The flight speed of parent birds feeding young


  • Alasdair I. Houston

Alasdair I. Houston, School of Biological Sciences, University of Bristol, Woodland Road, Bristol, BS8 1UG, UK. E-mail:


I review previous models of the speed at which parent birds should fly when delivering food to their young. Norberg gives a graphical method of finding a parent's best flight speed. This speed maximizes the overall rate at which energy is delivered to the young. An alternative assumption is that a parent maximizes the net rate of delivery of energy. I suggest that in general we cannot distinguish between net rate and overall rate on the basis of whether the parent feeds itself. The best way to distinguish between these currencies may be to use qualitative predictions. I present new results on the effect of a constraint on energy expenditure on the parent's optimal speed. I show that the optimal speed when foraging should be less than the optimal speed when traveling. I also analyse the advantage to a parent of flying faster than the maximum range speed and evaluate previous empirical studies of the speed at which parent birds fly. Only one study claims that parent birds fly at the speed identified by Norberg, but I raise doubts about this claim.

Models of optimal flight speed are based on the assumption that the rate of energy expenditure of a bird first decreases and then increases with its flight speed (e. g. Hedenström and Alerstam 1995). Denoting this rate of expenditure or power by P(v), where v is the flight speed, various speeds have been identified as candidates for the speed at which the bird should fly. One is the minimum power speed vmp. This is the speed at which power is minimized. Another is the maximum range speed vmr. This is the speed at which P(v)/v or equivalently energy spent per unit distance is minimized. As a consequence, the distance that can be flown (i.e. the range) for a given energy expenditure is maximized.

Norberg (1981) was the first person to consider how fast a parent bird should fly when delivering food to its young. Norberg assumed that the best flight speed would maximize the energy delivered to the young. Norberg's graphical procedure for finding this speed is as follows. Let the positive constant Pgain be the bird's net rate of gain. Pgain is added to the power curve to give a new curve P+Pgain. The tangent from the origin to this curve gives the optimal speed. Although Norberg defined Pgain as the rate of net energy gain during foraging, the first test of the model (McLaughlin and Montgomerie 1985) took Pgain to be the rate at which energy was delivered to the young. Subsequent work (Houston, 1986; Hedenström and Alerstam 1995) has extended Norberg's argument. The next section discusses various currencies and shows how the velocity characterised by Norberg maximizes the rate at which energy is delivered to the young.

Optimality and currencies

In their analysis of optimal flight speeds, Hedenström and Alerstam (1995) make a useful distinction between optimization criteria and immediate currencies. Optimization criteria are specifications such as “minimize the time spent foraging” or “minimize the total amount of energy spent while remaining in energy balance”. A given criterion is justified by relating it to fitness. An immediate currency is a function of behaviour that is maximized or minimized. It will typically be derived from an optimization criterion. Hedenström and Alerstam show that a given currency can be derived from more than one optimization criterion. This point is important in the context of evaluating flight speeds.

The behaviour of a parent bird feeding its young typically involves the following cyclic pattern of behaviour. A cycle starts when a parent bird leaves the nest. The bird flies to a foraging site, collects food and returns to feed the young. After the young are fed, the cycle is finished and the parent leaves to start the next cycle. Standard results in renewal theory show that the long-term rate of energy delivery is the expected energy delivered per cycle divided by the expected duration of a cycle (see Houston and McNamara 1999). In case of the parent bird, let L be the expected energy content of a load, let D be the round trip distance flown to the foraging site and back at speed v and let t be the time spent collecting the load (see Table 1 for a summary of the notation used in this paper). Then ignoring the time taken to transfer the load to the young, the expected duration of the cycle is (D/v)+t and so the gross rate of delivery is


It can be seen from this equation that the rate to the young is maximized by maximizing v, i.e. the parent should fly at its maximum possible speed. The trouble with this currency is that it ignores any costs to the parent associated with its flight speed. I now describe how a cost of flight speed based on energy expenditure leads to Norberg's result.

