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Parental effort and paternity
Anim. Behuv., 1995,50, 1635-1644
Parental effort and paternity ALASDAIR School of Biological
I. HOUSTON
Sciences, University
of Bristol
(Received 13 May 1994: initial ucceptance 2 August 1994; final acceptance 26 April 199.5; MS. number: 4660)
Abstract. When a male’s paternity differs from one breeding attempt to another, the male’s parental effort is expected to increase as paternity increases. Davies et al. (1992, Anim. Behav., 43, 729-745) and Whittingham et al. (1993, Anim. Behav., 46, 139-147) manipulated paternity in monogamous dunnocks, Prunella modularis, and tree swallows, Tachycineta bicolor, respectively and found that parental effort did not vary with paternity. Previous discussions of this topic are reviewed and various examples presented to illustrate that the optimal parental effort may not depend strongly on paternity. In contrast to Whittingham et al., it is argued that the existing data do not indicate a discontinuous relationship between effort and paternity. Reasons to be sceptical about attempts to infer the shape of the relationship between parental effort and offspring survival from the form of the relationship between :i: 1995 The Association for the Study of Animal Behaviour paternity and parental effort are also presented. The relationship between paternity and parental effort has frequently been the topic of both theoretical and empirical investigation. Maynard Smith (1978) and Grafen (1980) pointed out that if paternity is the same in all breeding attempts, then it should have no effect on the optimal form of parental behaviour. In contrast, when paternity varies from one attempt to another, the optimal effort of a male should be positively correlated with paternity (e.g. Houston & Davies 1985; Winkler 1987; Whittingham et al. 1992; Westneat & Sherman 1993). Experiments involving the manipulation of paternity have had mixed results. Maller (1988) found a positive correlation between effort and paternity in the barn swallow, Hirundo rustica, but Whittingham et al. did not find it for the tree swallow, Tachycineta bicolor. Davies et al. (1992) gave data on the parental effort of the dunnock, Prunella modularis: when more than one male is involved in the breeding attempt (polyandry), there is a positive correlation between paternity and parental effort, but there is no correlation when the breeding attempt involves one male and one female (monogamy). In this paper I expore the relationship between paternity and optimal parental effort. I show that under some circumstances the positive correlation between paternity and optimal effort may be
rather small, and hence hard to detect. I also evaluate various previous discussionsof this issue. In doing so I raise some doubts about the attempt by Whittingham et al. (1992, 1993) to infer the relationship between parental effort and offspring survival from the relationship between paternity and parental effort.
THE
FRAMEWORK
Models of optimal parental effort typically have the following form. A parent’s fitness W is the sum of its gain from the reproductive attempt under consideration (the current attempt) plus any fitness associated with other reproductive attempts. Concentrating on a male parent, W(x) =pC(x)
+ R(x)
(1)
where p is the probability that a randomly selected member of the brood under consideration is the offspring of the male. This parameter is known as paternity (e.g. Whittingham et al. 1992) or parentage (e.g. Westneat & Sherman 1993). x is the level of parental effort. Following Whittingham et al. (1992) and Westneat & Sherman (1993), I assume that 0
c 1995 The Association
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associated with extra-pair copulations, in which case the offspring are produced in the same season as the offspring represented by C(x), and reproduction that occurs if the animal survives to future breeding seasons (e.g. Whittingham et al. 1992; Westneat & Sherman 1993). C(x) is taken to be an increasing function of x and R(x) is taken to be a decreasing function of x. Instead of working with a function R(x) that decreases with x, we can work with a cost that increases with x (c.f. Westneat & Sherman 1993). As is shown below, the optimal parental effort does not change when a constant is added to equation (1). We can therefore use the fitness function W(x)=pC(x) - [R(O) - R(x)], where R(0) - R(x) is the cost associated with a level of parental care x. As Schwagmeyer & Mock (1993) pointed out, confidence of paternity is an over-worked term. I use ‘paternity’ to indicate genetic relatedness, as denoted by p in the model. Like Whittingham et al. (1992) I assume that a male can estimate p and respond accordingly, but cannot identify its own offspring within a brood. Westneat & Sherman (1993) refer to this as a facultative response to indiscriminate cues. The value of x that maximizes W is the optimal level of parental effort x*. The empirical relationship between effort and paternity will be called the effort-paternity function, and the relationship between x* and p will be called the optimal effort-paternity function. In some cases x*=0 if paternity is less than or equal to a critical level p, of paternity. As Whittingham et al. (1992) showed, sometimes the optimal effort-paternity function jumps. In such cases, x*=0 for ppc, x*>xO. (Because there are two values of x* at the jump, the optimal effort-paternity function in this case is not a function in the mathematical sense.) Whittingham et al. (1992, 1993) refer to this jump as a threshold response. As we shall see, the existence of a critical paternity p, does not necessarily mean that the elTort&paternity function jumps. I shall say that the optimal effort-paternity function is flat if x* is virtually independent of p. This is, of course, not a precise definition, but I think that in this context precision is misplaced.
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The condition that the function has a slope of zero is too restrictive. In empirical studies it is unlikely that a small slope could be detected, and hence an optimal effort-paternity function with such a slope is flat for practical purposes. RESULTS Why Might the Optimal Function be Flat?
Effort-Paternity
The male’s fitness R(x) from other reproductive attempts will depend on a male’s paternity in breeding attempts other than the current one. If paternity is the same for all reproductive attempts, then R(x) can be written as pR,(x), from which it is clear that x* does not depend on p. The fact that optimal male parental behaviour does not depend on p in this context was noted by Maynard Smith (1978) and Grafen (1980). The experiments of Davies et al. (1992) on dunnocks and Whittingham et al. (1993) on tree swallows involve manipulation of paternity. In both cases the parental effort of males did not change with the manipulation, despite the reduction of paternity to quite low levels. Assuming that the males are able to perceive the manipulation, why might such a result be obtained? I ignore the possibility that x* is up against the constraint imposed by the upper limit x= 1. If the functions C and R are differentiable, then a necessary condition for x* is pC’(x*)+R’(x*)=O
(2)
where ’ denotes differentiation with respect to x. It can be seen from equation (2) that only the slopes of C and R are relevant to the value of x* (cf. Westneat & Sherman 1993). Thus adding a constant to C or R will not change x*. Differentiating implicitly with respect to p, it is found that dx*ldp=
- C’(.u*)l[pC”‘(x*)+R”(x*)]
(3a)
cf. Winkler (1987). Using equation (2) to substitute can be seen that
for - C’(x*), it
dx*ldp=R’(x*)l(pIpC”(x*)+R”(x*)]}
(3b)
In order for equation (2) to give a maximum rather than a minimum, W”(x*)=pC”(x*)+ R”(x*)
Houston:
Parental
then dx*ldp, as given by these equations, must be closeto zero. One difficulty with checking whether or not this is the case is that the equations require the derivates to be evaluated at the optimal effort x*, which has to be determined from equation (2). The equations do, however, give some general conclusions about the optimal effort-paternity functions. These are discussed below, together with illustrative computations of x* as a function of p.
