The influence of contests on optimal clutch size: a game-theoretic model.

10 downloads 0 Views 126KB Size Report
Mar 24, 2004 - occur more frequently. We also show the existence of polymorphisms in clutch-size optima and that .... given host from 'static optimality' to 'game-theoretic' ... 156, lists 22 bird species and 68% of studies showing this pattern); ...
Received 6 August 2003 Accepted 17 December 2003 Published online 24 March 2004

The influence of contests on optimal clutch size: a game-theoretic model Mike Mesterton-Gibbons1* and Ian C. W. Hardy2 1

Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA School of Biosciences, University of Nottingham, Sutton Bonington Campus, Loughborough, Leicestershire LE12 5RD, UK

2

We develop a game-theoretic model to predict the effect of size-dependent contest outcomes on optimalclutch-size decisions. We consider the case where larger individuals develop from smaller clutches and, as adults, are advantaged in competition for limiting resources. The relationship between fitness and size thus depends on the sizes of other members of the population. We show that clutch-size optima are decreased by body-size-dependent contest outcomes, with larger effects when body size is most affected by clutch size, when prior resource ownership has less influence on contest outcome and when contests occur more frequently. We also show the existence of polymorphisms in clutch-size optima and that clutch-size driven changes in population density can, via an effect on the probability of host finding, further influence optimal clutch size. Our model is formulated to match the life history of a parasitoid wasp, in which clutch size affects offspring size and females engage in direct contests for host ownership, which larger females tend to win; we confirm that female–female competition is likely to influence clutch size in this species. However, the model is also relevant to clutch size in other taxa and supports recent suggestions concerning reproductive decisions in great tits. Keywords: clutch size; body-size-dependent contest outcomes; game theory; density dependence; Goniozus nephantidis; great tits 1. INTRODUCTION

summarizing the biology of the wasp to provide a specific context for our study. Goniozus nephantidis (Hymenoptera: Bethylidae) is a parasitoid wasp that lays clutches of up to 18 eggs on paralysed lepidopteran larvae. Clutch size is strongly and positively correlated with host size, and, for a given host size, smaller adults emerge from larger clutches (Hardy et al. 1992). Sex ratios are highly female biased (ca. 9% of adults are males). Adult females spend a relatively long, clutch-size independent, time producing each clutch and remain with the developing brood for several days to guard it against the detrimental actions of other parasitoids (Hardy & Blackburn 1991). Females that encounter a paralysed but unparasitized host, from which the initial ‘owner’ has been removed, generally lay their own eggs on it (Hardy & Blackburn 1991; Petersen & Hardy 1996). When such intruders encounter guarded hosts, brief aggressive owner–intruder contests usually ensue, resulting in the loser being driven from the vicinity of the host (Petersen & Hardy 1996; Stokkebo & Hardy 2000). Contests tend to be won by larger females, but prior ownership also confers an advantage (Petersen & Hardy 1996), possibly as a result of differences in egg load between owners and intruders leading to resource-value asymmetries (Stokkebo & Hardy 2000). It is thought that in nature individual females are unlikely to produce more than one clutch in their lifetime (Hardy et al. 1992). Under these conditions (i.e. semelparity), theory predicts that fitness returns should be maximized per clutch (e.g. Godfray et al. 1991; Mangel et al. 1994; Wilson & Lessells 1994); the clutch size that maximizes fitness per host is known as the Lack clutch size. In an attempt to calculate the Lack clutch size for G. nephantidis, Hardy et al. (1992) artificially created a range

Two successful areas of research within behavioural ecology are contest behaviour and clutch size. Each area has developed via a close association of theoretical developments with empirical studies on a wide variety of animal taxa (e.g. Godfray 1987; Godfray et al. 1991; Stearns 1992; Wilson & Lessells 1994; Riechert 1998; Renison et al. 2002). The two topics have largely developed independently of one another. Studies of clutch (or litter) size have examined how parental decisions concerning the size of the clutch influence the properties (e.g. body size) of the individual offspring that develop from it (e.g. Stearns 1992; Mangel et al. 1994; Wilson 1994; Zaviezo & Mills 2000). Meanwhile, studies of animal contests have examined how the properties (e.g. body size) of individuals influence their success in obtaining or retaining behaviourally disputed resources (e.g. Martin 1991; Lozano 1994; Schneider & Lubin 1997; Renison et al. 2002). Nevertheless, clutch-size optima have the potential to be influenced by the occurrence, frequency and mechanisms of contests in taxa where offspring develop in clutches (the size of which affects the individuals’ properties as adults) and adults compete for resources via contests (the outcomes of which are determined by the same properties); moreover, contest frequency may be reciprocally influenced by clutch-size via changes in population density. In this paper, we formally explore the influence of contests on clutch size optima using a game-theoretic model. Our model is intended to be broadly applicable, but its development was stimulated by investigations of clutch size and contest behaviour in a parasitoid wasp, and so we begin by

*

Author for correspondence ([email protected]).

