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Great tits and conveyor belts: a correction for non-random prey distribution
ANIMAL BEHAVIOUR, 29, 4
1276
sucrose solution than control bees exposed to air alone. In both analyses differences were significant between bees from different colonies (P < 0.005). Also, in both analyses no interaction occurred between the factors of type of air and the colony-source of bees. Thus, volatiles from empty comb at a temperature similar to that occurring in the brood area of bee nests increased the hoarding behaviour of bees. These volafiles were not given off in effective amounts by cold empty comb or warm comb that contained stored honey. Probably these volatiles are pheromones incorporated in comb by bees, since increased amounts of freshly made comb effectively increased hoarding (Rinderer & Baxter 1980). This research was done in cooperation with the Louisiana Agricultural Experiment Station. I thank J. R. Baxter for able technical assistance. THOMAS E. RINDERER Bee Breeding & Stock Center Laboratory, Agricultural Research, Science & Education Administration, U.S.D.A., Baton Rouge, LA 70808, U.S.A. References
Free, J. B. & Williams, H. I. 1972. Hoarding by honeybees (Apls mellifera L.). Anita. Behav., 20, 327-334. Kulin6evi6, J. M. & Rothenbuhler, W. C. 1973. Laboratory and field measurements of hoarding behaviour in the honeybee (Apis mellifera). J. Apie. Res., 12, 179-182. KulinEevi6, J. M., Rothenbuhler, W. C. & Stairs, G. C. 1973. The effect of presence of a queen upon outbreak of a hairless-black syndrome in the honey bee. J. lnvertebr. Pathol., 21, 241-247. Rinderer, T. E. & Baxter, J. R. 1978. Effect of empty comb on hoarding behavior and honey production of the honey bee. J. Econ. Entomol., 71, 757-759. Rinderer, T. E. & Baxter, 3. R. 1979. Honey bee hoarding behaviour: effects of previous stimulation by empty comb, Anim. Behav., 27, 426--428. Rinderer, T. E. & Baxter, J. R. 1980. Hoarding behavior of the honey bee: effects of empty comb, comb color, and genotype. Environ. Entomol., 9, 104-105.
(Received 20 August 1980; revised 6 June 1981 ; MS. number: AS-144) Great Tits and Conveyor Belts: a Correction for Non-random Prey Distribution
Basic optimal diet models generally assume that predators select prey to maximize their rate of energy intake. Krebs et al. (1977) set out to test this for a simple laboratory situation. Five great tits were offered large and small pieces of mealworm on a moving conveyor belt. A bird could specialize on one prey type, or take both. Krebs et al. worked out which of these strategies should be optimal for each bird at various presentation schedules, and compared their predictions with the birds' actual performance. However, the models used to determine the optimal strategies give slightly incorrect predictions for the conveyor belt experiment. Krebs et al. realized this for the first model they used: it was a random encounter model, whereas the conveyor belt items were arranged 'in a predictable order at fixed time intervals'. This difference is crucial because it alters the birds' expected waiting times at the belt. Let the time between successive items be a r a n d o m variable Ywith mean I1 = 1/7~(where %is the presentation
rate) and variance t~. If X Y is the time the belt takes to go round once, then the probability that a particular arrival of the bird will occur during a given interval j of length Yj is Yj/Z Y. If the bird arrives at r a n d o m during the interval, its expected waiting time is Yj/2. The expected waiting time over all intervals is therefore
Yy/~,Y) = 89
w = E( 89
(1)
Taking expectations in equation (1) gives w = 89
(2)
A model for non-random prey distribution can be derived using waithag times computed according to equations (1) and (2). For a prey type i, let E~ be the net energy gain and h, the handling time. Let Ws be the expected waiting time if the predator specializes on a more profitable prey type 1, computed from the intervals Ys (mean ~ts) between successive items of type 1; let we be the expected waiting time if the predator generalizes, computed from the intervals Y~ (mean Ix~) between all successive items. Let q be the probability that the first item the predator encounters in a search is of type 1. According to optimal diet theory, the predator should choose the foraging strategy with the highest rate of energy intake. It should therefore ignore prey type 2 if and only if E1 qEz +(1 --q)E2 - - > hlq-Ws qhlq-(1 --q)h~q-wa or, after cross-multiplying and cancelling, E1 - -
hl+ws
E~ >
(3)
h2+Q
where Q = (w~-qws)/(1-q). If both prey types are distributed randomly, equation (3) reduces to the r a n d o m encounter model used by Krebs et al., as explained in the following. In this case the interval lengths follow negative exponential distributions; therefore ~ts = Ors and rt~ = ~G. Then, according to equation (2), Ws = 89 = ~ts = 1/~1, and similarly wa = ~ta. Also, the probability of encountering an item of type 1 first in this case equals the proportion of type 1 items among the items offered, so that q = ~ta/~ts. It can therefore be concluded that w~ = ~t~ =q~ts =qws, and thus (wa--qws)= 0 and Q=O. Another case where equation (3) can be simplified is when all intervals between successive items of prey type 1 are equal and in addition all intervals between any two items are equal. Then er~ = Os = 0 and therefore according to equation (2), Ws = 89 = 1/(2LD and w~ = 89 Again q = Ix~/~ts, so that w~ = 89 = 89 = qws. This simplification applies for the presentation schedules A, B, D and E of Krebs et al. (1977). For the experiments performed by Krebs et al., a model accounting for non-random prey encounters according to equation (3) would predict selection for their treatments B, C, D and E in birds BW, R O and PW, and for their treatments D and E in bird YY. Rather than correcting their first model in some way such as that described above, Krebs et al. used a different approach to a model for non-random prey encounters. They defined t as the time interval between a currently encountered unprofitable item and the next profitable
SHORT C O M M U N I C A T I O N S
I00
*
setection
DO
*
*
OL
RO
BW
PW
trea~menfs
9 c aD
-.1
.1
.2
.3
[ meatworm
dlfterence
in
theoreticol
feeding
rates
Q~ 100 %
Rrebs 77
segments
ond 0 %
I s ]
selection
Fig. I. Relationship between the observed degree of selection and the benefit of selecting.