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tion to be flat, the condition for x* to jump is based on C” being positive for low values of x and negative for high values of x. This suggests that the condition for dx*ldp to be small can be divorced from the condition for x* to jump. One way to make this point is to compute the optimal effort-paternity function in some special cases. With R(x) given by equation (4), two casesare considered. In case (i), C(x)=k,x
Review of Previous Work In this section I review various papers that have discussed the slope of the effort-paternity function. Whittingham
et al. 1992
Whittingham et al. developed a model that they related to the flatness of the effort-paternity function. Within the framework used here, they assumed that x is the proportion of time allocated to the current attempt, and that fitness is gained at a rate b from other activities, so that R(x)=b(l -x)
, (4) Under this assumption, they showed that when C”O for small x and C”
(5)
We require that pC”(x*)+R”(x*)
- k,x2
(6)
where k, and k, are positive constants such that C’>O and C”
- blp)Rk,
(7) Provided that 2k,> k, - b, x* will be lessthan or equal to one when p= 1. For x* to be positive, p must be greater than b/k,. We can therefore define a critical value p, of p given by pc=blk,, such that below p,, x* =O and above p,, x*>O. From equation (7), dx*ldp= bl(2kg’)
(8)
In case (ii), C(x)=a,x+a2x2 - a3x3, where a,, a, and a3 are positive constants such that C’>O and C”>O for small x and C”
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0.8 *w
0
0.2
l-
0.4
0.6
I / , I 0.8
1
P
0.8 : 0.6 : "R 0.4 :
Figure 1. Optimal parental effort x* as a function of paternity p when R(x)=h(l - x) for different values of h. In (a), C(x)=k,x - kg?, with k, =2 and k,= I. In (b), C(x)=c~,x+a~x* - a$, with ~1,=0.85, a,=0.35, rr,=O.5. C(x) is the number of young from the current brood that survive to independence, R(x) is the fitness from other reproductive attempts, h is the rate of gain of fitness from other activities, and the as and ks are positive constants.
x* will still be given by equation (7). k, will, however, have an effect on the fitness W(x*) that results from using the optimal effort. C(O)=&, so when Q-0 some young survive from the current brood even when the male provides no help. When p is below pc, x*=0 and so W(x*)=pk,+b. This shows that when k,zO, W(x*) increases with paternity even when the male is providing no help. If k,,=O, then no young survive from the current brood if the male does not help and the male’s fitness for p
(I 993)
Westneat & Sherman (1993) addressed the general issue of how p influences optimal parental care. They stated ‘Reduced parentage will have little effect when there are large differences between the benefit curve and the cost curve
E-
0.6
0.6 0.8 1 P Figure 2. Optimal parental effort x* as a function of paternity p for C(X)=(I,X+U~X* -a3x3 and R(x)= b( 1 - x). From left to right, the curves are for the following parameter values: a, =O, az= 1.5, a3 = 1.O, b=O.l; rr,=O.9, a,=0.8, u,=O.8, h=0.35; a,=1.2, a,=0.25, u,=O.S, h=0.5; a,=1.2, a,=0.15, a,=0.5, b=0.6; a,=1.5, ti2=o,1, a,=0.5, b=0.9; a,=l.5, a,=0.05, u,=o.5, h=l.O. See Fig. 1 for definitions of C(x) and R(x). 0
0.2
0.4
Conversely, reduced parentage will have a dramatic effect when the difference between the benefit curve and the cost curve is slight’ (page 73). Equation (2) shows that the optimal effort x* depends on just the first derivate of R and C. It can be seen from equation (3b) that how x* changes with p depends on the first and second derivative of R and on the second derivative of C. It follows that arbitrary constants can be added to R and C without changing dx*ldp, and hence the difference between R and C is irrelevant to dx*ldp. Westneat & Sherman illustrated their argument with reference to a figure (their Figure 4). The caption to this figure presents an explanation of the effect of p that seems to be different from the one in the text that I have just quoted. As far as I can see, it is not correct. Figure 3 is based on Figure 4a of Westneat & Sherman. Using PE to stand for parental effort, they state in the caption to this figure: ‘Increasing PE has a dramatic effect on offspring survival, with a minimal effect on the costs. Because optimal PE is at the top of a steep part of the benefit curve, reducing a parentage to 0.5 has only a slight effect on optimal parental effort’. The first sentence of this quotation might mean that the slope of the gain curve is greater than the slope of the cost curve, but at the optimum, the slopes are equal. The second sentence suggests that the steep rise of C(x) followed by a less steep region is important. It is obvious, however, that the benefit curves can be varied for
Houston:
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0.8 ++X
0.6 0.4
0.2
0
0.4 Parental
0.6 effort x
0.8
1
Figure 3. The effect of a change in paternity p from 1.Oto 0.5 on the optimal parental effort x* (based on Figure 4a of Westneat & Sherman 1993). The top two curves are PC(X) for p=l.O and 05, the bottom curve is the cost R(0) - R(x). When p is reduced from 1.0 to 0.5, the optimal effort falls from x*(1.0) to x*(0.5). See Fig. 1 for definitions of C(x) and R(x).
x below the lower x* (provided that C(x) increases in x) without changing how p influences x*. Equations (3a) and (3b) show that dx*ldp depends on the second derivative of C and R at the optimum. These derivatives do not seem to be included in the argument of Westneat & Sherman. Davies
et al.
(I 992)
Davies et al. (1992) suggested that the flat effort-paternity function in monogamous dunnecks is related to the fact that future reproductive success is low. This idea is plausible. Let us assume that R(x) is based on future matings, so that R(x)=S(x)F, where S(x) is the probability of surviving until the next breeding attempt as a function of x and F is the future reproductive success. Then from equation (3b) dx*ldp=
S’(x*)l@[pC”(x*)lF+
s”(x*)]}
which suggests that a decrease in F will act somewhat like an increase in p and hence will tend to make the optimal effort-paternity function flat. If we identify 1 - x in the model of Whittingham et al. (1992) with S(x) and identify b with F, then when C(x) is given by equation (6), it is clear from equation (8) that a decrease in F will decrease dx*ldp and hence will tend to make the optimal effort-paternity function flat. As the following examples show, this effect is not general.