Proc. R. Soc. Lond. B (2004) 271, 971–978 DOI 10.1098/rspb.2003.2670

971

 2004 The Royal Society

972 M. Mesterton-Gibbons and I. C. W. Hardy Influence of contests on optimal clutch size

of clutch sizes on hosts of standard size and assessed the fitness consequences for female clutch members in relation to their adult body size, using longevity and fecundity as estimators of fitness. The Lack clutch size (ca. 18 eggs per host) calculated using a static optimality approach (in which a given clutch size generates offspring of a particular size and thus fitness) was considerably larger than clutch sizes naturally laid on hosts of the chosen standard size (ca. nine eggs). If the argument that G. nephantidis should produce the Lack clutch size is correct, the discrepancy between observed and predicted clutch sizes suggests that the fitness consequences of body size were underestimated, possibly because some components of the fitness–size relationship were not measured. Petersen & Hardy (1996) subsequently found that larger females are more successful in host-ownership contests, and thus identified a fitness component based on relative rather than absolute body size. They suggested that size-dependent contest outcomes change the approach required to calculate the Lack clutch size on a given host from ‘static optimality’ to ‘game-theoretic’ (sensu Maynard Smith 1982), with the optimal decisions of one individual depending on decisions made by other members of the population. Petersen & Hardy argued (verbally) that this dependence provides a potentially powerful explanation for the observed clutch-size discrepancy. The logic is that the fitness of females of a given size will be influenced by the relative sizes of any competitors for host ownership that they encounter. Females that develop from smaller clutches will tend to be larger and tend to win contests. Therefore the occurrence of contests can select for a reduction in clutch size, with the optimal value dependent on the size of clutches laid on other hosts by other mothers in the population. Our aim in this paper is to formalize and hence verify this argument. To elaborate: there are two extremes at which an optimality approach suffices. If owners always win contests or contests never occur, then there is logically no influence of relative body size on clutch size; whereas, if contests always occur and the larger female always wins, then the only stable clutch size is just one egg (or the lowest clutch size that can be predicted). Neither extreme accords with observations (Petersen & Hardy 1996), and intermediate frequencies of contest occurrence and varying degrees of ownership advantage require a game-theoretic model. While the model we develop employs assumptions grounded in the biology of G. nephantidis, readers who are more interested in vertebrates may prefer to think of the issue we explore in avian terms. Imagine a generalized bird species in which three properties co-occur: (i) larger individuals develop from smaller clutches (Stearns 1992, p. 156, lists 22 bird species and 68% of studies showing this pattern); (ii) nesting sites or territories are of limited abundance (e.g. Minot & Perrins 1986; Lozano 1994; Renison et al. 2002; Both & Visser 2003); and (iii) success in contests for these sites is body-size related (e.g. Martin 1991; Lozano 1994; Renison et al. 2002). Parallel verbal arguments to those of Petersen & Hardy (1996) have been made for clutch-size decisions in great tits, Parus major (Both et al. 1999); to our knowledge these suggestions have not previously been formalized. Proc. R. Soc. Lond. B (2004)

