bad year (Fig. 5). We assume in the model that a updates its expectation of a good food year according to a Bayesian process as described in Eq. (21) and Table I (e.g., if live food items are delivered in the first 5 days of the nestling period, a becomes quite conlident that it is a good food year). The problem for a is twofold. First, after n food items have been delivered by Day t, u must assessthe fitness payoffs (~co, mCB,e&o, mEB, where w denotes fitness and the subscripts E, C, G, and B denote execution and clemency, good year and bad year, respectively) derived from clemency and execution in good and bad years. Of course if execution has already occurred earlier in the nestling period, there is no further decision to be made. Second, u must assessthe probability that the current year is either good, Pr(G I n, t), or bad, Pr(BI n, t) (where Pr(BI n, t) = 1 - Pr(GI n, t)). Clemency thus will be favored over execution whenever oCG Pr(G 1n, t) + oCB Pr(B 1n, t) > oEG Pr(G I n, t) + oEB Pr(B I n, t).

(22)

That is, clemency is favored when the payoffs for clemency in good and bad years weighted by the Bayesian probabilities of good and bad years exceed the similarly weighted payoffs for execution. Results of the Dynamic Model

We begin by illustrating the optimal behavior for CIif it knows (i.e., it has perfect information about the year type) if it is a good (Fig. 6, top) or bad year (Fig. 6, bottom). We have fixed the nestling period at 50 days, and the optimal decision, execution or clemency, is presented as a function of n and t. Three patterns emerge.First, on any given day clemency is favored only at higher levels of 12.Fitness is a function of the total quantity of provisioning received over the entire nestling period and early food shortages (due to a run of bad luck) mean that it is less likely that sufftcient food will be delivered by the end of the nestling period to favor clemency (e.g., if no food items are delivered on the first 5 days of the nestling period, the maximum possible number of food items that could be delivered over the nestling period declines from 50 to 45). Second, clemency is more likely in a good year. This result is simply due

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GOOD YEAR

BAYESIAN 40

n

BAD YEAR

10

40

20

30

40

t FIG. 6. Top. Optimal strategy for a in a good year (execution [dotted region] or clemency [cross-hatched region]) as a function of the number of food deliveries, n, on a given day of the nestling period, r (n < t), which is 50 days long. Black area indicates the region where the level of provisioning is insullicient to sustain even one chick and hence a’s fitness is zero. Clemency is favored at higher levels of n. Middle. Optimal strategy for a (execution or clemency) using Bayesian updating to estimate the year type (good or bad) as a function of n and t. Note that the execution region is intermediate between those in good and bad years. Bottom. Optimal strategy for a in a bad year (execution or clemency) as a function of n and t. Note that execution is more likely than in a good year.

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to the higher mean rate of food delivery in a good year than in a bad year. For example, in a good year at the beginning of Day 40, a chick expects on average 7 more deliveries (0.7 deliveries/day x 10 days) whereas in a bad year a chick expects only three more deliveries (0.3 deliveries/day x 10 days). Third, execution is more likely early in the nestling period. The longer GI delays execution of 8, the less it has to gain. It will receive all food delivered over the remainder of the nestling period (contrasted to 59% on days where it shares the nest with /I), but fewer days remain on which food can be delivered. Thus, early execution gives the senior sibling potential access to more food than late execution. Moreover, by killing the junior sib, the senior sib loses accessto all food already invested in the junior sib, and the potential benefits in inclusive fitness c1might have derived had it allowed /I to survive the fledge. We note parenthetically that in cannibalistic species, 0: might reclaim some of this “lost food” by recycling nutrients invested in fl (see Perrigo, 1987). A chick that is uncertain about the type of the current year must strike a balance between the costs and benefits of clemency and execution in good and bad years; the use of a Bayesian algorithm to calculate the odds of good and bad years enhances the senior sib’s ability to make the right choice. Figure 6 (middle) illustrates the case where a uses Bayesian updating to choose between clemency and execution. In comparison with the case where c1has perfect information about year type (Fig. 6, top and bottom), three patterns emerge. First, early in the nestling period, the boundary between execution and clemency lies intermediate between that for good (Fig. 6, top) and bad years (Fig. 6, bottom). Second, later in the nestling period, the boundary between execution and clemency converges toward that of the good year. These two patterns reflect U’Searly uncertainty about whether it is a good or bad year. Such uncertainty diminishes as the nestling period progresses and c1 obtains further information about the type of year from the pattern of food deliveries. Third, in the example given here u favors obligate execution even though it incorporates Bayesian updating into its decision. Parents cannot deliver sufficient food to deter c1from early execution. For example, execution is still favored even if four deliveries occur in the first 4 days (Fig. 6, middle). This early in the nestling period, the risks (from a’s perspective) associated with a bad food year outweigh the potential benefits of a good food year, even though clemency is the preferred policy in a good year (Fig. 6). a favors clemency after five deliveries in 5 days (Fig. 6), as that is sufficiently compelling information to convince o! that it is a good food year, but by then it is too late for the senior sib to reverse its verdict of execution.