Table 1.  Summary of notation.
vFlight speedm/sLT−1
DRound trip flight distancemL
P(v)Power curveJ/sML2T−3
LEnergy content of loadJML2T−2
vmpMinimum power speedm/sLT−1
vmrMaximum range speedm/sLT−1
tTime taken to collect a loadsT
bParent's gross rate of gain while foragingJ/sML2T−3
cParent's rate of energy expenditure while foragingJ/sML2T−3
γ=b−c, net rate at which parent can acquire energyJ/sML2T−3
crParent's rate of energy expenditure while restingJ/sML2T−3
KParent's maximum overall rate of energy expenditureJ/sML2T−3
j(v)Energetic cost of take-offJML2T−2
Y(v)Overall rate of deliveryJ/sML2T−3
N(v)Net rate of deliveryJ/sML2T−3
vYSpeed that maximises the overall rate of delivery to the youngm/sLT−1
vNSpeed that maximises the net rate of delivery to the youngm/sLT−1
vminLowest speed at which an energy-constrained parent should flym/sLT−1

Overall rate

Houston (1986) obtained Norberg's result by considering a rate of delivery to the young which includes the time required by the parent to replace the energy spent. If the parent flies at speed v and collects food for the young for a time t, then it spends an amount of energy inline image on the flight and an amount of energy ct while collecting food. If the parent is to maintain energy balance, then it must replace the sum of these energies. (The energy spent per trip need not be replaced on every trip. The argument from renewal theory caries through if we consider the average time per trip required to replace the energy.) If the parent can acquire energy at a net rate γ, then the time to replace the energy is


We can then define the overall rate of delivery to be


It can be seen that Y is maximized by the speed that minimizes the time


From the condition θ′(v)=0, it follows that the optimal speed satisfies


which is the equation given by Norberg (1981), with Norberg's Pgain corresponding to γ. I will refer to the speed that maximizes Y (i.e. that satisfies equation (4)) as the Norberg speed vY. The graphical solution to equation (4) is shown in Fig. 1. Rather than constructing a new curve by adding γ to P as Norberg (1981) did, I have followed the equivalent procedure adopted by McLaughlin and Montgomerie (1985). In this procedure the power axis is extended to −γ, and a tangent from (0, −γ) is constructed to the power curve. An advantage of this version is that the tangent intersects the speed axis at the speed v0, which can be thought of as an overall speed given that the bird flies at speed vY. This overall speed ignores any time associated with collecting the load. To elaborate, consider a bird flying at speed v and draw a straight line connecting (0, −γ) with (v, P (v)). The broken line in Fig. 1 is an example. The line intersects the velocity axis at speed z. By elementary geometry,


which means that


Equation (3) gives the total time to fly a distance D and replace the energy spent on flight. It follows that the overall speed D/θ(v) is


which is equal to z. Thus z is the overall speed if the bird flies at speed v. The best value of v maximizes z. It follows that this best value of v can be found by taking the tangent from (0, −γ) to the power curve, and that v0 is the resulting maximum overall speed. The graphical determination of the optimal overall speed was introduced by Alerstam (1991) in the context of migration. In this context the overall speed gives the overall speed at which the bird travels, including the time taken to refuel. In the context of a parent feeding young, the overall speed is the overall speed of travel between the nest and the foraging area if the time to collect the load is ignored.

Figure 1.

graphical solution to equation (4). The tangent from (0, −γ) to the power curve P(v) gives the flight speed vY that maximizes overall rate. The tangent intersects the flight speed axis at the speed v0, where v0=D/θ(vY) is the overall speed (ignoring the time to collect the load).

The derivation that I have given makes it clear that Pgain is the rate at which the parent can replace the energy that it spends. The same conclusion is reached by Hedenström and Alerstam (1995), who refer to the currency of overall speed as the speed of transport.

This currency gives us a simple way of comparing the cost and benefit of increasing flight speed. An increase in speed always reduces the travel time D/v. If speed is below the maximum range speed vmr then an increase in speed decreases the energy spent per trip inline image and hence decreases the time required to replace this energy. The total time is the sum of these two times. It follows that if speed is below the maximum range speed an increase in speed reduces the total time. Thus the speed at which total time is minimized will be greater than vmr. Above the maximum range speed an increase in speed increases inline image and hence increases the time required to replace this energy (an example is given in Fig. 2). When the rate at which the parent can replace energy is low, the time required to replace energy will dominate and the optimal speed will by only slightly greater than vmr (Fig. 2a). When the rate at which energy can be replaced is high, the time to replace energy is relatively less important and the optimal speed may be considerably greater than vmr (Fig. 2b).

Figure 2.

The components (expressed as time per unit distance) that determine time spent flying inline image and time spent replacing energy inline image together with their sum. (i) inline image (ii) inline image.γ is low in 2a and high in 2b. The replacement time inline image is minimized at vmr and the total time inline image is minimized at vY.

