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Vigilance and scanning patterns in birds
Anim. Behav., 1984, 32, 1216-1224
VIGILANCE AND SCANNING PATTERNS IN BIRDS A N D R E W HART* & DENNIS WILLIAM L E N D R E M ~
Animal Behaviour Research Group, Department of Zoology, South Parks Road, Oxford, OX1 3PS, U.K. Abstract. We present a new approach to the analysis of scanning patterns in feeding birds. We estimate the probability of a bird detecting a predator from the frequency distribution of inter-scan intervals and the proportion of time spent scanning. This method avoids several unrealistic assumptions implied in earlier analyses of vigilance data, and can accommodate predators attacking randomly or using prey behaviour to time an attack. The practical application of this approach is illustrated using data for feeding and vigilance in the ostrich (Struthio camelus). The analysis is discussed with reference to the hunting tactics of predators and Pulliam's (1973) model of feeding and vigilance in birds. intervals is then wholly described by the rate. Unfortunately there is growing evidence that these assumptions are frequently violated. In a direct test of Pulliam's model Elgar & Catterall (1981) found departures from predicted scanning rates in flocks of house sparrows, which they attributed to the fact that scans are not instantaneous but of finite duration. Moreover Elcavage & Caraco (1983) and Lendrem (1982, 1983) report that birds rarely scan randomly, and Lazarus (t979) reports that birds scan synchronously rather than independently of each other. The method which we present here avoids all three of these unrealistic assumptions and encourages a more detailed examination of scanning patterns. We illustrate its application using data on feeding and vigilance in the ostrich kindly made available by Dr B. C. R. Bertram.
Scanning rates are widely used as indices of vigilance in feeding birds (e.g. Lazarus 1979; Barnard 1980; Bertram 1980; Caraco et al. 1980; Jennings & Evans 1980; Elgar & Catterall 1981 ; Caraco 1982). It is assumed that the probability of detecting an approaching predator increases as scanning rate (or time spent scanning) increases. However this ignores the pattern of scanning behaviour. Two birds may have identical scanning rates but very different scanning patterns, and the probability of predator detection will depend on those patterns. This is because what determines whether an attack at a particular time is detected is not average scanning behaviour, but whether the prey looks up before it is too late; that is, detection depends on the length of the particular inter-scan interval in which the attack occurs. This in turn depends on both the scanning pattern of the prey and the predator's choice of when to attack. In this paper we present a method for the analysis of scanning patterns in birds. The method estimates the probability of predator detection from the proportion of time spent scanning and the frequency distribution of interscan intervals, and it takes account of the tactics used by the predator. Like Pulliam (1973) we consider the probability of predator detection in an interval, % during which the predator breaks cover to come within striking distance of its prey. Pulliam's model assumes that birds scan randomly, instantaneously and, when in flocks, independently of one another. If these assumptions hold, mean scanning rate is indeed an adequate measure of vigilance, since the distribution of inter-scan
The Model Consider a lion hunting an ostrich. It stalks the ostrich unseen. Suddenly it breaks cover and makes a final uncovered dash to capture the ostrich. Let this dash take ~-s. The probability of our ostrich detecting the lion is the probability that it will make a scan before the lion completes its final uncovered dash. In order to calculate this probability we need to know the frequency distribution of inter-scan intervals, the proportion of time spent scanning and the tactics employed by the lion. In the approach outlined below we consider lions using three different kinds of hunting tactic. 'Random' lions do not use ostrich behaviour to time an attack; an attack is initiated regardless of whether an ostrich is feeding or scanning. In contrast, both 'check-wait' and 'double-check' lions delay their attack if an ostrich is scanning. The former (check-wait) attack if ostriches are feeding but otherwise delay until the start of a new feeding bout. The double-check lion always
Present addresses: *Mammals and Birds Department, Worplesdon Laboratory, Tangley Place, Guildford GU3 3LQ, U.K. ]'Department of Psychology, The Ridley Building, Claremont Place, Newcastle upon Tyne, NE1 7RU, U.K. 1216
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR waits until the start of the next feeding bout whether or not the ostrich is currently feeding. We first consider attacks against solitary ostriches. Solitary Prey Let the proportion of time spent scanning be ps and the probability density of inter-scan intervals bef(t). Both ps and the discrete form of f(t) may easily be obtained empirically (see later). If the lion attacks randomly then on a proportion (l-ps) of occasions the ostrich's head will be down, and the start of the attack will be undetected. How long will it be before the ostrich raises its head ? A randomly-timed attack is proportionately more likely to fall into a long inter-scan interval than into a short one, and more likely to fall into a frequent length of interval than an infrequent length. In fact, the proportion of all attacks which will fall into intervals of a certain length, t s, is given by the total time for all intervals of that length, as a proportion of all time. The total time in intervals of length t is proportional to tf(t), and thus the probability density of all attacks falling in intervals of this length is
tf(t)
g(t) -
(1) 0f ~176 tJ(t)dt
The function g(t) describes the frequency distribution of the lengths of intervals which are interrupted by a randomly attacking lion. The probability of detection depends on the length of the interrupted interval. If it is less than % the time taken for the final uncovered dash, then the probability of detection is one. If the interval is greater than z then detection depends on the point at which it has been interrupted; on .r/t occasions the attack will begin less than -r s before the next scan and the predator will be detected. Thus the probability of detecting an attack started at random within an inter-scan interval is given by + oo
G(-r) =o
g(t) dt
~-
tg(t) dt
(2)
However, on a proportion ps of occasions an ostrich will be scanning when an attack begins. The overall probability of detecting an attack by a 'random' lion is therefore pl(~) -- ps + (1 - ps) a(~)
(cf. Lendrem 1984).