2. MATHEMATICAL MODEL We are interested in the effect of contest behaviour on clutch size. In our model, contest behaviour itself is not under selection. Sex ratio is also not under selection, so we consider a thelytokous population of females (which is a close approximation to the highly female-biased sex ratios observed in G. nephantidis; Hardy et al. 1992). Let the number of eggs in a clutch laid by an individual in the general population be V, and let U be the clutch size laid by a potential mutant; daughters are assumed to inherit their mothers’ strategies. Although U and V are in practice integers, we assume for simplicity that they vary continuously over positive real numbers, thus following a longstanding tradition in the literature on Lack clutch size (e.g. Wilson & Lessells 1994). For simplicity we assume that there are only two adult body sizes, ‘small’ and ‘large’. In contests between two small individuals or between two large individuals or between a large owner and a small intruder, the owner always wins. However, in a contest between a small owner and a large intruder, which individual wins depends upon the advantage of ownership versus the advantage of size. Accordingly, we introduce a measure of the first advantage relative to the second by defining ␳ to be the probability that a small owner defeats a large intruder. We will refer to the parameter ␳ as the owner’s relative advantage. In the absence of any contest behaviour, the expected number of offspring from a clutch of a given size—or fitness per clutch—equals the size of the clutch multiplied by the fitness per egg, which we assume to decrease with clutch size (because, for example, egg-to-adult survival is not perfect). We further assume that all hosts are of equal size (quality), that each surviving egg becomes a large adult or a small adult independently of any other egg, and that the probability of becoming a large adult also decreases with clutch size. More formally, for an egg laid in a clutch of size U, let S(U ) denote its fitness in the absence of contest behaviour, and let ⌽(U ) denote its probability of developing into a large adult; S⬘(U ) and ⌽⬘(U) are both assumed to be negative (where a prime denotes differentiation with respect to U ). In the absence of any contest behaviour, the Lack clutch size is by definition the U that maximizes US(U ), i.e. fitness per clutch. We will denote this optimal clutch size by L and refer to it as the ‘traditional Lack clutch size’, because it is the interpretation of optimal clutch size that is usually implied in discussions of Lack’s theory (e.g. Godfray et al. 1991; Wilson & Lessells 1994). We assume throughout that US(U) is a unimodal function. Because of contest behaviour, however, the traditional Lack clutch size does not in general maximize fitness per clutch in the sense of expected number of offspring from a suitable (unparasitized, though possibly already paralysed) host. In other words, it is not the ‘true Lack clutch size’. Traditional Lack clutch size is a simple optimum, whereas contest behaviour makes true Lack clutch size depend on the behaviour of other animals (Petersen & Hardy (1996) and see § 1), and it is therefore an evolutionarily stable strategy (ESS) (sensu Maynard Smith 1982). Our purpose is to predict this ESS as the solution of a simple population game. The analysis is simplified if true Lack clutch size is measured as a proportion of the traditional Lack

Influence of contests on optimal clutch size M. Mesterton-Gibbons and I. C. W. Hardy

clutch size. We therefore scale clutch size with respect to L by defining u = U/L and v = V/L and rewriting the fitness per egg and the probability of developing into a large adult as s(u) = S(Lu) and ␾(u) = ⌽(Lu), respectively. Thus our assumptions become s⬘(u) ⬍ 0 and ␾⬘(u) ⬍ 0, and the traditional fitness of a clutch whose size is proportion u of the traditional Lack clutch size becomes F(u) = Lus(u).

(2.1)

By assumption, F is a unimodal function whose maximum occurs where u = 1. For the sake of simplicity, we now further assume that F ⬙(u) ⬍ 0 for all u ⭐ 1. Let Z(T ) be the probability that a suitable host located at time T is being guarded, i.e. already has an owner; let Y(T ) be the probability that an owner will subsequently be intruded upon by another insect; and let k be the probability that a host is never found (during the entire vulnerable period of its development). Then (1 ⫺ Z(T ))(1 ⫺ Y (T )) = k,

(2.2)

because, from our protagonist’s point of view, k is simply the probability 1⫺Z(T) that a host has not already been found times the probability 1⫺Y(T ) that it is not subsequently discovered: the bigger the value of Z(T ), the smaller the value of Y(T ). In particular, if the vulnerable period of development has length p and the time to the next arrival follows an exponential distribution with parameter a, then Z(T ) = 1 ⫺ e⫺aT, Y (T ) = 1 ⫺ e⫺a( p⫺T ) and k = e⫺aTe⫺a( p⫺T) = e⫺a p.

(2.3)

Here, we assume for simplicity that a, and hence k, is independent of the population clutch size, v. We relax this assumption in § 4. We assume that there is a very narrow time window in which an insect can actually acquire a host and thus that each host is the subject of at most one contest. To be able to reproduce, females must either find an unguarded host and defend it against at most one intruder or take over a guarded host in a contest. To compute the reward to a u-strategist in a population of v-strategists, let f(u,v) denote this reward, i.e. the expected number of offspring from a host that has just been discovered. First, we compute the pay-off from a host located at time T, conditional on being large. With probability 1 ⫺ Z(T ) the host is unguarded, in which case the owner is guaranteed to keep the host, because, even if it is subsequently intruded upon, the owner will win the contest by virtue of being large. However, with probability Z(T ) the host is guarded, in which case the intruding ustrategist wins (and hence keeps) the host only if its owner is small, and then only with probability 1 ⫺ ␳. The owner is a v-strategist, and therefore small with probability 1 ⫺ ␾(v). So the probability that a large u-strategist retains a host is 1 ⫺ Z(T ) ⫹ Z(T )(1 ⫺ ␾(v))(1 ⫺ ␳), and so its payoff, if it is large, is (1 ⫺ Z(T ) ⫹ Z(T )(1 ⫺ ␾(v))(1 ⫺ ␳))F(u) (see the left-hand branch of the decision tree in figure 1). Next we compute the pay-off from a host located at time T for a small adult. Then, with probability 1 ⫺ Z(T) the host is unguarded, in which case the owner will keep the Proc. R. Soc. Lond. B (2004)