It can be seen from the above analysis that the argument for flying at vY rests on the assumption that the parent bird can replace any amount of energy that it spends. Houston (1993) considers the consequences of an upper limit on the rate at which a bird can expend or acquire energy and shows that if the limit forces the bird to rest then delivery rate is maximized by minimizing inline image, where cr is the rate of energetic expenditure while resting. Ydenberg et al. (1994) reach a slightly different conclusion; see Hedenström and Alerstam (1995) and Houston (1995) for discussion. Rather than repeat the formal arguments of Hedenström and Alerstam (1995) and Houston (1993, 1995) I will now discuss energy expenditure using a graphical analysis that is similar to the approach used by McNamara and Houston (1997).

Constraint on energy expenditure

An intuitive understanding of the effect of a constraint on energy expenditure can be obtained by considering the extended set of options available to the bird if it combines flying and resting (cf. the analysis of foraging and resting in McNamara and Houston 1997). Fig. 3 shows the average power versus average speed for various combinations of resting and flying. If the bird rests all the time, its speed=zero and its power=metabolic rate cr. If the bird flies at speed v all the time, its power is P(v). For any combination of resting and flying at speed v, the average speed and average power will lie on the line joining the resting option and the point (v, P (v)) – see Fig. 3 for examples. Now assume that a bird's only decisions are its flight speed and whether or not it rests. Let K be the maximum average rate at which the bird can spend energy and let K1 be the average rate of energy expenditure if the bird flies at vY. If K≥K1, then the bird can adopt the speed vY that is given by equation (4). When K is less than K1, this is no longer possible. The energy delivered to the young increases with speed, so the bird should fly as fast as possible given the constraint. Should this involve a mixture of flying and resting? It can be seen from Fig. 3 that when the speed that can be achieved is high, it is better not to rest. The options that involve resting (shown by the line (ii)) have (in the region of interest) a higher value of P(v) for a given v than the option of flying at speed v without resting. This is true until K reaches a value K2 at which the speed is vmin. As is shown in Fig. 3, this speed can be found by constructing a tangent to the power curve from the resting option, i.e. vmin satisfies the equation


A mixture of flying at vmin and resting has a lower rate of expenditure than flying at speed less than vmin without resting. Thus the bird should never fly at speed less than vmin. (This minimum speed is analogous to the minimum net rate at which an animal should forage that is discussed by McNamara and Houston 1997.) We can now summarize the effect of a constraint as follows.

Figure 3.

Resting v. flying. When the bird rests, it uses energy at a rate cr. The tangent from (0, cr) to the power curve (line (i)) determines a flight speed vmin. If a bird is constrained by average energy expenditure, it should not fly at a speed below vmin, because its rate of energy expenditure will be given by the power curve, and a lower average rate of expenditure could be achieved by alternating between resting and flying at vmin. Line (ii) shows that for speeds above vmin, the average rate of expenditure if the bird alternates between flying and resting is above the power curve, so in this region the bird should not rest.

There are two critical values of K, K1 and K2. If K≥K1, then the bird can fly at the Norberg speed without resting, and should do so. Below K1, the bird cannot fly at speed greater then vmin without resting. If K2≤K<K1, the bird can fly at a speed greater than vmin without resting, and should do so. If K<K2, then the bird should spend some time resting and some time flying at speed vmin. The exact amount of time spent resting will depend on the value of K. The value of K1 and K2 will in general depend on details of the foraging process (e.g. distance to be flown, size of load). Once K<K2, it is optimal to fly at vmin, as given by equation (5). It can be seen from this equation and from Fig. 3 that vmin<vmr, but as Hedenström and Alerstam (1995) point out, vmin is likely to be very close to vmr. A striking feature of both vmin and vmr is that they are independent of the foraging process.

McLaughlin and Montgomerie (1985, 1990) suggested that Lapland longspurs might often be able to meet their own need and those of their young without having to fly at speeds above vmr. As McLaughlin and Montgomerie (1990) suggest, flying fast may incur a long-term cost in terms of physiological damage. We can formalize this idea by suggesting that a parent should minimize the energy expenditure associated with the delivery of a given amount of energy. In the context of flight speed, this means minimizing


which means flying at vmin. This shows that vmin can be the optimal speed under two different criteria:

  • aMaximize energy to the young in circumstances that require resting in order to keep energy expenditure to some critical level.
  • bMinimize the energy expenditure associated with delivering a given amount of energy to the young.