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ENVIRONMENT

Increasing Length of the Nestling Period.

Figure 7 shows the effect of changing the length of the nestling period on the threshold for execution. Clemency becomes increasingly favored as the nestling period is lengthened. This is an intuitive result, as a longer nestling period offers more opportunities for parents to deliver food (in the model all else remains equal). Thus, there is a greater likelihood that sufficient food will be delivered over the entire nestling period to favor clemency. Note that with a 70-day nestling period (Fig. 7), execution is not obligatory

50 40 30 20 10

0

n

60

50

T=50d 40 30 20 10

0 10

20

30

40

10

20

30

40

50

60

t FIG. 7. Effect of the length of the nestling period, T, on the optimal strategy (clemency [cross-hatched area] or execution [dotted area]) for G(as a function of t and n. Black area indicates region where n is insufficient for any chicks to survive. Nestling periods 40 (T= 40), 50 (T= 50), 60 (T= 60), and 70 (T= 70) days long are shown. Note that as T increases, clemency is increasingly favored earlier in the nestling period. At T= 70, execution is facultative-i.e., parents can deliver enough food at t= 1 to deter execution. At T=40, 50, and 60 execution is obligatory-i.e., execution occurs at t = 1 no matter how much food is delivered.

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as it is with shorter nestling periods (Figs. 7): parents can deliver sufficient food to satisfy c1early on. However, facultative execution still remains a possibility if parents do not continue to deliver sufficient food. The Cost of Execution Presumably /I becomes more difficult to kill as it grows older and larger. For tl, an increased risk of injury during fighting may be the principal cost of execution (Lamey and Mock, 1991). Injuries affecting locomotor or foraging efficiency might result in decreased survivorship beyond fledging, although we know of no empirical estimates of such costs. We explore the potential effects of delayed execution by modeling the cost of injury, inj,,,,, as an increasing linear function of the number of food items delivered to date, n (which is proportional to chick mass), of the form

n - 40

- 30

- 20

; 10

-

0

FIG. 8. Effect of risk of injury on the optimal strategy (clemency [cross-hatched area] or execution [dotted area]) for a as a function of t and n (T= 50). Top. The cost of an injury is 0. Bottom. The maximum cost of an injury during execution is 20% of the expected fitness of a after execution.

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153

where wE is the expected fitness of a after p’s execution (excluding any cost of execution) and 4, the total number of time periods, is also the maximum possible number of deliveries over the entire nestling period, and is a positive constant (4 > 1). Note that inj..,, decreasesas 4 increases,and that if 4 = 1, then the maximum possible cost is equal to wE (i.e., the fitness of LYafter execution is zero). The fitness of CIafter execution when the cost of injury is incorporated, oinj, thus becomes Winj

= 0~ - inj.,,,.

We explored a range of values for 4 and obtained similar results for each. Figure 8 illustrates a typical case in which the maximum cost of injury is 20% of oE. Surprisingly, the prospect of injury risk had little effect on the optimal decision (Fig. 8). However, the reason becomesclear in light of our earlier results. Above we show that execution is favored early in the nestling period, when the risk of injury is small, and clemency is favored late in the nestling period, when the risk of injury is large. In neither case will the prospect of injury change the behavioral outcome.