(3)
1217
Cued Attacks on Solitary Prey We now derive the probability of detection if a lion uses the ostrich's behaviour to time its attack. As outlined above, two tactics of this sort are considered. In the first, check-wait, the lion does not attack randomly but delays if the ostrich is scanning. If the ostrich is not scanning it begins its attack immediately. The detection probability on the proportion (1-ps) of occasions when the ostrich is not scanning is given by G(~-), as in equation (2). On theps occasions when the attack is delayed until the end of a scan the probability of detection is given not by G(z), but by f(~-),
F(~') = 0
f
,r
f(t) dt
(4)
F(~') is the probability that the next scan will start within a period r when the attack begins at the start of an inter-scan interval rather than at a random point within the interval. Thus for this kind of tactic the overall detection probability is given by pz(~') = ps F(~-) q- (1 -- ps) G(~')
(5)
The second type of delaying tactic (doublecheck) is for a lion always to wait until the beginning of an inter-scan interval. For this kind of predator the overall detection probability is simply p~(z) = F(~-)
(6)
Attacks on Groups of Prey Calculating levels of corporate vigilance (the probability that at least one member of a group will detect an approaching predator) is fairly straightforward if we assume that individual birds scan independently of each other (see Appendix). However, it is even easier if we sidestep the question of independence altogether by treating the group as a unit and asking directly what is the probability of the group detecting an approach predator? Levels of corporate vigilance can then be calculated using equations (3), (5) and (6) above, provided we derive F(~-) and G(~-) from the distribution of intervals when all birds have their heads down and ps is the observed proportion of time when one or more birds are scanning. The raw data must therefore consist at least of timed records of the beginning and end of each period for which no member of the group is scanning.
1218
ANIMAL
BEHAVIOUR,
Calcuiating Detection Probabilities from Real Data Our analysis provides a measure of vigilance behaviour which takes account of both the amount and the .pattern of scanning behaviour. This measure is the detection probability, calculated from the data for one or more predator types and over a range of values for -r. Basic procedures for maldng comparisons are applied to data on ostriches in the next section. Here we deal with the use of the formulae derived above in calculating detection probabilities and plotting vigilance curves from real data. The first step is to construct the frequency distribution f(d) (the discrete analogue of f(t) in the model) by sorting the observed inter-scan intervals into classes. These classes must be arranged such that the values of ~- for which detection probabilities are to be calculated fall on the class boundaries. The choice of the number of classes to use is critical, and the usual guidelines for constructing histograms apply. If the classes are too narrow then the number of observations in each class will be small, and the general form of the distribution is more likely to be obscured by sampling fluctuations. If, on the other hand, classes are too wide, then resolution is lost and detailed features of the distribution may be overlooked. If it is only possible to obtain a reasonably 'smooth' histogram by using a very small number of classes then more data are required. Suppose that the number of classes is c, denoted by i-= 1, 2, 3. . . . . c. If the xth interval class is the one whose upper limit is -r then the discrete form of F(-r) is obtained as X
F(,)
=
Xf(d~) i=1
(7)
where di is an inter-scan interval length representative of the ith class. In this case, classes are best represented by the median of the observed values which they contain. If the distribution is not strongly skewed or uneven, and if the classes are not too wide, then the midpoint will be an adequate measure. Using the same notation, the discrete form of G(~-)is obtained as x
G(T)=
Z g(dl) + i=1
C
~-
X --g(dO i~-x+l d~
(8)
32,
4
where g(dQ is obtained as
d, f(di) g(&) =
(9) r
Z d, f(d~) i--1 Equations (7), (8) and (9) are used to calculate G(~-) and F(~-) for successive values o f - r and substituted into the appropriate equation (from 3, 5, and 6) to obtain the respective detection probabilities. These are best presented graphically, plotted against ~-.