973

host unless subsequently disturbed by a large intruder who wins. The intruder is a v-strategist, and therefore large with probability ␾(v). The small owner will thus keep the host with probability 1 ⫺ Y (T )␾(v)(1 ⫺ ␳). With probability Z(T ), however, the host is guarded, in which case the intruding u-strategist has no chance of winning because it is small. So, the probability that a small u-strategist retains a host it discovers is (1 ⫺ Z(T ))(1 ⫺ Y (T )␾(v)(1 ⫺ ␳)), and its pay-off, if it is small, is (1 ⫺ Z(T ))(1 ⫺ Y (T )␾(v)(1 ⫺ ␳))F(u). However, the probabilities of being large and small (conditional upon survival) are ␾(u) and 1 ⫺ ␾(u), respectively. Thus the pay-off from a host located at time T is (1 ⫺ Z(T ) ⫹ Z(T )(1 ⫺ ␾(v))(1 ⫺ ␳))F(u)␾(u) ⫹ (1 ⫺ Z(T ))(1 ⫺ Y(T ) ␾(v)(1 ⫺ ␳))F(u)(1 ⫺ ␾(u)). But T is a random variable, and so we compute the expected value of this pay-off over the distribution of T to obtain the animal’s reward. Thus, if E denotes the expected value, on using equation (2.2) we obtain f (u,v) = E[{(1 ⫺ ␳)(Z(T)(1 ⫺ ␾(v)) ⫹ Y (T)(1 ⫺ Z(T ))␾(v))␾(u) ⫹ (1 ⫺ Z(T ))(1 ⫺ Y (T )(1 ⫺ ␳)␾(v))} F (u)] = E[1 ⫺ Z(T ) ⫹ (1 ⫺ ␳){Z(T )(1 ⫺ ␾(v))␾(u) ⫺ (1 ⫺ k ⫺ Z(T ))(1 ⫺ ␾(u))␾(v)}] F (u)

or f (u,v) = (1 ⫺ z ⫹ (1 ⫺ ␳){z(1 ⫺ ␾(v))␾(u) ⫺ (1 ⫺ k ⫺ z)(1 ⫺ ␾(u))␾(v)}) F (u),

(2.4)

where z = E[Z(T )].

(2.5)

In particular, if time to next arrival follows an exponential distribution with parameter a and T is uniformly distributed between 0 and p (the length of the vulnerable period of development), then





1 p (ap ⫺ 1)eap ⫹ 1 1 Z(t)dt = {1 ⫺ e⫺at}dt = p0 apeap 0 p 1⫺k , =1⫹ ln(k)

z=

p

(2.6)

on using equation (2.3). Note that f = (1 ⫺ z)F when ␳ = 1. Thus (because z is independent of u regardless of whether it depends on v), if the advantage of ownership is sufficiently strong that even small owners cannot lose, then the ESS invariably agrees with the traditional Lack clutch size (as it does when contests cannot occur, because f = F when z = 0 and k = 1). For ␳ ⬍ 1, however, it follows from our assumptions that f is decreasing for u ⭓ 1, so that f has its maximum where u ⬍ 1 for any given v, and, if this maximum occurs where u = v, so that v is a best reply to itself, then v is an ESS. We will denote this ESS by v∗. Thus v∗ ⬍ 1: the true Lack clutch size is smaller than the traditional Lack clutch size, and, because v∗ increases with ␳ and k (as demonstrated in § 3), the true Lack clutch size is smallest in the limit as ␳ → 0 and k → 0. A contest is then inevitable, and f(u,v) approaches (1 ⫺ ␾(v))␾(u)F(u), so that the minimum optimal clutch size is the u that maximizes largeoffspring fitness ␾(u)F(u). We denote this minimum optimal clutch size by ␧.