DISCUSSION

For the senior sibling there are two possible behavioral tactics, execution and clemency, from which three strategies can be derived: obligate execution, facultative execution. and obligate clemency. From the senior sib’s perspective, should the total level of provisioning over the nestling period (M) never exceed the execution threshold (mCRIT)obligate execution will always be favored. Conversely, if M always exceeds mCRITexecution will never be favored. We expect that in most casesfood availability will vary around mCRIT: in some years parents could easily deliver sufficient food for two chicks, whereas in other years sufficient food may only exist for one chick. In a variable environment, the threshold for obligate execution, M,, necessarily lies above mCRIT: thus obligate execution can be favored even if M is greater than mCRITin most years. Although the argument here has been developed with a two-chick model, exactly the same logic applies to larger brood sizes. Facultative execution will be favored when the fitness benefits (a sibling in good years, no competition in bad years) outweigh the costs. These costs arise from the possibility, due to sampling effects, that the senior sib arrives at an incorrect decision, killing its sib in a good year, or perhaps more importantly, allowing it to live in a bad year (see Stephens (1989) for a general discussion of the costs and benefits of information in variable environments). Lengthening the period of clemency reduces the

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chance of an error, but has the disadvantage that the possible victim becomes ever larger and more able to oppose (or even reverse) a verdict of execution. During the clemency period the junior sib also consumes food that, if the eventual decision is execution, the senior sib could better have consumed itself. Note that it is irrelevant whether food is abundant at the time of the junior sib’s demise. Stinson (1979) articulated the pending competition hypothesis, and suggested that although food may not be limiting at the time of the demise of the junior sib, it may become so later in the nestling period. Forbes (1991b), using a dynamic programming model, shows how execution may be favored by natural selection before the onset of proximate food shortage. However, it may even be irrelevant whether food exists over the entire season to support two chicks. All that is necessary for obligate execution to be favored is that individuals that always eliminate junior sibs do better over the long run than individuals that do not. Type of Information

Used during Sampling

Exactly what information can provide a senior sib with information about whether or not it should perform execution is unclear, but clues that relate to the rate of food delivery are likely candidates. A chick might tally the number of deliveries, as in our dynamic model, or the quantity of food it receives over some interval (e.g., a day). Other proximate cues of food availability might also be used (e.g., early growth rate, proportion of time crop is full). Even the size or species composition of the diet alone could provide useful information about the state of the environment and the corresponding parental behavior (in good food years a particular type of prey might be prevalent). In species such as the Peruvian guanay cormorants (Phalacrocorax bougainvilli), piqueros (Ma variegata), and brown pelicans (Pelecanus occidentalis), which rear chicks primarily on Peruvian anchovetta (Engraulis ringens), the presence of many nonanchovetta prey in the diet might reliably indicate to chicks a failure of the anchovetta (e.g., an El Niiro event), and hence a “bad” food year. Thus a simple decision rule, independent of the rate of prey delivery might be favored: perform execution if diet not primary prey (anchovetta), do not perform execution if diet primary prey. Similarly, other cues not directly related to food abundance may influence the decisions of offspring regarding execution. Hatching date (which might be judged by photoperiod) could provide an offspring with useful information about its future prospects for several reasons. Postfledging survival of late-hatched chicks is often lower than early-hatched chicks (Perrins, 1963, 1970; Birkhead and Nettleship, 1982; Roskaft and Slagsvold, 1985; Poole, 1989), thus appropriating more resources from parents at the expense of junior sibs may become more important.

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Hatching date and parental quality are negatively correlated in many species (Askenmo and Unger, 1986) (young breeders nest later in the season), thus offspring could expect a lower rate of parental provisioning: once again obligate execution may be increasingly favored. Further Questions Here we have addressed the effects of interannual variability in food supplies on brood reduction decisions by the offspring. To do so, we have assumed that food supplies within a year are unchanging, but how might the prospect of within season variation in food supplies (e.g., the potential effect of a prolonged storm) affect the behavioral decisions of the senior sib? We have also assumed here that food delivered to the brood is a strict function of the amount of food in the environment. Parents, though, may vary how hard they work to gather food, and could either dampen or exaggerate the effects of variability in food supplies for the offspring (see Evans, 1990). As well, parents may adjust their effort according to brood size, once again affecting the decisions of the offspring. Such questions remain open and important areas for future study.