The example is instructive because the two criteria differ in terms of the level of energy expenditure that we expect to see. Under (a), the parent will be working at the maximum possible level, whereas under (b) the parent's level of expenditure can be much lower. This gives us an escape from the apparent paradox posed by birds flying at vmin even though the energy expenditure is below a reasonable estimate of the maximum value.

Net rate

The net rate of energetic gain is a currency that is widely used in the context of an animal foraging for its own benefit; see Stephens and Krebs (1986) for a review. A problem with using it in the case of a bird delivering food to its young is that neither the young nor the parent gets the net rate (Houston 1987). We can, of course, simply test net rate as a possible currency without offering a specific justification in terms of fitness, cf. Welham and Ydenberg (1993), Welham and Beauchamp (1997). But it is perhaps worth noting that under special assumptions net rate can be given a justification.

Assume that the parent spends a fixed time t in the foraging area, and that during this time the parent gains energy at a constant rate b and spends energy at a constant rate c. Thus its net rate of gain γ=b−c. The parent flies a round-trip distance D at speed v, so the energy expended on flight is inline image. Now assume that the parent replaces the energy spent on the trip by consuming some of the food that it obtained. Then the energy gained by the young per cycle is


and the expected duration of a cycle is t+D/v. It follows that under these assumptions the net rate


is the rate at which energy is delivered to the young. On this view, the parent is balancing its energy budget, and in doing so it imposes a cost on the young. In contrast to the overall rate model, the parent imposes the cost not by increasing the duration of the cycle to forage for itself, but by reducing the amount of energy delivered per cycle. It can be shown (see Appendix 1) that for any flight speed this results in a lower delivery rate than if the young get all the energy that is collected during the time t and the parent spends an extra amount of time foraging for itself.

As well as showing how net rate can be related to the rate at which the young obtain energy, this simple model can be used to illustrate how the speed that maximizes net rate depends on the round trip distance D. The equation for net rate (equation (6) above) can be written as


where S=D/t is the round trip distance divided by the time spent collecting food. Then from the condition that N′=0 we have


This equation can be solved graphically by an obvious extension of the method introduced by Norberg (1981). It can be seen from the equation that when S=0, we have the standard Norberg equation. As S increases from zero, the resulting optimal speed decreases from the speed vY towards the minimum power speed. This analysis shows that when rate of energetic gain is constant, optimal speed decreases as distance increases. This effect is general provided that there are no take-off costs.

In this simple net rate model, I have taken the parent to gain energy at the same rate while foraging for itself and while foraging for its young. This need not be the case (see discussion and references in Houston 1993 and Houston et al. 1996). Models that distinguish between the two foraging processes (Houston 1986, Hedenström and Alerstam 1995) show that the speed that maximizes net rate depends on the process of gathering energy for the young, whereas the speed that maximizes overall rate does not.

Daily energy balance

Blake (1985) introduced a currency called daily energy budget. de la Cueva and Blake (1997) call it daily energy balance, and use it as a currency in the context of swallows feeding their young. The currency is defined as the energy gathered during the foraging period divided by the total energy spent during the day. There is a problem with this currency. As Hedenström and Alerstam (1995) point out, there is no obvious relationship between DBAL and fitness. This means that the use of DBAL as a currency lacks a justification.

Currencies and cost of take off

In their discussion of net rate of gain and overall delivery rate, McLaughlin and Montgomerie (1990) suggest that these currencies differ in their assumptions about whether or not the parent feeds itself on a foraging trip. In particular, they suggest that if a parent is maximizing overall rate, it will feed itself during every foraging trip. This argument does not seem to give a reliable way to distinguish between currencies. The justification of overall rate as a currency is based on expected times (see above) so even if a bird is replacing energy it need not do so every trip. Thus absence of self-feeding is consistent with the maximization of overall rate. The best way to distinguish between the currencies may be to use qualitative predictions. These are summarized in Table 2.