item on the belt, and p as the probability of handling the current prey within time t. For simplicity they assumed E1 = 2E~. They argued that a bird ignoring a current unprofitable item gains 2Ea in the time (t+hl), and a bird taking a current item gains (Ea+2pEa) in the same time; it follows that a bird should ignore a current item if p< 89 But there is a flaw here. A bird taking a current unprofitable item and missing the next profitable one can be expected to return to the belt after a handling time of ha', the mean of all type-2 handling times greater than t. If ha' < (hi+t), a bird that takes the current unprofitable item will, with a probability of (1--p), gain the time (hi+t--h2'), which can be used to feed; the model then tends erroneously to predict specializing. Similarly, if h a ' > (hi+t), the model tends erroneously to predict generalizing. Krebs et al. introduce a further error in applying the model. The predictions on pp. 34-35 do not follow from the model as given: in addition to the cases listed, the model predicts specializing for treatment B in birds B W and PW, and for treatments C, D and E in bird GBW. Two other recent studies examined great tits foraging from conveyor belts: Erichsen et al. (1980) and Houston et al. (1980). In both papers the models are based on random prey encounters, although in fact the intervals between items were regular and the sequence of prey types was pseudo-random. For Erichsen et al., a correction of the waiting times for the specific schedules employed does not change the predicted optimal strategies. For Houston et al., the predictions given for their
1277
treatments A and C are correct, but for their treatment B selection should be predicted for Olivia. Correcting the models for non-random prey encounters does not improve the fit between predictions and results. However, the method of comparing the results with deterministic predictions may not be the best way of assessing them. For instance, the prediction that a bird should forage selectively might be based on a n insignificantly small difference between the theoretically predicted feeding rates for specializing and generalizing. Equation (3) allows the calculation of potential feeding rates for each bird and treatment. This information is used in Fig. 1 to plot the degree of selection found against the potential benefit of selecting, i.e. against the difference in feeding rates for selective and unselective foraging (data from Krebs et al. (1977) and Houston et al. (1980)). I n Fig. 1 the degree of selection found clearly increases with the benefit of selecting. The cases in which selection was predicted but not observed are ones in which the benefit of selecting is relatively small. The deviations from the predicted behaviour could therefore be within the range of experimental error and are not necessarily strong evidence against the model. A more important discrepancy between the predicted and observed behaviour is that the birds do not show an all-or-nothing change from generalizing to selecting (Krebs et al. 1977). A satisfactory model would have to explain why partial preferences occur. C. RECH~N* ~. J. R. ICrtEBSt A. I. HOUSTON* *Animal Behaviour Research Group, and
t Edward Grey Institute, South Parks Road, Oxford OX1 3PS. :~Present address: Max-Planck-Institnt f'tir Verhaltensphysiologie, D-8131 Seewiesen, W. Germany.
References Erichsen, J. T., Krebs, J. R. & Houston, A. I. 1980. Optimal foraging and cryptic prey. J. Anim. Ecol., 49, 271-276. Houston, A. I., Krebs, J. R. & Erichsen, J. T. 1980. Optimal prey choice and discrimination time in the great tit (Parus major L.). Behav. Ecol. SociobioL, 6, 169-175. Krebs, J. R., Erichsen, J. T., Webber, M. I. & Charnov, E. L. 1977. Optimal prey choice in the great tit. Anim. Behav., 25, 30-38.