974 M. Mesterton-Gibbons and I. C. W. Hardy Influence of contests on optimal clutch size

1 – φ (u)

φ (u) protagonist large

protagonist small Z(T)

Z(T)

host free (and accepted)

φ (v)

1 – φ (v) owner small

ρ 0

1–ρ

intruder loses

host 0 guarded

1 – Z(T)

host guarded F(u)

owner large 0

F(u)

1 – Z(T)

Y(T)

1 – Y(T)

not intruded intruded on F(u) on 1 – φ (v) φ (v) intruder intruder large F(u) small ρ 1– ρ owner wins F(u)

intruder wins

host free (and accepted)

0

owner loses

Figure 1. The decision tree for the protagonist (u-strategist). The left-hand branch is for a large protagonist; the right-hand branch is for a small protagonist. The (conditional) probability associated with each arc is written over it; the root and all internal nodes are represented by dots; terminal nodes are represented by squares; and pay-offs are indicated in the rectangular boxes. To obtain the reward, multiply each pay-off by the product of the probabilities associated with the arcs that lead to that pay-off, then sum.

3. TRUE LACK CLUTCH SIZE To calculate the true Lack clutch size for ␳ ⬍ 1, we must first choose explicit forms for both s and ␾. Accordingly, we now set s(u) = e⫺u;

(3.1)

because s(0) cannot affect where f is maximized, no generality is lost by assuming s(0) = 1, and it follows from equation (2.1) that all of our assumptions about F are satisfied. Specifically, F is unimodal with its maximum at u = 1, and F ⬙(u) ⬍ 0 for all u ⭐ 1. We note in passing that other such forms of s yield qualitatively similar results. We now turn to ␾. On the one hand, intuition strongly suggests that increasing clutch size will cause a smaller reduction in the probability of being large when clutch size is small than when clutch size is somewhat larger, i.e. ∂2␾/∂u2 ⬍ 0 when u is small. On the other hand, the probability of being large must eventually approach zero as clutch size increases, and so ∂2␾/∂u2 must eventually become positive. Let c denote the critical clutch size at which the graph of ␾ thus has an inflection point. Then we need a form satisfying ∂2␾/∂u2 ⬍ 0 for u ⬍ c but ∂2␾/∂u2 ⬎ 0 for u ⬎ c. We choose to satisfy these conditions by setting 2

␾(u) = ␾0e⫺1/2(u/c) ,

(3.2)

with ␾ 0 ⭐ 1. It now follows from the end of § 2 that true Lack clutch size satisfies ␧ ⬍ v∗ ⬍ 1 where

冉冑

1 ␧= c 2



c2 ⫹ 4 ⫺ c .

(3.3)

We have no data from which the critical clutch size c can reliably be estimated. Nevertheless, results from an empirical study (Hardy et al. 1992) suggest that the probability of becoming large in a traditional Lack clutch is significantly greater than zero, so that c cannot be much smaller than 1. The case where c ⭓ 1 is straightforward. Proc. R. Soc. Lond. B (2004)

Then ∂2␾/∂u2 ⬍ 0 for all u ⬍ 1, and it follows from our prior assumptions that ∂2f/∂u2 ⬍ 0 for all u ⬍ 1. So, for any given v, f has a unique local maximum at u = u˜, defined by ∂f = 0, ∂u u = u˜

|

and this local maximum is also the global maximum of f. Thus the ESS, v∗, is the unique value of v for which u˜ = v and 0 ⬍ v ⬍ 1. Note that u˜ = v is a transcendental equation; its explicit analytical form is readily calculated, but in practice it is too cumbersome to be useful (and so we do not present it). The case where c ⬍ 1 is somewhat more complicated, because for small enough c there arises the possibility that ∂2f/∂u2 changes sign twice (from negative to positive and back) for 0 ⬍ u ⬍ 1, so that f has two local maxima. (In passing, we briefly explain why. When c is small, ␾ falls rapidly to zero as u increases, so that f becomes approximately F(u) times a positive function of v and hence has approximately the shape of F(u) with a maximum near 1—except when u is small. In that region, f is the product of increasing F(u) and a function that decreases rapidly because ␾ does so, thus generating a second local maximum nearer to 0.) Nevertheless, it is straightforward to determine which of these local maxima yields the global maximum, and, where this clutch size yields a best reply to itself, there is again a unique ESS. Broadly speaking, the clutch size at the ESS is small or large according to whether ␳ is small or large; however, there is an intermediate range of values of ␳ for which no monomorphic ESS exists, because a small clutch size and a large clutch size are both stable to small perturbations in clutch size, although a population of either is readily invaded by the other. For such intermediate values of ␳, the evolutionarily stable state is a polymorphic mixture of the two clutch sizes, whose stable proportions may be found by standard

Influence of contests on optimal clutch size M. Mesterton-Gibbons and I. C. W. Hardy

clutch size

(a)

(b)

v*

v*

1.0

1.0

0.5

0.5

ρ

0 0 (c)