Table 2.  Qualitative predictions about the effect of various features on the optimal flight speed for three currencies in the absence of take-off costs or an effect of load on flight costs.
 LoadParent's rate of gainDistanceYoung's needs
Net rateIncreaseIndepDecreaseIndep
Overall rateIndepIncreaseIndepIndep
Energy spentIndepIndepIndepIndep

The qualitative results presented in Table 2 are based on models that do not include any energetic cost associated with take-off. McLaughlin and Montgomerie (1985) suggest that such costs might be important, and Houston (1986) investigates these costs in the case of net rate and overall rate. I now present an analysis that unifies the minimization of energetic expenditure and the maximization of overall rate. Let j(v) be the take-off cost, which is assumed to be an increasing function of flight speed. Then the total energy spent on a trip is


and the total time for a trip (including time to replace energy) that depends on speed is


This time can be written as


which shows that the minimization of the energy spent on a trip can be thought of as a limiting case of minimizing overall time as γ tends to 0.

By differentiating the total trip time with respect to flight speed and setting the result equal to zero, it follows that the optimal speed satisfies


It can be seen from this equation that if there are no take off costs or j′=0 then it is optimal to fly at vY. When j′>0 then (i) because P″>0 the optimal speed with take-off cost is less than the optimal speed without take-off cost and (ii) the optimal speed with take-off cost tends to vY as D tends to infinity. These general features can be brought out by developing a graphical argument. Equation (8) can be written as


which can be solved graphically by subtracting the curve v2j′(v)/D from the power curve and then constructing the tangent to the resulting curve from (0, −γ).

The data

The flight speed of parent birds feeding young has been studied systematically in three species, the Lapland longspur Calcarius lapponicus (McLaughlin and Montgomerie, 1985, 1990), the black tern Chlidonias niger (Welham and Ydenberg 1993) and the swallow Hirundo rustica (Blake et al. 1990, de la Cueva and Blake 1997). The results of these studies are summarized in Table 3. The analysis of flight speeds of swallows by de la Cueva and Blake is the only study that claims to find evidence in support of birds flying at the Norberg speed vY. There is, however, a fundamental problem with the analysis presented by de la Cueva and Blake. They identify Norberg's Pgain (Edel in their Fig. 1 and Pnest in their text) with the rate at which the parent is delivering energy. As is noted by Houston (1986) and Hedenström and Alerstam (1995), Pgain is the rate at which the parent can replace energy. This rate will typically be higher than the delivery rate. In the case of the swallow, there is evidence (e.g. Turner 1983, Bryant and Tatner 1991) which suggests that the value of Pgain (γ in my notation) used by de la Cueva and Blake is too low. Because vY increases as γ increases, this means that de la Cueva and Blake (1997) have probably underestimated vY.

Table 3.  Data on flight speed of parent birds.
StudyEffect of distanceAuthors’ conclusion
Lapland longspur McLaughlin and MontgomerieSpeed increasesMinimize energy
Black tern Welham and YdenbergNo clear effectMinimize energy
Swallow de la Cueva and BlakeNot knownMaximize overall rate

The obvious way to estimate γ from the data in de la Cueva and Blake is to use Pharv, which is the parent's rate of harvesting energy in flight. de la Cueva and Blake say that


de la Cueva and Blake give units of W/kg for E. Even when we assign units of J/kg to E, the equation for Pharv does not have the right units; a term for the mean mass of the insects is missing.

Flight speed while foraging

So far, we have been concerned with the speed at which a parent bird flies while traveling to and from the foraging area. Some species forage on the wing (e.g. swifts, swallows and martins). In these cases there are three speeds that we can consider


Assume that the parent collects food at a rate b that is independent of food already collected but increases with flight speed, and that the maximum possible load is L. Then the overall rate of delivery is


It is clear from this equation that when the parent is foraging for itself it should fly at the speed w* that maximizes the net rate of gain γ. The resulting net rate γ(w*) will be denoted by γ*. Then the speed while traveling should minimize


which (as we would expect) is the Norberg condition. The speed when collecting food for the young should minimize


which is equivalent to minimizing


This condition also follows from the analysis given by McNamara and Houston (1997). The condition implies that u* satisfies


The optimal speed while the parent is collecting for itself satisfies


If equation (11) holds then u*=w* is a solution of equation (10). It can be shown that this solution is unique, and so we can conclude that the optimal speed is the same when parents are foraging for themselves and foraging for their young (cf. Hedenström and Alerstam 1995, p. 478).

I now show that if b(0)=0, b′>0 and b″≤0 then u*=w*<v*, i.e. the optimal speed when foraging is less than the optimal speed when traveling. To see this, note that the optimal speed v* while traveling satisfies


whereas the speed u* while collecting satisfies the equation


(The conditions on b mean that inline image, so inline image.)


which is positive (b″≤0 and P″>0) and so u*<v*.