(Received 19 December 1980; revised 16 March 1981 ; MS. number: sc-94)
1276
sucrose solution than control bees exposed to air alone. In both analyses differences were significant between bees from different colonies (P < 0.005). Also, in both analyses no interaction occurred between the factors of type of air and the colony-source of bees. Thus, volatiles from empty comb at a temperature similar to that occurring in the brood area of bee nests increased the hoarding behaviour of bees. These volafiles were not given off in effective amounts by cold empty comb or warm comb that contained stored honey. Probably these volatiles are pheromones incorporated in comb by bees, since increased amounts of freshly made comb effectively increased hoarding (Rinderer & Baxter 1980). This research was done in cooperation with the Louisiana Agricultural Experiment Station. I thank J. R. Baxter for able technical assistance. THOMAS E. RINDERER Bee Breeding & Stock Center Laboratory, Agricultural Research, Science & Education Administration, U.S.D.A., Baton Rouge, LA 70808, U.S.A. References
Free, J. B. & Williams, H. I. 1972. Hoarding by honeybees (Apls mellifera L.). Anita. Behav., 20, 327-334. Kulin6evi6, J. M. & Rothenbuhler, W. C. 1973. Laboratory and field measurements of hoarding behaviour in the honeybee (Apis mellifera). J. Apie. Res., 12, 179-182. KulinEevi6, J. M., Rothenbuhler, W. C. & Stairs, G. C. 1973. The effect of presence of a queen upon outbreak of a hairless-black syndrome in the honey bee. J. lnvertebr. Pathol., 21, 241-247. Rinderer, T. E. & Baxter, J. R. 1978. Effect of empty comb on hoarding behavior and honey production of the honey bee. J. Econ. Entomol., 71, 757-759. Rinderer, T. E. & Baxter, 3. R. 1979. Honey bee hoarding behaviour: effects of previous stimulation by empty comb, Anim. Behav., 27, 426--428. Rinderer, T. E. & Baxter, J. R. 1980. Hoarding behavior of the honey bee: effects of empty comb, comb color, and genotype. Environ. Entomol., 9, 104-105.
(Received 20 August 1980; revised 6 June 1981 ; MS. number: AS-144) Great Tits and Conveyor Belts: a Correction for Non-random Prey Distribution
Basic optimal diet models generally assume that predators select prey to maximize their rate of energy intake. Krebs et al. (1977) set out to test this for a simple laboratory situation. Five great tits were offered large and small pieces of mealworm on a moving conveyor belt. A bird could specialize on one prey type, or take both. Krebs et al. worked out which of these strategies should be optimal for each bird at various presentation schedules, and compared their predictions with the birds' actual performance. However, the models used to determine the optimal strategies give slightly incorrect predictions for the conveyor belt experiment. Krebs et al. realized this for the first model they used: it was a random encounter model, whereas the conveyor belt items were arranged 'in a predictable order at fixed time intervals'. This difference is crucial because it alters the birds' expected waiting times at the belt. Let the time between successive items be a r a n d o m variable Ywith mean I1 = 1/7~(where %is the presentation
rate) and variance t~. If X Y is the time the belt takes to go round once, then the probability that a particular arrival of the bird will occur during a given interval j of length Yj is Yj/Z Y. If the bird arrives at r a n d o m during the interval, its expected waiting time is Yj/2. The expected waiting time over all intervals is therefore
Yy/~,Y) = 89
w = E( 89
(1)
Taking expectations in equation (1) gives w = 89
(2)
A model for non-random prey distribution can be derived using waithag times computed according to equations (1) and (2). For a prey type i, let E~ be the net energy gain and h, the handling time. Let Ws be the expected waiting time if the predator specializes on a more profitable prey type 1, computed from the intervals Ys (mean ~ts) between successive items of type 1; let we be the expected waiting time if the predator generalizes, computed from the intervals Y~ (mean Ix~) between all successive items. Let q be the probability that the first item the predator encounters in a search is of type 1. According to optimal diet theory, the predator should choose the foraging strategy with the highest rate of energy intake. It should therefore ignore prey type 2 if and only if E1 qEz +(1 --q)E2 - - > hlq-Ws qhlq-(1 --q)h~q-wa or, after cross-multiplying and cancelling, E1 - -
hl+ws
E~ >
(3)
h2+Q
where Q = (w~-qws)/(1-q). If both prey types are distributed randomly, equation (3) reduces to the r a n d o m encounter model used by Krebs et al., as explained in the following. In this case the interval lengths follow negative exponential distributions; therefore ~ts = Ors and rt~ = ~G. Then, according to equation (2), Ws = 89 = ~ts = 1/~1, and similarly wa = ~ta. Also, the probability of encountering an item of type 1 first in this case equals the proportion of type 1 items among the items offered, so that q = ~ta/~ts. It can therefore be concluded that w~ = ~t~ =q~ts =qws, and thus (wa--qws)= 0 and Q=O. Another case where equation (3) can be simplified is when all intervals between successive items of prey type 1 are equal and in addition all intervals between any two items are equal. Then er~ = Os = 0 and therefore according to equation (2), Ws = 89 = 1/(2LD and w~ = 89 Again q = Ix~/~ts, so that w~ = 89 = 89 = qws. This simplification applies for the presentation schedules A, B, D and E of Krebs et al. (1977). For the experiments performed by Krebs et al., a model accounting for non-random prey encounters according to equation (3) would predict selection for their treatments B, C, D and E in birds BW, R O and PW, and for their treatments D and E in bird YY. Rather than correcting their first model in some way such as that described above, Krebs et al. used a different approach to a model for non-random prey encounters. They defined t as the time interval between a currently encountered unprofitable item and the next profitable
SHORT C O M M U N I C A T I O N S
I00
*
setection
DO
*
*
OL
RO
BW
PW
trea~menfs
9 c aD
-.1
.1
.2
.3
[ meatworm
dlfterence
in
theoreticol
feeding
rates
Q~ 100 %
Rrebs 77
segments
ond 0 %
I s ]
selection
Fig. I. Relationship between the observed degree of selection and the benefit of selecting.