0.5

0 (d) 1.0

0.5

0.5

ρ 0

0.5

1.0

0.5

1.0

0.5

1.0

v*

1.0

0

ρ

0

1.0

v*

975

ρ

0 0

owner’s relative advantage Figure 2. The relationship between true Lack clutch size (as a proportion of the traditional value) v∗ and the owner’s relative advantage ␳ for ␾0 = 1 in equation (3.2) and various values of c (the critical clutch size at which the probability of being large has an inflection point): (a) c = 1; (b) c = 0.8; (c) c = 0.6; and (d ) c = 0.4. In each case, the ESS is shown for various values of the probability k that a host is never found: namely, k = 0.5 (uppermost curve), k = 0.25, k = 0.1, k = 0.01, k = 10⫺6 and k → 0 (lowermost curve, corresponding to equation (3.3)). The dashed curves are identical to the dashed curves in figure 5.

techniques (e.g. Mesterton-Gibbons 2001). These results are illustrated in figures 2–4.

reducing k in the model where k is independent of v, and figure 5 confirms this: clutch-size optima decrease with increasing strength of density dependence.

4. EFFECT OF DENSITY DEPENDENCE We have so far assumed that a (and hence the probability of a host not being found) is independent of the population clutch size, v. In nature, however, the mean time to next arrival at a site, i.e. 1/a, will be lower at higher density. Because density will increase with fitness per clutch, F(v), we can incorporate the effect of density dependence by assuming that a increases with F(v) according to a = a0 ⫹

␦F(v) , pL

(4.1)

where L denotes traditional Lack clutch size, p denotes the vulnerable period of host development and ␦ ( ⬎ 0) represents the strength of density dependence. Then, from equations (2.1), (2.3) and (3.1), the probability of a host not being found is ⫺v

k = k0e⫺␦ve ,

(4.2)

where k0 = e is the probability of a host not being found in the absence of density dependence. Intuition suggests that the effect of increasing ␦ in the model where k depends on v will be similar to that of holding ␦ = 0 but ⫺a0 p

Proc. R. Soc. Lond. B (2004)

5. DISCUSSION We have used a biologically based, yet idealized, gametheoretic model to elucidate the effect of contests on clutch size. This model predicts that clutch-size optima are decreased by body-size-dependent contest outcomes, with larger effects when: (i) the body size of clutch members is most affected by clutch size; (ii) prior ownership has less influence on contest outcome; and (iii) contests occur more frequently. We have also shown: (iv) the existence of polymorphisms in clutch-size optima; and (v) that clutch-size-driven changes in population density can, via an effect on the probability of host finding, further influence optimal clutch size. These results support verbal arguments put forward by Petersen & Hardy (1996), specifically concerning clutch-size decisions in G. nephantidis. Laboratory studies have provided empirical estimates of the relationships highlighted in (i) and (ii) above (Hardy et al. 1992; Petersen & Hardy 1996; Stokkebo & Hardy 2000), but our model reaffirms the need to assess the unknown frequency at which contests occur in nature (which is inversely related to parameter k); without such an estimation, we cannot predict the true Lack clutch size.

976 M. Mesterton-Gibbons and I. C. W. Hardy Influence of contests on optimal clutch size

clutch size

(a)

(b)

v*

v*

1.0

1.0

0.5

0.5

ρ

0 0 (c)

0.5

0 (d)

v*

1.0

0.5

0.5

ρ

0 0.5

0.5

1.0

0.5

1.0

v*

1.0

0

ρ

0

1.0

ρ

0

1.0

0

owner’s relative advantage Figure 3. The relationship between true Lack clutch size (as a proportion of the traditional value) v∗ and the owner’s relative advantage ␳ for ␾0 = 1 and c = 0.2 in equation (3.2) and various values of the probability k that a host is never found: (a) k = 0.02; (b) k = 0.01; (c) k = 0.0001; and (d ) k = 10⫺6. In each case, the dotted line corresponds to equation (3.3). The shading indicates an intermediate range of values of ␳ where the evolutionarily stable state is a polymorphic mixture of the clutch sizes above and below the shading. Note that the upper curves, which appear to be horizontal, in fact rise extremely slowly with ␳.

(b)

(a)

f

.

.

f

.

fitness

.