APPENDIX

I

A Stochastic Dynamic Programming Model According to the dynamic programming principle (Bellman 1957, Mange1 and Clark, 1988), the optimal sequence of decisions to attain a desired end state from state x at time t is the same, regardless of the sequence of decisions by which state x at time t was arrived at. Thus the best decision for every state x at every time t can be found by computing from the end state backward. In our model, two sibs, cxand p, feed and grow over a finite nestling period T days long. All feeding and growth occurs on Days 1 to T- 1, and on day T, the chicks fledge. A chick’s mass changesaccording to how much it eats, and fitness is assumed to be a function of mass at fledging. We assume that u, the senior sib, is dominant and can choose whether B lives (clemency) or dies (execution). fi has no choice but to accept a’s decision. The fitness of u at time t is specified by two state variables, the number of food deliveries, n (n < t), and the presence(S = 1) or absenceof a junior sib (S= 0), and its behavioral decision, d (execution [d= 0] or clemency [d= l]), and is denoted F[n, S, t, T]. At the terminal period, T, CI’Sfitness from a strategy of clemency is specified by

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AND

YDENBERG

(Np-N,i,)+lN(l-p)-N,i”

N(l-p~,~

2

NP

N(l -P)

(NP - Nmin) NP

F[n, 1, T, T] = i

mm

Np>N,i”>N(l-p)

L0

NP G Nmin9 (25)

where N is the total number of food deliveries over the entire nestling period, Nmin is the minimum provisioning a chick requires to be viable, and p and (1 - p) are the proportions of food consumed by c1and /I, respectively. Note that N and Nmi, are analogous to M and mmin in the analytical model presented above. Nmin was set at 10, and in accordance with Eq.(7) we setp=0.59 and (l-p)=O.41. At the terminal period T, GI’Sfitness from a strategy of execution is specified by

F[n, 0, T, T] =

(NPm- Nmin ) NP,

NP, ’ Nmin

where pz is the proportion of N taken by c1over the entire nestling period. For example, if execution occurs on Day 10 after five food items have been delivered, and 20 additional food items are delivered over the remainder of the nestling period, then p, = ((p .5) + 20)/25. Senior sibs therefore lose accessto food invested in j? prior to execution. On the penultimate day, T- 1, there are three possibilities: a may have executed fi at some earlier point in the nestling period, a may choose to execute /I on day T- 1, or a may allow /I to live (clemency). Whether or not clemency or execution is favored depends upon the fitness payoffs for the two strategies, and a’s Bayesian estimation of the probability of a good food year, y, given n deliveries over t days and a prior expectation of a good year, Pr(Good year) = 0.5 = Pr(Bad year). If execution occurs on day T- 1 or has already occurred, a’s fitness is given by F[n, 0, T-

1, T]

= ~“Y,f’Cnp, + (1 -Y)

+ to, ‘Yd’Cnp,

T,

Tl + ~(1 - Y’c) F[np,, 0, T, T] + ho,

T,

Tl + (1 - y)( 1 - ul,) F[np,, 0, T, T-J, (27)

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SIBLING RIVALRY IN VARIABLEENVIRONMENT

where Yi is the probability of a food item being delivered on a given day (0.7 in a good year [ lu,], 0.3 in a bad year [ ul,]), and (1 - !PJ is the corresponding probability of no food item being delivered on a given day. We deliberately chose values of Yi in the range where both clemency and execution might be favored. If clemency is favored, the fitness of c1on day T - 1 is given by F[n, 1, T- 1, T]

= ~Yc+Tnp, + 1, 1, T, Tl + 141- vl,) FCnp,, 1, T, T] +(~-Y)YY,FC~~,+~,~,T,TI+(~-~)(~-Y~)FC~~,,~,T,T] (28)

To generalize for the remainder of the nestling period, replace T- 1 with t - 1 and T with t in Equations 27 and 28. Modelling the Risk of Injury The injury cost is simply a proportion of the expected fitness CIwould receive from execution on day t, and rises linearly with n (as the chicks, and specifically 8, become older and larger, the cost of execution rises) where

inj,,,, = FCn,0, t, Tl$,

(29)

and where 4 is a positive constant, set equal to 5 in the model. The fitness of a senior sib killing its junior sib on day t is simply F[n, 0, t- 1, t ] - inj..,,.

ACKNOWLEDGMENTS We thank L. Dill, D. Mock, B. Munro, R. Roitberg comments on earlier versions of this manuscript.

and an anonymous

reviewer

for

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