item on the belt, and p as the probability of handling the current prey within time t. For simplicity they assumed E1 = 2E~. They argued that a bird ignoring a current unprofitable item gains 2Ea in the time (t+hl), and a bird taking a current item gains (Ea+2pEa) in the same time; it follows that a bird should ignore a current item if p< 89 But there is a flaw here. A bird taking a current unprofitable item and missing the next profitable one can be expected to return to the belt after a handling time of ha', the mean of all type-2 handling times greater than t. If ha' < (hi+t), a bird that takes the current unprofitable item will, with a probability of (1--p), gain the time (hi+t--h2'), which can be used to feed; the model then tends erroneously to predict specializing. Similarly, if h a ' > (hi+t), the model tends erroneously to predict generalizing. Krebs et al. introduce a further error in applying the model. The predictions on pp. 34-35 do not follow from the model as given: in addition to the cases listed, the model predicts specializing for treatment B in birds B W and PW, and for treatments C, D and E in bird GBW. Two other recent studies examined great tits foraging from conveyor belts: Erichsen et al. (1980) and Houston et al. (1980). In both papers the models are based on random prey encounters, although in fact the intervals between items were regular and the sequence of prey types was pseudo-random. For Erichsen et al., a correction of the waiting times for the specific schedules employed does not change the predicted optimal strategies. For Houston et al., the predictions given for their
1277
treatments A and C are correct, but for their treatment B selection should be predicted for Olivia. Correcting the models for non-random prey encounters does not improve the fit between predictions and results. However, the method of comparing the results with deterministic predictions may not be the best way of assessing them. For instance, the prediction that a bird should forage selectively might be based on a n insignificantly small difference between the theoretically predicted feeding rates for specializing and generalizing. Equation (3) allows the calculation of potential feeding rates for each bird and treatment. This information is used in Fig. 1 to plot the degree of selection found against the potential benefit of selecting, i.e. against the difference in feeding rates for selective and unselective foraging (data from Krebs et al. (1977) and Houston et al. (1980)). I n Fig. 1 the degree of selection found clearly increases with the benefit of selecting. The cases in which selection was predicted but not observed are ones in which the benefit of selecting is relatively small. The deviations from the predicted behaviour could therefore be within the range of experimental error and are not necessarily strong evidence against the model. A more important discrepancy between the predicted and observed behaviour is that the birds do not show an all-or-nothing change from generalizing to selecting (Krebs et al. 1977). A satisfactory model would have to explain why partial preferences occur. C. RECH~N* ~. J. R. ICrtEBSt A. I. HOUSTON* *Animal Behaviour Research Group, and
t Edward Grey Institute, South Parks Road, Oxford OX1 3PS. :~Present address: Max-Planck-Institnt f'tir Verhaltensphysiologie, D-8131 Seewiesen, W. Germany.
References Erichsen, J. T., Krebs, J. R. & Houston, A. I. 1980. Optimal foraging and cryptic prey. J. Anim. Ecol., 49, 271-276. Houston, A. I., Krebs, J. R. & Erichsen, J. T. 1980. Optimal prey choice and discrimination time in the great tit (Parus major L.). Behav. Ecol. SociobioL, 6, 169-175. Krebs, J. R., Erichsen, J. T., Webber, M. I. & Charnov, E. L. 1977. Optimal prey choice in the great tit. Anim. Behav., 25, 30-38.
(Received 19 December 1980; revised 16 March 1981 ; MS. number: sc-94)