0 0

0.5

0 u 1.0 0 clutch size

u 0.5

1.0

Figure 4. The relationship between fitness and clutch size when c = 0.2 and k = 0.0001 (as in figure 3c). (a) Shows f(u,v∗) for ␳ = 0.5 (solid curve) and ␳ = 0.9 (dashed curve); in either case, the dot represents the unique ESS. These curves illustrate the unshaded regions in figure 3c. (b) Shows f(u,v∗1 ) as a solid curve and f(u,v∗2 ) as a dashed curve for ␳ = 0.7 in a stable polymorphism of clutch sizes v∗1 and v∗2 , which are represented by dots. These curves illustrate the shaded region in figure 3c.

We note, however, that, under a number of model parameter combinations, the true Lack clutch size is predicted to be around half of the traditional Lack clutch size (e.g. figure 2), which fits well with the results of the original empirical study (Hardy et al. 1992); contest behaviour Proc. R. Soc. Lond. B (2004)

remains a good candidate explanation for the observed discrepancy. Game-theoretic models of clutch size have previously been developed to investigate oviposition decisions when multiple females exploit a single resource (e.g. host)

Influence of contests on optimal clutch size M. Mesterton-Gibbons and I. C. W. Hardy

clutch size

(a)

(b)

v*

v*

1.0

1.0

0.5

0.5

ρ

0 0

977

0.5

ρ

0

1.0

0

(c)

(d)

v* 1.0

0.5

1.0

0.5

1.0

v*

1.0

0.5

0.5

0 0

0.5

1.0

ρ

ρ

0 0

owner’s relative advantage Figure 5. The relationship between true Lack clutch size (as a proportion of the traditional value) v∗ and the owner’s relative advantage ␳ for ␾0 = 1 in equation (3.2) and various values of c (the critical clutch size at which the probability of being large has an inflection point): (a) c = 1; (b) c = 0.8; (c) c = 0.6; and (d ) c = 0.4. In each case, k0 = 0.1 in equation (4.2) and the ESS is shown for various values of the strength of density dependence: namely, ␦ = 0 (uppermost curve, shown dashed), ␦ = 5, ␦ = 20, ␦ = 50 and ␦ → ⬁ (lowermost curve, corresponding to equation (3.3)). The dashed curves are identical to the dashed curves in figure 2.

(reviewed by Godfray (1987); Wilson & Lessells (1994); and see references in Petersen & Hardy (1996)). Our model differs in that each host is used by at most one female. In our case the game-theoretic approach is required because the fitness of individuals in a clutch is influenced by the clutch-size decisions of females laying eggs on other hosts in the population; offspring developing in those clutches are the future, rather than current, competitors of offspring in a given focal clutch. We are aware of several empirical analogues of the circumstances we model. For example, Cowan (1981) studied eumenid (caterpillar hunting) wasps in which males engage in male–male contests for ownership of a mating resource (the nest area) and male owners are larger than non-owners. Offspring body size is positively related to food supply during development, and mothers provisioned developing males with more food (analogous to reducing clutch size on a fixed resource) when nests were clumped together (making contests likely) than when they were isolated. These data support our prediction of a densitydependent effect of contest behaviour on reproductive decisions. Similarly, in great tits, larger offspring are produced from smaller clutches (references in Stearns 1992), larger offspring can be advantaged in contests (Garnett 1981; Sandell & Smith 1991) and smaller clutches are Proc. R. Soc. Lond. B (2004)

produced during periods of high population density (Krebs & Perrins, 1978; Both 1998a,b; Both et al. 2000). Recently, Both et al. (1999) have shown that the relative size of fledglings within a cohort, and not just their absolute size, affects recruitment into the breeding population, and also that the mass of fledglings is more important to their recruitment when population densities are higher and competition is more severe. This evidence from great tits thus accords with the thrust of our model’s predictions. Like Petersen & Hardy (1996), Both et al. (1999) concluded that clutch-size optimization should be considered using a game-theoretic approach (see also Krebs & Perrins 1978). To conclude, the model we have developed examines clutch-size optima in relation to contest behaviour. Even though only one female lays in each host, a game-theoretic approach is required because clutch size affects a property (body size) of the offspring that subsequently plays a role in determining the outcomes of contests with individuals from the wider population. The fitness associated with a particular clutch size or body size depends on the sizes of other clutches and individuals in the population, and is thus relative, not absolute. Clutch-size optima can be greatly influenced by the occurrence of size-dependent contest outcomes.

978 M. Mesterton-Gibbons and I. C. W. Hardy Influence of contests on optimal clutch size We thank S. D. Mylius and S. A. H. Geritz for discussions during the early stages of this work, and M. P. Gammell and T. P. Batchelor for comments at the end.