0
0.2
0.4
0
0.2
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1
0.6
0.8
1
*x
P
Figure 4. Optimal parental effort x* as a function of paternity p when C(x)=k,x- kzx2 and R(x)= F(1 - I~x-ax’). In both (a) and (b), k,=2, k,=l and a=O.l. In (a), S=O, in (b), S=O.l. The curves are (from top to bottom) for F=0.25, 0.5, 1, 2 and 4. F=future reproductive success; a and I3 are non-negative constants. See Fig. 1 for definitions of other parameters. Case (iii) Assume that C(x)=k,xk,x2, and S(x)= (1 - Bx - ux2), where a and l3 are non-negative constants, then x* = (pk, - Ff3)/[2@k, + Fa)]. When a=O, the model is very similar to case (i). x* decreases and dx*ldp increases with F. An interesting feature of the model when fl=O is that there is no positive value ofp for which x* is zero, i.e. p,=O. It can be shown that when both a and l3 are nonnegative, dx*ldp increases with F when pk,>Fa. Thus if F
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v is a positive constant. Then x*=(u+v)- ’ In [(pNv)l(Fu)] and dx*/dp= ll[p(u+v)], which shows that dx*tdp decreaseswith p but is independent of F. Note that as in the previous models, there is a critical value pE below which x*=0. In this case, P,=(@WV). Another Way of Looking at x* A graphical analysis of whether or not p has a large effect on x* can be based on equation (2). If pC’ and - R’ are plotted on the same graph, then any intersections are candidates for x*. (An intersection may correspond to a minimum rather than a maximum.) In case (i), C(x)=k,x - k,x2 and R(x)=b(l - x). It follows that C’=k, - 2k,x and - R’= b and so at most only one intersection for a given parameter value is possible. An example of this case is given in Fig. 5a. The figure shows pC’ for p=l.O and 0.5 and b=O.2 and 0.7. When b=0.2, the intersection with C is much closer to the intersection with 0.X than when b=O.l. Figure 5b shows a similar plot for case (ii). Because C is non-monotonic, there may be two intersections. As we have already seen, at x*, C” must be negative, i.e. C’ must be decreasing. (The intersection that occurs when c’ is increasing corresponds to a local minimum.) Like Fig. 5a, Fig. 5b shows that when p is fairly high and b is low, p will have a small effort on x*. The pattern shown in the figures can be extended to casesin which R’ is not constant. Provided that p is fairly high and R’ is low in the region where the intersections that determine the optimal effect occur, x* will not be very sensitive top. The same conclusion is suggested by equation (3b). Infanticide When a male tree swallow is very unlikely to be the father of a brood, he may kill the young (Robertson & Stutchbury 1988; Robertson 1990). This infanticide is not explicitly incorporated in the models of Whittingham et al. (1992) or Westneat & Sherman (1993). One way to include it is to assume that a male can either not kill the current brood (this includes caring and not caring) resulting in fitness IV,, or can kill the brood and perhaps breed again, resulting in fitness IV,. If the male’s paternity of the current brood is p and he adopts the optimal effort, then IV,,= w[x*@)]. If the male’s paternity has the same value in all breeding attempts, there does not seem to be any
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reason for the male to base his decision about caring versus killing on paternity. I assume that the male’s paternity can vary from one breeding attempt to another, but its value in any attempt is independent of its value in previous attempts. (This does not preclude the possibility that different males have different mean values of paternity.) Under this assumption Wi is independent of the paternity p associated with the brood that the male might kill. Wi may, of course, depend on the male under consideration and the time of year, A male should commit infanticide if IV,> W,. As p decreases, W, either decreasesor remains constant. This means that if infanticide is optimal for some value pi of paternity, it is optimal for all p
where rO,r,, r2 and r3 are chosen such that R’O for small x and R”
Houston:
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effort and paternity
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(a)
Paiental
0
0.2
0.4
effort x
0.6 0.8 Parental effort x
3
Figure 5. Graphical illustrations of the condition PC’(X)= - R’(x). This is a necessary condition for parental effort to maximize fitness. In both (a) and (b), R(x)=&1 - x), so - R’=b. (a) is based on case (i), with k,=2, k,= 1. Curves are shown for ~~0.5 and 1.0 and b=0.2 and 0.7. When b=0.2, x*(1.0)=0.9 and x*(0.5)=0.8. When bzO.7, x*(1.0)=0.65 and x*(0.5)=0.3. (b) is based on case (ii), with a,=0.85, a,=0.35, u,=O.5. Curves are shown forp=O.S and 1.0 and b=0.15 and 0.45. When b=0.15, x*(1.0)=0.955 and x*(0.5)=0.88. When b=0.45, x*(1.0)=0.89 and x*(0.5)=0.379. See Fig. 1 for definitions of C(x) and R(x). ’ denotes differentiation with respect to x.
infer that the function C(x) is S-shaped. The figure also shows that the optimal effort-paternity functions can be rather flat. The form of R(x) used in this case can be very close to a straight line and yet the optimal effort-paternity function may jump.
There is also a practical problem with using the presence of a jump in parental effort to establish the form of C(x). Whether or not the parental effort function jumps may be very difficult to establish. Figure la shows that even when this function is continuous, it can be zero for low
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0.8 0.6 *H 0.4
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I
I
I
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PC
Paternity Figure 6. Graphical analysis of infanticide. The solid curve gives the fitness if the male does not kill the young. Above the critical level of paternity pc he helps to care for the young (i.e. optimal parental effort x*>O), whereas below p= he does not (i.e. x*=0). The three broken lines correspond to three values of the fitness W, if the male kills the young. (1) IV, is quite low; as paternity is decreased the pattern is care, no care. (2) W, is higher; the pattern is care, no care, infanticide. (3) U: is still higher; the pattern is care, infanticide. values of p and then rise very steeply once p is greater than pc. This suggests that a continuous function might be mistaken for a discontinuous one. It is also possible for the jump to be quite small (see Fig. 2), in which case it might be overlooked, so that a discontinuous function would be mistaken for a continuous one. It turns out that in their attempt to infer the form of C(x), Whittingham et al. (1992, 1993) identified a flat effort-paternity function with a discontinuous effort response function (a threshold response in their terminology). For example, Whittingham et al. (1993) wrote: We predicted that in many monogamous species of birds, males will not decrease their level of parental care until their confidence of paternity is very low (threshold response). (page 144) . . we tested the prediction that male parental care will exhibit a threshold (discontinuous) response to decreasing confidence of paternity in the monogamous tree swallow . We predicted that experimental changes in confidence of paternity would have no effect on the level of male parental care until confidence of paternity was very low. (page 140)
0.2
0.4
0.6
0.8
1
P
Figure 7. Optimal parental effort x* as a function of paternity p when C(x)=k,x - kzx2 and R(x)=r, r,x - r2x - r3x3, with constants k,=2, k,=l, r,=2, r, =0.52, rz= - 0.7 and (from top to bottom) r,=0.4, 0.5, 0.7, 1.1, 1.9. See Fig. 1 for definitions of C(x) and R(x).
In summarizing data from Whittingham et al. (1993) wrote:
tree
swallows,
. males do not decrease their level of parental care when their confidence of paternity is reduced by up to 50% (this study); but provide no parental care and usually kill the nestlings (Robertson 1990) when their confidence of paternity in a brood is very low or zero. From these studies, we suggest that male tree swallows exhibit a threshold response to decreasing confidence of paternity. (page 144) I would argue that what the data show is a flat effort-paternity function for relatively high values of p. The occurrence of infanticide at low values of p is not the form of discontinuity predicted by Whittingham et al. (1992) and rather than supporting the model, it obscures the effort-paternity function in this region, so that the presence of a jump in parental effort cannot be established. In discussing the work of Davies et al. (1992), Whittingham et al. (1992) wrote: ‘when dunnocks mate monogamously, reductions in paternity . do not influence the level of male parental care . Therefore, the behavior of monogamously mated male dunnocks appears to be consistent with the threshold relationship between male parental care and paternity’ (page 1122). In fact, the dunnocks’ behaviour is no more consistent with a threshold relationship than with a continuous relationship.