An Example: Vigilance in Ostriches Bertram (1980) observed ostriches (Struthio camelus) feeding in Tsavo West National Park, Kenya between July and October, 1977. The habitat in the park consists of open, undulating grasslands with many scattered bushes and a few trees. Observations were made of adults without young, and birds were considered to be part of a group if they were within 65 m of one another. Birds were usually 30-40 m apart. Further details of the study area and data collection methods are published in Bertram (1980). Bertram observed that, in pairs of ostriches, males seemed more vigilant than females. Accordingly we have analysed male-female and female-female pairs separately. In fact, further significant variation exists within group types for Bertram's data, but for the purposes of illustration we have ignored it. Frequency distributions of intervals when all heads were down in four different types of group are illustrated in Fig. 1. There are sufficient data in each group type for fairly narrow class intervals, of 5 s throughout. Levels of corporate vigilance were calculated without assuming independence using equations (3), (5) and (6). F(~-) and G(~-)were derived from the frequency distributions shown in Fig. 1 according to equations (7) and (8) as described above. The proportion of time for which one or more birds were scanning, ps, was obtained directly from the pooled data for each group type. The median values of ps over instances of each group type are given alongside the respective histograms in Fig. 1. Vigilance curves (the probability of predator detection as a function of predator approach time, ~-) were plotted for each type of group. Randomly attacking lions were at a serious disadvantage compared with their more patient counterparts (Fig. 2). In fact, unless scans are instantaneous (so ps = zero), the detection
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR probabilities for check-wait and double-check lions must always be lower than for randomly attacking lions, other things being equal. In order to show a difference between check-wait and double-check tactics it is sufficient to show that the birds are not raising their heads at random. Only if head-raising is random (a stationary Poisson process) will the expected waiting time to the end of an inter-scan interval be constant and independent of the time since the interval began. It is easy to see that if this is true then the probability of detecting an attack starting at any point during an inter-scan interval is the same, and therefore check-wait and double-check predators will be detected with equal probability. We tested the fit of the observed distributions in Fig. 1 to the negative exponential using the Kolmogorov-Smirnov one-sample test, as given by Pearson & Hartley (1972). This test does not involve grouping into interval classes and so is more sensitive to deviations. Using this test we found significant deviations from the negative exponential for all groups. This means that for all groups the differences (shown in Fig. 2) between
80 i60140L
solitary birds p. =0 343 s
2ot
N=253
OH
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pairs
60
ps= 0.520
2
N=249
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o
the vigilance curves for double-check and checkwait attacks are significant. The nature of deviations from the negative exponential may not be clear from a simple histogram of the data. If the sample size is not large (np to 120) then a graphical method described by Pearson & Hartley (1972) may be useful. This method involves plotting the ranked observations against the expected values of the order statistic. If the distribution conforms to the negative exponential then the points should fall on a straight line with a slope that is estimated by the mean of the observations. Figure 3 shows the plot obtained using the data collected by Bertram for one instance of a solitary male ostrich. The Kolmogorov-Smirnov one-sample test shows that the distribution of inter-scan intervals for this ostrich deviates significantly from the negative exponential (Dmax = 1.44, N = 25, P < 0.01). The points in the plot against the expected values of the order statistic suggest that the shortest intervals are larger than expected, and the longest intervals shorter than expected on the basis of the negative exponential distribution. This result is typical of the majority of those for instances of solitary male ostriches, and indeed for solitary females and for groups.
1.0 0.8I 0.7 O.E ?, 0.~= 0.4 0.3 o~. 0.1
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=9
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15
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1219
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male-female '
. . . . .
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0 10 20 30 40 50 0 10 20 30 40 50 55
Inter-scan intervals (s) Fig., 1. Frequency distributions of intervals when all birds were feeding, and the median proportion of time spent scanning (ps) in groups of ostriches.
Predator approach time (s) Fig. 2. Corporate vigilance in groups of ostriches. Randomly attacking lions (P1) were more likely to be detected than either check-wait (P2) or double-check (Pa) lions. Double-check lions were slightly but consistently more likely to pass undetected than check-wait lions.