REFERENCES Both, C. 1998a Density dependence of clutch size: habitat heterogeneity or individual adjustment? J. Anim. Ecol. 67, 659–666. Both, C. 1998b Experimental evidence for density dependence of reproduction in great tits. J. Anim. Ecol. 67, 667–674. Both, C. & Visser, M. E. 2003 Density dependence, territoriality, and divisibility of resources: from optimality models to population processes. Am. Nat. 161, 326–336. Both, C., Visser, M. E. & Verboven, N. 1999 Density-dependent recruitment rates in great tits: the importance of being heavier. Proc. R. Soc. Lond. B 266, 465–469. (DOI 10.1098/rspb.1999.0660.) Both, C., Tinbergen, J. M. & Visser, M. E. 2000 Adaptive density dependence of avian clutch size. Ecology 81, 3391–3403. Cowan, D. P. 1981 Parental investment in two solitary wasps Ancistrocerus adiabatus and Euodynerus foraminatus (Eumenidae: Hymenoptera). Behav. Ecol. Sociobiol. 9, 95– 102. Garnett, M. C. 1981 Body size, its heritability and influence on juvenile survival among great tits. Ibis 23, 31–42. Godfray, H. C. J. 1987 The evolution of clutch size in invertebrates. Oxf. Surv. Evol. Biol. 4, 117–154. Godfray, H. C. J., Partridge, L. & Harvey, P. H. 1991 Clutch size. A. Rev. Ecol. Syst. 22, 409–429. Hardy, I. C. W. & Blackburn, T. M. 1991 Brood guarding in a bethylid wasp. Ecol. Entomol. 16, 55–62. Hardy, I. C. W., Griffiths, N. T. & Godfray, H. C. J. 1992 Clutch size in a parasitoid wasp: a manipulation experiment. J. Anim. Ecol. 61, 121–129. Krebs, J. R. & Perrins, C. M. 1978 Behaviour and population regulation in the great tit (Parus major). In Population control by social behaviour (ed. F. J. Ebling & D. M. Stoddard), pp. 23–47. London: Institute of Biology. Lozano, G. A. 1994 Size, condition, and territory ownership in male tree swallows (Tachycineta bicolor). Can. J. Zool. 72, 330–333. Mangel, M., Rosenheim, J. A. & Adler, F. R. 1994 Clutch size, offspring performance and intergenerational fitness. Behav. Ecol. 5, 412–417.

Proc. R. Soc. Lond. B (2004)

Martin, K. 1991 Experimental evaluation of age, body size, and experience in determining territory ownership in willow ptarmigan. Can. J. Zool. 69, 1834–1841. Maynard Smith, J. 1982 Evolution and the theory of games. Cambridge University Press. Mesterton-Gibbons, M. 2001 An introduction to game-theoretic modelling, 2nd edn. Providence, RI: American Mathematical Society. Minot, E. & Perrins, C. M. 1986 Interspecific interference competition: nest sites for blue and great tits. J. Anim. Ecol. 55, 331–350. Petersen, G. & Hardy, I. C. W. 1996 The importance of being larger: parasitoid intruder–owner contests and their implications for clutch size. Anim. Behav. 51, 1363–1373. Renison, D., Boersma, D. & Martella, M. B. 2002 Winning and losing: causes for variability in outcome of fights in male Magellanic penguins (Spheniscus magelanicus). Behav. Ecol. 13, 462–466. Riechert, S. E. 1998 Game theory and animal contests. In Game theory and animal behavior (ed. L. A. Dugatkin & H. K. Reeve), pp. 64–93. New York: Oxford University Press. Sandell, M. & Smith, H. G. 1991 Dominance, prior occupancy, and winter residency in the great tit (Parus major). Behav. Ecol. Sociobiol. 29, 147–152. Schneider, J. M. & Lubin, Y. 1997 Infanticide by males in a spider with suicidal maternal care, Stegodyphus lineatus (Eresidae). Anim. Behav. 54, 305–312. Stearns, S. C. 1992 The evolution of life histories. Oxford University Press. Stokkebo, S. & Hardy, I. C. W. 2000 The importance of being gravid: egg load and contest outcome in a parasitoid wasp. Anim. Behav. 59, 1111–1118. Wilson, K. 1994 Evolution of clutch size in insects. II. A test of static optimality models using the beetle Callosobruchus maculatus (Coleoptera: Bruchidae). J. Evol. Biol. 7, 365– 386. Wilson, K. & Lessells, C. M. 1994 Evolution of clutch size in insects. I. A review of static optimality models. J. Evol. Biol. 7, 339–363. Zaviezo, T. & Mills, N. 2000 Factors influencing the evolution of clutch size in a gregarious insect parasitoid. J. Anim. Ecol. 69, 1047–1057. As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.