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DISCUSSION In this paper I have investigated the optimal response of a male to its level of paternity p. I have assumed that the value of p associated with the current reproductive attempt is not necessarily the value associated with future attempts. I have also assumed that the male can accurately assess p and set its level of parental effort accordingly. I have ignored any explicit representation of the parental effort of the female. When the adaptive efforts of both parents are analysed, a gametheoretic approach is required (e.g. Houston & Davies 198.5). Although I have not presented a rigorous analysis of models of parental effort based on equation (1) general trends have emerged in the results presented here and in related calculations. There is sometimes a critical level of paternity p, below which x* =O. p, can be thought of as a threshold level of paternity that is necessary for a male to care for the young. Given that this threshold occurs both when the optimal effort-paternity function is continuous and when it jumps, the use by Whittingham et al. (1992, 1993) of ‘threshold response’ for the case in which the function jumps might cause confusion. ? suggest that in future the term ‘threshold response’ should not be used to mean that the optimal effort-paternity function jumps. In the region where x* is a continuous function of p, the slope dx*/dp decreases as p increases. The model of Whittingham et al. (1992) assumes that R(x)=b(l - x). In this model these two properties hold, provided that C(0) is not infinite and C’(x) decreases for large x. Furthermore, dx*ldp increases with increasing b. Thus in this model, a flat optimal effort-paternity function is associated with high values of p and low values of b. Davies et al. (1992) suggested that a flat effort-paternity function is likely to be associated with a low value of future reproductive success. Although this relationship holds in some models, it is not universal. In case (v), the sign of R” changes when the optimal effort-paternity function jumps. This case, together with the results of Whittingham et al. (1992) suggests that a jump can occur when either R” or C” changes sign. Westneat & Sherman (1993) claimed that the effect of p on x * is determined by whether the curve describing the costs of parental care and the benefits of parental care are close together or
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far apart. This claim is based on a single graph (their Figure 4, repeated as Figure 2 in Owens 1993). I have pointed out that the absolute values of the cost and benefit curves are irrelevant to x* and dx*ldp. The general implication of equation (3a) or (3b) is that dx*/dp depends on the second derivatives C” and R” evaluated at x*. Any explanation of why x* is flat as a function ofp that does not mention these second derivatives is unlikely to be a correct general explanation. Whittingham et al. (1992) stated that it might be difficult to decide whether C is concave-down (i.e. C” always negative) or ‘S-shaped’ (i.e. C” positive for small x and negative for large x) ‘because in both cases birds may be operating in the concave-down portion of the curve’ (page 1120). Both the analytical argument that I have presented and the graphical argument of Whittingham et al. (1992) show that if R(x)= b( 1 - x) and birds are adopting the optimal parental effort then, when C” is positive for small x and negative for large x, they must be operating in the concave-down region, i.e. C” must be negative. Whittingham et al. (1992, 1993) suggested that C is ‘S-shaped’ for monogamous dunnocks and tree swallows. The argument can be summarized as follows. (1) If R(x)=b(l - x), then a threshold response (i.e. a jump in x*) means that C is ‘S-shaped’. (2) The data from tree swallows and dunnocks show a threshold response. There are major problems with both steps of this argument. (1) The assumption that R(x)=b(l -x) is crucial. If R(x) is not linear in x, then as shown above, a jump can occur when C” is negative for all x. Despite the importance of this assumption, it is not mentioned by Whittingham et al. (1993). The fact that the conclusions of Whittingham et al. (1992) are not robust to changes in the form of R(x) does not invalidate their model. It means, however, that their result cannot be used to make inferences about C unless there are good reasons to believe that R(x)=b( 1 - x). They present an argument in favour of this relationship (page 1117) which I do not understand. I find it hard to see how such a particular relationship can be expected to hold in general, especially as future reproductive success can be based on a range of activities, including searching for mates, achieving extra-pair copulations and surviving to breed at a later time (Whittingham et al. 1992, page 1116). Whittingham et al. (1992, 1993) provided
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no empirical evidence for this form of R(x) in the case of monogamous dunnocks or tree swallows. (2) The data from monogamous dunnocks and tree swallows do not show a threshold response (i.e. a jump). What they do show is a flat effortpaternity function. Infanticide in tree swallows is not an example of a jump as envisaged in the model of Whittingham et al. (1992). In fact infanticide can be thought of as masking the effortpaternity function at low values of p. Even in the absence of infanticide, a jump is likely to be very hard to detect. Given these problems, it seems premature to draw any conclusions about the form of the relationship between parental effort and offspring survival.
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proportional to b. The constant of proportionality is C’(x,) which by (A7) does not depend on b.
ACKNOWLEDGMENTS This work was begun while I was in the NERC Unit of Behavioural Ecology, Department of Zoology, Oxford. I thank Tim Birkhead, Jose Garcia, Ben Hatchwell, John Lazarus, John McNamara, Wolfgang Weisser, Nicky Welton, David Westneat and Jon Wright for comments on an early version of the manuscript and NERC for financial support.
REFERENCES APPENDIX Analysis of the Jump in the Optimal Effort-Paternity Function If the optimal effort-paternity function jumps at x=x0, then x0 can be found from the following two conditions Hqx,) = 0
641)
Davies, N. B., Hatchwell, B. J., Robson, T. & Burke, T. 1992. Paternity and parental effort in dunnocks Prunellu modularis: how good are male chick-feeding rules? Anim. Behav., 43, 129-145. Grafen, A. 1980. Opportunity cost, benefit and degree of relatedness. Anim. Behav., 28, 967-968. Houston, A. I. & Davies, N. B. 1985. The evolution of cooperation and life-history in the dunnock. In: Behavioural Adaptive I I
and From equation
va (Al),
= Wxo)
pC’(x,)+R’(x,)=0 and from equation
642) (A3)
(A2),
PC(Xo) + N%) =Pc(o) + W) (A4) In the model of Whittingham et al. (1992) R(x)=b(l - x), and so, from equation (A3), pC’x,=b,
and from equation It follows
(AS)
(A4)
PCW - bxo=pW (‘46) from equations (A5) and (A6) that
[C(%)- mw% = @b,> In case (ii), C(x)=aox+a,x2 It follows
from equation x,=a,/2a,
(A71
- a2x3.
(A7) that (A@
It can be seen from this equation that x0 is independent of b. The value of p at which the jump occurs does depend on b. From equation (A5) it can be seen that paternity at the jump is
Ecology:
Ecological
Consequences
of
Behavio&(Ed. by k. M. Sibly & R. G. Smith). DD. 471488. Oxford: Blackwell Scientific Publications. Maynard Smith, J. 1978. The Evohaion of Sex. Cambridge: Cambridge University Press. Moller, A. P. 1988. Paternity and parental care in the swallow, Hirundo rustica. Anim. Behav., 36,9961005. Owens, I. P. F. 1993. When kids just aren’t worth it: cuckoldry and parental care. Trends Ecol. Evol., 8, 269-27 1, Robertson, R. J. 1990. Tactics and counter-tactics of sexually selected infanticide in tree swallows. In: Population Approach
_ .
Biology
of Passerine
Birds:
an Integrated
(Ed. by J. Blondel, A. Gosler, J. D. Lebreton & R. McCleery),. __ pp. 381-390. Berlin: Springer-Verlag. Robertson. R. J. & Stutchburv. B. J. 1988. Exaerimental evidence for sexually selected infanticide in tree swallows. Anim. Behav., 36, 749-753. Schwagmeyer, P. L. & Mock, D. W. 1993. Shaken confidence of paternity. Anim. Behav., 46, 1020-1022. Westneat, D. F. & Sherman, P. W. 1993. Parentage and the evolution of parental behavior. Behav. Ecol., 4, 6671. Whittingham, L. A., Dunn, P. 0. & Robertson, R. J. 1993. Confidence of paternity and male parental care: an experimental study in tree swallows. Anim. Behav., 46, 139-147. Whittingham, L. A., Taylor, P. D. & Robertson, R. J. 1992. Confidence of paternity and male parental care. Am. Nut., 139, 1115-1125. Winkler, D. W. 1987. A general model for parental care. Am.