For example, if inline image where inline image then inline image, so that u* approaches the Norberg speed vY from below as a increases towards 1.

Advantage of flying faster than vmr

McLaughlin and Montgomerie (1985) discuss the advantage in terms of time saved by flying faster than vmr. They suggest that the advantage for a Lapland longspur may be small and justify this claim with the following example. Female Lapland longspurs often forage within 100 m of their nest. Assuming that D=200 m, then if the female flies at vmr (=9.2 m/s) then the trip takes 22 s, while if the female flies at 11.6 m/s (the value of vY obtained by McLaughlin and Montgomerie – probably an underestimate) then the time taken would be 17 s, so five seconds could be saved. McLaughin and Montgomerie then express this saving as an increase in potential foraging time. Given that females spend about 3 min per trip at the foraging site, this gives a potential increase in foraging time of 3%. McLaughlin and Montgomerie go on to argue that the time saved and hence the increase in foraging time could be substantial for a bird traveling a long distance. The problem with this argument is that it simply considers that time saved by flying fast can be added to foraging time. On this argument the bird should fly as fast as possible. The logic behind flying at the velocity vY requires a consideration of the time required to replace the energy spent, and because McLaughlin and Montgomerie do not include this their estimate is conservative. We can say something general about time saved in these circumstances. Let


be the total cycle duration. Then


At vmr, P′=P/v and so


This equation gives us a simple and general expression for the advantage of flying faster than vmr. It shows that at vmr the rate of reduction in trip time with speed is proportional to the round-trip distance. If we wish to compare different species, it may be better to use the dimensionless elasticity inline image (cf. Byrd et al. 1991, Houston 2000, Houston et al. 1996). Evaluating this at vmr we find that


This shows that all else being equal elasticity increases with D and decreases with t, which is in line with the suggestions of McLaughlin and Montgomerie.


The analysis presented by Norberg (1981) did much to stimulate work on the flight speed of parent birds feeding their young. Unfortunately, the tests of Norberg's prediction carried out by McLaughlin and Montgomerie (1985) and de la Cueva and Blake (1997) have taken Pgain to be the rate at which energy is delivered to the young instead of the net rate at which the parent gains energy. It follows from equation (12) of Houston (1986) that McLaughlin and Montgomerie (1985) were actually testing whether birds flew at the speed predicted if net rate of delivery to the young is maximized. Given that delivery rate is less than the parent's gain rate, this speed will be less than vY. In fact vY may be above the maximum possible speed at which birds can fly (Houston 1986).

There are various models for the dependence of power on flight speed (e.g. Pennycuick 1975, 1989, Tucker 1975, Rayner 1979, 1999), so it is perhaps unwise to place emphasis on the quantitative predictions of any one model. This does not mean, however, that we have to abandon the development and testing of methods that predict the best flight speed. Two lines of attack are available. One approach, recommended by Hedenström and Alerstam (1995), is to measure the flight speed of a species (or, if possible, the same birds) in different circumstances. Hedenström and Alerstam (1996) illustrate this approach by comparing the flight speed of the skylark Alauda arvensis when singing with its flight speed when migrating. The difference in flight speed in the two contexts refutes the claim made by Tolkamp et al. (2002) that speed always minimizes P(v)/v. Another approach is to make qualitative predictions. For example, Houston (1986) establishes various general features of the speed that maximizes the rate at which energy is delivered to the young and the speed that maximizes the net rate of energetic gain. Some of these results are summarized in Table 2, which also includes predictions based on the minimization of energy expenditure. All three currencies predict that flight speed should be independent of the number of young to be fed or their condition. Thus a clear demonstration that flight speed depends on the state of the brood would require us to abandon all current models.


My thanks to Thomas Alerstam, John McNamara and two anonymous referees for helpful comments on the ms and to John McNamara for discussing the equations. Completion of this paper was facilitated by a Leverhulme Research Fellowship.


Appendix A: Appendix 1. For any flight speed, overall rate is greater than net rate.

The parent flies at speed v and spends a time t collecting food at a gross rate b. The net rate=b−c. The parent spends an amount of energy inline image. Then




inline image if and only if


But from the definition of x


Thus μ(v) is always negative and so Y(v)>N(v).