Nut.,
130, 526543.
Parental effort and paternity ALASDAIR School of Biological
I. HOUSTON
Sciences, University
of Bristol
(Received 13 May 1994: initial ucceptance 2 August 1994; final acceptance 26 April 199.5; MS. number: 4660)
Abstract. When a male’s paternity differs from one breeding attempt to another, the male’s parental effort is expected to increase as paternity increases. Davies et al. (1992, Anim. Behav., 43, 729-745) and Whittingham et al. (1993, Anim. Behav., 46, 139-147) manipulated paternity in monogamous dunnocks, Prunella modularis, and tree swallows, Tachycineta bicolor, respectively and found that parental effort did not vary with paternity. Previous discussions of this topic are reviewed and various examples presented to illustrate that the optimal parental effort may not depend strongly on paternity. In contrast to Whittingham et al., it is argued that the existing data do not indicate a discontinuous relationship between effort and paternity. Reasons to be sceptical about attempts to infer the shape of the relationship between parental effort and offspring survival from the form of the relationship between :i: 1995 The Association for the Study of Animal Behaviour paternity and parental effort are also presented. The relationship between paternity and parental effort has frequently been the topic of both theoretical and empirical investigation. Maynard Smith (1978) and Grafen (1980) pointed out that if paternity is the same in all breeding attempts, then it should have no effect on the optimal form of parental behaviour. In contrast, when paternity varies from one attempt to another, the optimal effort of a male should be positively correlated with paternity (e.g. Houston & Davies 1985; Winkler 1987; Whittingham et al. 1992; Westneat & Sherman 1993). Experiments involving the manipulation of paternity have had mixed results. Maller (1988) found a positive correlation between effort and paternity in the barn swallow, Hirundo rustica, but Whittingham et al. did not find it for the tree swallow, Tachycineta bicolor. Davies et al. (1992) gave data on the parental effort of the dunnock, Prunella modularis: when more than one male is involved in the breeding attempt (polyandry), there is a positive correlation between paternity and parental effort, but there is no correlation when the breeding attempt involves one male and one female (monogamy). In this paper I expore the relationship between paternity and optimal parental effort. I show that under some circumstances the positive correlation between paternity and optimal effort may be
rather small, and hence hard to detect. I also evaluate various previous discussionsof this issue. In doing so I raise some doubts about the attempt by Whittingham et al. (1992, 1993) to infer the relationship between parental effort and offspring survival from the relationship between paternity and parental effort.
THE
FRAMEWORK
Models of optimal parental effort typically have the following form. A parent’s fitness W is the sum of its gain from the reproductive attempt under consideration (the current attempt) plus any fitness associated with other reproductive attempts. Concentrating on a male parent, W(x) =pC(x)
+ R(x)
(1)
where p is the probability that a randomly selected member of the brood under consideration is the offspring of the male. This parameter is known as paternity (e.g. Whittingham et al. 1992) or parentage (e.g. Westneat & Sherman 1993). x is the level of parental effort. Following Whittingham et al. (1992) and Westneat & Sherman (1993), I assume that 0
c 1995 The Association
1635
for the Study
of
Animal Behaviour
1636
Animal
Behaviour,
associated with extra-pair copulations, in which case the offspring are produced in the same season as the offspring represented by C(x), and reproduction that occurs if the animal survives to future breeding seasons (e.g. Whittingham et al. 1992; Westneat & Sherman 1993). C(x) is taken to be an increasing function of x and R(x) is taken to be a decreasing function of x. Instead of working with a function R(x) that decreases with x, we can work with a cost that increases with x (c.f. Westneat & Sherman 1993). As is shown below, the optimal parental effort does not change when a constant is added to equation (1). We can therefore use the fitness function W(x)=pC(x) - [R(O) - R(x)], where R(0) - R(x) is the cost associated with a level of parental care x. As Schwagmeyer & Mock (1993) pointed out, confidence of paternity is an over-worked term. I use ‘paternity’ to indicate genetic relatedness, as denoted by p in the model. Like Whittingham et al. (1992) I assume that a male can estimate p and respond accordingly, but cannot identify its own offspring within a brood. Westneat & Sherman (1993) refer to this as a facultative response to indiscriminate cues. The value of x that maximizes W is the optimal level of parental effort x*. The empirical relationship between effort and paternity will be called the effort-paternity function, and the relationship between x* and p will be called the optimal effort-paternity function. In some cases x*=0 if paternity is less than or equal to a critical level p, of paternity. As Whittingham et al. (1992) showed, sometimes the optimal effort-paternity function jumps. In such cases, x*=0 for p
50, 6
The condition that the function has a slope of zero is too restrictive. In empirical studies it is unlikely that a small slope could be detected, and hence an optimal effort-paternity function with such a slope is flat for practical purposes. RESULTS Why Might the Optimal Function be Flat?
Effort-Paternity
The male’s fitness R(x) from other reproductive attempts will depend on a male’s paternity in breeding attempts other than the current one. If paternity is the same for all reproductive attempts, then R(x) can be written as pR,(x), from which it is clear that x* does not depend on p. The fact that optimal male parental behaviour does not depend on p in this context was noted by Maynard Smith (1978) and Grafen (1980). The experiments of Davies et al. (1992) on dunnocks and Whittingham et al. (1993) on tree swallows involve manipulation of paternity. In both cases the parental effort of males did not change with the manipulation, despite the reduction of paternity to quite low levels. Assuming that the males are able to perceive the manipulation, why might such a result be obtained? I ignore the possibility that x* is up against the constraint imposed by the upper limit x= 1. If the functions C and R are differentiable, then a necessary condition for x* is pC’(x*)+R’(x*)=O
(2)
where ’ denotes differentiation with respect to x. It can be seen from equation (2) that only the slopes of C and R are relevant to the value of x* (cf. Westneat & Sherman 1993). Thus adding a constant to C or R will not change x*. Differentiating implicitly with respect to p, it is found that dx*ldp=
- C’(.u*)l[pC”‘(x*)+R”(x*)]
(3a)
cf. Winkler (1987). Using equation (2) to substitute can be seen that
for - C’(x*), it
dx*ldp=R’(x*)l(pIpC”(x*)+R”(x*)]}
(3b)
In order for equation (2) to give a maximum rather than a minimum, W”(x*)=pC”(x*)+ R”(x*)
Houston:
Parental
then dx*ldp, as given by these equations, must be closeto zero. One difficulty with checking whether or not this is the case is that the equations require the derivates to be evaluated at the optimal effort x*, which has to be determined from equation (2). The equations do, however, give some general conclusions about the optimal effort-paternity functions. These are discussed below, together with illustrative computations of x* as a function of p.
effort and paternity
1637
tion to be flat, the condition for x* to jump is based on C” being positive for low values of x and negative for high values of x. This suggests that the condition for dx*ldp to be small can be divorced from the condition for x* to jump. One way to make this point is to compute the optimal effort-paternity function in some special cases. With R(x) given by equation (4), two casesare considered. In case (i), C(x)=k,x
Review of Previous Work In this section I review various papers that have discussed the slope of the effort-paternity function. Whittingham
et al. 1992
Whittingham et al. developed a model that they related to the flatness of the effort-paternity function. Within the framework used here, they assumed that x is the proportion of time allocated to the current attempt, and that fitness is gained at a rate b from other activities, so that R(x)=b(l -x)
, (4) Under this assumption, they showed that when C”
(5)
We require that pC”(x*)+R”(x*)
- k,x2
(6)
where k, and k, are positive constants such that C’>O and C”
- blp)Rk,
(7) Provided that 2k,> k, - b, x* will be lessthan or equal to one when p= 1. For x* to be positive, p must be greater than b/k,. We can therefore define a critical value p, of p given by pc=blk,, such that below p,, x* =O and above p,, x*>O. From equation (7), dx*ldp= bl(2kg’)
(8)
In case (ii), C(x)=a,x+a2x2 - a3x3, where a,, a, and a3 are positive constants such that C’>O and C”>O for small x and C”
Animal
1638
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50, 6
0.8 *w
0
0.2
l-
0.4
0.6
I / , I 0.8
1
P
0.8 : 0.6 : "R 0.4 :
Figure 1. Optimal parental effort x* as a function of paternity p when R(x)=h(l - x) for different values of h. In (a), C(x)=k,x - kg?, with k, =2 and k,= I. In (b), C(x)=c~,x+a~x* - a$, with ~1,=0.85, a,=0.35, rr,=O.5. C(x) is the number of young from the current brood that survive to independence, R(x) is the fitness from other reproductive attempts, h is the rate of gain of fitness from other activities, and the as and ks are positive constants.
x* will still be given by equation (7). k, will, however, have an effect on the fitness W(x*) that results from using the optimal effort. C(O)=&, so when Q-0 some young survive from the current brood even when the male provides no help. When p is below pc, x*=0 and so W(x*)=pk,+b. This shows that when k,zO, W(x*) increases with paternity even when the male is providing no help. If k,,=O, then no young survive from the current brood if the male does not help and the male’s fitness for p
(I 993)
Westneat & Sherman (1993) addressed the general issue of how p influences optimal parental care. They stated ‘Reduced parentage will have little effect when there are large differences between the benefit curve and the cost curve
E-
0.6
0.6 0.8 1 P Figure 2. Optimal parental effort x* as a function of paternity p for C(X)=(I,X+U~X* -a3x3 and R(x)= b( 1 - x). From left to right, the curves are for the following parameter values: a, =O, az= 1.5, a3 = 1.O, b=O.l; rr,=O.9, a,=0.8, u,=O.8, h=0.35; a,=1.2, a,=0.25, u,=O.S, h=0.5; a,=1.2, a,=0.15, a,=0.5, b=0.6; a,=1.5, ti2=o,1, a,=0.5, b=0.9; a,=l.5, a,=0.05, u,=o.5, h=l.O. See Fig. 1 for definitions of C(x) and R(x). 0
0.2
0.4
Conversely, reduced parentage will have a dramatic effect when the difference between the benefit curve and the cost curve is slight’ (page 73). Equation (2) shows that the optimal effort x* depends on just the first derivate of R and C. It can be seen from equation (3b) that how x* changes with p depends on the first and second derivative of R and on the second derivative of C. It follows that arbitrary constants can be added to R and C without changing dx*ldp, and hence the difference between R and C is irrelevant to dx*ldp. Westneat & Sherman illustrated their argument with reference to a figure (their Figure 4). The caption to this figure presents an explanation of the effect of p that seems to be different from the one in the text that I have just quoted. As far as I can see, it is not correct. Figure 3 is based on Figure 4a of Westneat & Sherman. Using PE to stand for parental effort, they state in the caption to this figure: ‘Increasing PE has a dramatic effect on offspring survival, with a minimal effect on the costs. Because optimal PE is at the top of a steep part of the benefit curve, reducing a parentage to 0.5 has only a slight effect on optimal parental effort’. The first sentence of this quotation might mean that the slope of the gain curve is greater than the slope of the cost curve, but at the optimum, the slopes are equal. The second sentence suggests that the steep rise of C(x) followed by a less steep region is important. It is obvious, however, that the benefit curves can be varied for
Houston:
Parental
effort
and paternity
1639
0.8 ++X
0.6 0.4
0.2
0
0.4 Parental
0.6 effort x
0.8
1
Figure 3. The effect of a change in paternity p from 1.Oto 0.5 on the optimal parental effort x* (based on Figure 4a of Westneat & Sherman 1993). The top two curves are PC(X) for p=l.O and 05, the bottom curve is the cost R(0) - R(x). When p is reduced from 1.0 to 0.5, the optimal effort falls from x*(1.0) to x*(0.5). See Fig. 1 for definitions of C(x) and R(x).
x below the lower x* (provided that C(x) increases in x) without changing how p influences x*. Equations (3a) and (3b) show that dx*ldp depends on the second derivative of C and R at the optimum. These derivatives do not seem to be included in the argument of Westneat & Sherman. Davies
et al.
(I 992)
Davies et al. (1992) suggested that the flat effort-paternity function in monogamous dunnecks is related to the fact that future reproductive success is low. This idea is plausible. Let us assume that R(x) is based on future matings, so that R(x)=S(x)F, where S(x) is the probability of surviving until the next breeding attempt as a function of x and F is the future reproductive success. Then from equation (3b) dx*ldp=
S’(x*)l@[pC”(x*)lF+
s”(x*)]}
which suggests that a decrease in F will act somewhat like an increase in p and hence will tend to make the optimal effort-paternity function flat. If we identify 1 - x in the model of Whittingham et al. (1992) with S(x) and identify b with F, then when C(x) is given by equation (6), it is clear from equation (8) that a decrease in F will decrease dx*ldp and hence will tend to make the optimal effort-paternity function flat. As the following examples show, this effect is not general.
0
0.2
0.4
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
*x
P
Figure 4. Optimal parental effort x* as a function of paternity p when C(x)=k,x- kzx2 and R(x)= F(1 - I~x-ax’). In both (a) and (b), k,=2, k,=l and a=O.l. In (a), S=O, in (b), S=O.l. The curves are (from top to bottom) for F=0.25, 0.5, 1, 2 and 4. F=future reproductive success; a and I3 are non-negative constants. See Fig. 1 for definitions of other parameters. Case (iii) Assume that C(x)=k,xk,x2, and S(x)= (1 - Bx - ux2), where a and l3 are non-negative constants, then x* = (pk, - Ff3)/[2@k, + Fa)]. When a=O, the model is very similar to case (i). x* decreases and dx*ldp increases with F. An interesting feature of the model when fl=O is that there is no positive value ofp for which x* is zero, i.e. p,=O. It can be shown that when both a and l3 are nonnegative, dx*ldp increases with F when pk,>Fa. Thus if F
1640
Animal
Behaviour,
v is a positive constant. Then x*=(u+v)- ’ In [(pNv)l(Fu)] and dx*/dp= ll[p(u+v)], which shows that dx*tdp decreaseswith p but is independent of F. Note that as in the previous models, there is a critical value pE below which x*=0. In this case, P,=(@WV). Another Way of Looking at x* A graphical analysis of whether or not p has a large effect on x* can be based on equation (2). If pC’ and - R’ are plotted on the same graph, then any intersections are candidates for x*. (An intersection may correspond to a minimum rather than a maximum.) In case (i), C(x)=k,x - k,x2 and R(x)=b(l - x). It follows that C’=k, - 2k,x and - R’= b and so at most only one intersection for a given parameter value is possible. An example of this case is given in Fig. 5a. The figure shows pC’ for p=l.O and 0.5 and b=O.2 and 0.7. When b=0.2, the intersection with C is much closer to the intersection with 0.X than when b=O.l. Figure 5b shows a similar plot for case (ii). Because C is non-monotonic, there may be two intersections. As we have already seen, at x*, C” must be negative, i.e. C’ must be decreasing. (The intersection that occurs when c’ is increasing corresponds to a local minimum.) Like Fig. 5a, Fig. 5b shows that when p is fairly high and b is low, p will have a small effort on x*. The pattern shown in the figures can be extended to casesin which R’ is not constant. Provided that p is fairly high and R’ is low in the region where the intersections that determine the optimal effect occur, x* will not be very sensitive top. The same conclusion is suggested by equation (3b). Infanticide When a male tree swallow is very unlikely to be the father of a brood, he may kill the young (Robertson & Stutchbury 1988; Robertson 1990). This infanticide is not explicitly incorporated in the models of Whittingham et al. (1992) or Westneat & Sherman (1993). One way to include it is to assume that a male can either not kill the current brood (this includes caring and not caring) resulting in fitness IV,, or can kill the brood and perhaps breed again, resulting in fitness IV,. If the male’s paternity of the current brood is p and he adopts the optimal effort, then IV,,= w[x*@)]. If the male’s paternity has the same value in all breeding attempts, there does not seem to be any
50, 6
reason for the male to base his decision about caring versus killing on paternity. I assume that the male’s paternity can vary from one breeding attempt to another, but its value in any attempt is independent of its value in previous attempts. (This does not preclude the possibility that different males have different mean values of paternity.) Under this assumption Wi is independent of the paternity p associated with the brood that the male might kill. Wi may, of course, depend on the male under consideration and the time of year, A male should commit infanticide if IV,> W,. As p decreases, W, either decreasesor remains constant. This means that if infanticide is optimal for some value pi of paternity, it is optimal for all p
where rO,r,, r2 and r3 are chosen such that R’
Houston:
&rental
effort and paternity
I641
(a)
Paiental
0
0.2
0.4
effort x
0.6 0.8 Parental effort x
3
Figure 5. Graphical illustrations of the condition PC’(X)= - R’(x). This is a necessary condition for parental effort to maximize fitness. In both (a) and (b), R(x)=&1 - x), so - R’=b. (a) is based on case (i), with k,=2, k,= 1. Curves are shown for ~~0.5 and 1.0 and b=0.2 and 0.7. When b=0.2, x*(1.0)=0.9 and x*(0.5)=0.8. When bzO.7, x*(1.0)=0.65 and x*(0.5)=0.3. (b) is based on case (ii), with a,=0.85, a,=0.35, u,=O.5. Curves are shown forp=O.S and 1.0 and b=0.15 and 0.45. When b=0.15, x*(1.0)=0.955 and x*(0.5)=0.88. When b=0.45, x*(1.0)=0.89 and x*(0.5)=0.379. See Fig. 1 for definitions of C(x) and R(x). ’ denotes differentiation with respect to x.
infer that the function C(x) is S-shaped. The figure also shows that the optimal effort-paternity functions can be rather flat. The form of R(x) used in this case can be very close to a straight line and yet the optimal effort-paternity function may jump.
There is also a practical problem with using the presence of a jump in parental effort to establish the form of C(x). Whether or not the parental effort function jumps may be very difficult to establish. Figure la shows that even when this function is continuous, it can be zero for low
1642
Animal
Behaviour,
50,
6
0.8 0.6 *H 0.4
I
I
I
I
0.2
0.4
0.6
0.8
0
1
PC
Paternity Figure 6. Graphical analysis of infanticide. The solid curve gives the fitness if the male does not kill the young. Above the critical level of paternity pc he helps to care for the young (i.e. optimal parental effort x*>O), whereas below p= he does not (i.e. x*=0). The three broken lines correspond to three values of the fitness W, if the male kills the young. (1) IV, is quite low; as paternity is decreased the pattern is care, no care. (2) W, is higher; the pattern is care, no care, infanticide. (3) U: is still higher; the pattern is care, infanticide. values of p and then rise very steeply once p is greater than pc. This suggests that a continuous function might be mistaken for a discontinuous one. It is also possible for the jump to be quite small (see Fig. 2), in which case it might be overlooked, so that a discontinuous function would be mistaken for a continuous one. It turns out that in their attempt to infer the form of C(x), Whittingham et al. (1992, 1993) identified a flat effort-paternity function with a discontinuous effort response function (a threshold response in their terminology). For example, Whittingham et al. (1993) wrote: We predicted that in many monogamous species of birds, males will not decrease their level of parental care until their confidence of paternity is very low (threshold response). (page 144) . . we tested the prediction that male parental care will exhibit a threshold (discontinuous) response to decreasing confidence of paternity in the monogamous tree swallow . We predicted that experimental changes in confidence of paternity would have no effect on the level of male parental care until confidence of paternity was very low. (page 140)
0.2
0.4
0.6
0.8
1
P
Figure 7. Optimal parental effort x* as a function of paternity p when C(x)=k,x - kzx2 and R(x)=r, r,x - r2x - r3x3, with constants k,=2, k,=l, r,=2, r, =0.52, rz= - 0.7 and (from top to bottom) r,=0.4, 0.5, 0.7, 1.1, 1.9. See Fig. 1 for definitions of C(x) and R(x).
In summarizing data from Whittingham et al. (1993) wrote:
tree
swallows,
. males do not decrease their level of parental care when their confidence of paternity is reduced by up to 50% (this study); but provide no parental care and usually kill the nestlings (Robertson 1990) when their confidence of paternity in a brood is very low or zero. From these studies, we suggest that male tree swallows exhibit a threshold response to decreasing confidence of paternity. (page 144) I would argue that what the data show is a flat effort-paternity function for relatively high values of p. The occurrence of infanticide at low values of p is not the form of discontinuity predicted by Whittingham et al. (1992) and rather than supporting the model, it obscures the effort-paternity function in this region, so that the presence of a jump in parental effort cannot be established. In discussing the work of Davies et al. (1992), Whittingham et al. (1992) wrote: ‘when dunnocks mate monogamously, reductions in paternity . do not influence the level of male parental care . Therefore, the behavior of monogamously mated male dunnocks appears to be consistent with the threshold relationship between male parental care and paternity’ (page 1122). In fact, the dunnocks’ behaviour is no more consistent with a threshold relationship than with a continuous relationship.
Houston:
Parental
DISCUSSION In this paper I have investigated the optimal response of a male to its level of paternity p. I have assumed that the value of p associated with the current reproductive attempt is not necessarily the value associated with future attempts. I have also assumed that the male can accurately assess p and set its level of parental effort accordingly. I have ignored any explicit representation of the parental effort of the female. When the adaptive efforts of both parents are analysed, a gametheoretic approach is required (e.g. Houston & Davies 198.5). Although I have not presented a rigorous analysis of models of parental effort based on equation (1) general trends have emerged in the results presented here and in related calculations. There is sometimes a critical level of paternity p, below which x* =O. p, can be thought of as a threshold level of paternity that is necessary for a male to care for the young. Given that this threshold occurs both when the optimal effort-paternity function is continuous and when it jumps, the use by Whittingham et al. (1992, 1993) of ‘threshold response’ for the case in which the function jumps might cause confusion. ? suggest that in future the term ‘threshold response’ should not be used to mean that the optimal effort-paternity function jumps. In the region where x* is a continuous function of p, the slope dx*/dp decreases as p increases. The model of Whittingham et al. (1992) assumes that R(x)=b(l - x). In this model these two properties hold, provided that C(0) is not infinite and C’(x) decreases for large x. Furthermore, dx*ldp increases with increasing b. Thus in this model, a flat optimal effort-paternity function is associated with high values of p and low values of b. Davies et al. (1992) suggested that a flat effort-paternity function is likely to be associated with a low value of future reproductive success. Although this relationship holds in some models, it is not universal. In case (v), the sign of R” changes when the optimal effort-paternity function jumps. This case, together with the results of Whittingham et al. (1992) suggests that a jump can occur when either R” or C” changes sign. Westneat & Sherman (1993) claimed that the effect of p on x * is determined by whether the curve describing the costs of parental care and the benefits of parental care are close together or
efSort and paternity
1643
far apart. This claim is based on a single graph (their Figure 4, repeated as Figure 2 in Owens 1993). I have pointed out that the absolute values of the cost and benefit curves are irrelevant to x* and dx*ldp. The general implication of equation (3a) or (3b) is that dx*/dp depends on the second derivatives C” and R” evaluated at x*. Any explanation of why x* is flat as a function ofp that does not mention these second derivatives is unlikely to be a correct general explanation. Whittingham et al. (1992) stated that it might be difficult to decide whether C is concave-down (i.e. C” always negative) or ‘S-shaped’ (i.e. C” positive for small x and negative for large x) ‘because in both cases birds may be operating in the concave-down portion of the curve’ (page 1120). Both the analytical argument that I have presented and the graphical argument of Whittingham et al. (1992) show that if R(x)= b( 1 - x) and birds are adopting the optimal parental effort then, when C” is positive for small x and negative for large x, they must be operating in the concave-down region, i.e. C” must be negative. Whittingham et al. (1992, 1993) suggested that C is ‘S-shaped’ for monogamous dunnocks and tree swallows. The argument can be summarized as follows. (1) If R(x)=b(l - x), then a threshold response (i.e. a jump in x*) means that C is ‘S-shaped’. (2) The data from tree swallows and dunnocks show a threshold response. There are major problems with both steps of this argument. (1) The assumption that R(x)=b(l -x) is crucial. If R(x) is not linear in x, then as shown above, a jump can occur when C” is negative for all x. Despite the importance of this assumption, it is not mentioned by Whittingham et al. (1993). The fact that the conclusions of Whittingham et al. (1992) are not robust to changes in the form of R(x) does not invalidate their model. It means, however, that their result cannot be used to make inferences about C unless there are good reasons to believe that R(x)=b( 1 - x). They present an argument in favour of this relationship (page 1117) which I do not understand. I find it hard to see how such a particular relationship can be expected to hold in general, especially as future reproductive success can be based on a range of activities, including searching for mates, achieving extra-pair copulations and surviving to breed at a later time (Whittingham et al. 1992, page 1116). Whittingham et al. (1992, 1993) provided
1644
Animal
Behaviour,
no empirical evidence for this form of R(x) in the case of monogamous dunnocks or tree swallows. (2) The data from monogamous dunnocks and tree swallows do not show a threshold response (i.e. a jump). What they do show is a flat effortpaternity function. Infanticide in tree swallows is not an example of a jump as envisaged in the model of Whittingham et al. (1992). In fact infanticide can be thought of as masking the effortpaternity function at low values of p. Even in the absence of infanticide, a jump is likely to be very hard to detect. Given these problems, it seems premature to draw any conclusions about the form of the relationship between parental effort and offspring survival.
50, 6
proportional to b. The constant of proportionality is C’(x,) which by (A7) does not depend on b.
ACKNOWLEDGMENTS This work was begun while I was in the NERC Unit of Behavioural Ecology, Department of Zoology, Oxford. I thank Tim Birkhead, Jose Garcia, Ben Hatchwell, John Lazarus, John McNamara, Wolfgang Weisser, Nicky Welton, David Westneat and Jon Wright for comments on an early version of the manuscript and NERC for financial support.
REFERENCES APPENDIX Analysis of the Jump in the Optimal Effort-Paternity Function If the optimal effort-paternity function jumps at x=x0, then x0 can be found from the following two conditions Hqx,) = 0
641)
Davies, N. B., Hatchwell, B. J., Robson, T. & Burke, T. 1992. Paternity and parental effort in dunnocks Prunellu modularis: how good are male chick-feeding rules? Anim. Behav., 43, 129-145. Grafen, A. 1980. Opportunity cost, benefit and degree of relatedness. Anim. Behav., 28, 967-968. Houston, A. I. & Davies, N. B. 1985. The evolution of cooperation and life-history in the dunnock. In: Behavioural Adaptive I I
and From equation
va (Al),
= Wxo)
pC’(x,)+R’(x,)=0 and from equation
642) (A3)
(A2),
PC(Xo) + N%) =Pc(o) + W) (A4) In the model of Whittingham et al. (1992) R(x)=b(l - x), and so, from equation (A3), pC’x,=b,
and from equation It follows
(AS)
(A4)
PCW - bxo=pW (‘46) from equations (A5) and (A6) that
[C(%)- mw% = @b,> In case (ii), C(x)=aox+a,x2 It follows
from equation x,=a,/2a,
(A71
- a2x3.
(A7) that (A@
It can be seen from this equation that x0 is independent of b. The value of p at which the jump occurs does depend on b. From equation (A5) it can be seen that paternity at the jump is
Ecology:
Ecological
Consequences
of
Behavio&(Ed. by k. M. Sibly & R. G. Smith). DD. 471488. Oxford: Blackwell Scientific Publications. Maynard Smith, J. 1978. The Evohaion of Sex. Cambridge: Cambridge University Press. Moller, A. P. 1988. Paternity and parental care in the swallow, Hirundo rustica. Anim. Behav., 36,9961005. Owens, I. P. F. 1993. When kids just aren’t worth it: cuckoldry and parental care. Trends Ecol. Evol., 8, 269-27 1, Robertson, R. J. 1990. Tactics and counter-tactics of sexually selected infanticide in tree swallows. In: Population Approach
_ .
Biology
of Passerine
Birds:
an Integrated
(Ed. by J. Blondel, A. Gosler, J. D. Lebreton & R. McCleery),. __ pp. 381-390. Berlin: Springer-Verlag. Robertson. R. J. & Stutchburv. B. J. 1988. Exaerimental evidence for sexually selected infanticide in tree swallows. Anim. Behav., 36, 749-753. Schwagmeyer, P. L. & Mock, D. W. 1993. Shaken confidence of paternity. Anim. Behav., 46, 1020-1022. Westneat, D. F. & Sherman, P. W. 1993. Parentage and the evolution of parental behavior. Behav. Ecol., 4, 6671. Whittingham, L. A., Dunn, P. 0. & Robertson, R. J. 1993. Confidence of paternity and male parental care: an experimental study in tree swallows. Anim. Behav., 46, 139-147. Whittingham, L. A., Taylor, P. D. & Robertson, R. J. 1992. Confidence of paternity and male parental care. Am. Nut., 139, 1115-1125. Winkler, D. W. 1987. A general model for parental care. Am.
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130, 526543.