ANIMAL
1220
BEHAVIOUR,
The vigilance curves are replotted for betweengroup comparison in Fig. 4. The probability of detecting a double-check predator does not depend on ps (see equation 6), and so a test for differences among the f(t) is a sufficient test for differences between the vigilance curves in Fig. 4C. The Kruskal-Wallis one-way non-parametric analysis of variance (Siegel 1956) revealed significant variation between the group types in the distribution of inter-scan intervals ( H = 28.11, c.lf= 3, P<0.0001; see Fig. 1 for histograms). The Kolmogorov-Smirnov two-sample test could be used for pairwise comparisons of the f(t), but it is in fact often less sensitive (Pearson & Hartley 1972). To compare the probability of detecting random or check-wait predators for different .group types it is necessary to test for differences m ps as well as f(t). Again the Kruskal-Wallis test indicates significant variation between grouptypes ( H = 9.21,df = 3, P < 0 . 0 5 ; medians forps are shown in Fig. 1). Figures 4A and 4B show how these significant differences in ps and f(t) combine to give differences in corporate vi.gilance. Unfortunately contradictions may arise. If the between-group-types variation in ps a n d f ( t ) have contrary implications for vigilance then the reliability of the between-groups differences in vigilance for random and checkwait predators becomes questionable. This is because the effects ofps andf(t), though individually significant, may effectively cancel each other out. In the present case this complication arises in the comparisons of pairs of females with solitary ostriches and with groups of three and
32,
4
four. The remaining comparisons are not affected, as may be seen by comparing the trends inf(t) andps (Fig. 1). Examination of Fig. 4 indicates that corporate vigilance does not simply increase with group size, but rather that male-female pairs were the most vigilant groups, and solitary birds and pairs of females least vigilant. Clearly it would be very useful if confidence limits could be attached to the vigilance curves, both for visual pairwise comparisons and to resolve the difficulties caused by contradictory differences i n f ( t ) and ps.
1.0A 0.9 0.8 0.7 0.6
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1.0
.
.
.
.
.
i
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i
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i
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=1.44,N=25 ; P < 0 . 0 1
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Expected order statistic
Fig. 3. Ranked inter-scan intervals for a solitary male ostrich plotted against the expected values of the order statistic. If the distribution of inter-scan intervals conforms to the negative exponential then the points should fall along the line whose slope (6.84) is given by the mean of the observations. The deviations shown by this individual are typical of the majority of Bertram's ostriches.
0.4 0.3 0.2 0.1 0,0
i
,
,
,
, . ,
=
,
i
,
0 10 20 30 4(3 50 Predator approach time (s)
Fig. 4. Corporate vigilance as a function of group size given (A) random, (B) check-wait and (C) double-check attacks. 1: solitary; 2 ~ : male-female pairs; 2~9: female-female pairs; 3-t-4: three and four birds.
HART & L E N D R E M : V I G I L A N C E A N D S C A N N I N G BEHAVIOUR
In fact it seems that confidence limits might be obtained by 'bootstrapping', a technique lucidly explained and demonstrated by Diaconis & Efron (1983), though some thought would have to be given to the way in which variation in ps andf(t) was combined. We have explored the use of the 'bootstrap' for double-check predators, which involves only fit). The results for a predator approach time of 5 s are presented in Fig. 5, which shows for each group type the distribution of detection probabilities obtained from 1000 bootstrap samples. Each bootstrap sample is of the same size as its parent dataset, and is computergenerated by sampling at random with replacement from the parent dataset. The results we obtained correspond broadly to the result of the Kruskal-Wallis analysis of variance: the confidence intervals shown in Fig. 5 indicate that at least some of the group types differ significantly in detection probability.
400-
~
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~_~
3 and 4 birds
I
--
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pairs
400
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400.
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200-
i
solitary
[L
0.0
015
1.0
Detection probability Fig. 5. Distribution of double-check predator detection probabilities (corresponding to ~ = 5 s in Fig. 4C) obtained from t000 'bootstrap' samples for each grouptype (see text). The arrows indicate the detection probabilities obtained from the original data. The 95 ~ confidence interval is given by the unshaded portion under each histogram, drawn to exclude the 25 most extreme bootstrap samples at each end.
1221
Discussion Interpretation Both Pulliam's (1973; Pulliam et al. 1982) model and our own employ the concept of 'predator approach time'. The interpretation of the models, and of the results of calculations based on them, depends upon what is meant by ~-. What we actually calculate is the probability that the ostriches will look up before the end of a period -r, given various assumptions about when ~- begins (corresponding to various predator tactics). In interpreting this as the probability of detection during ~- we make an assumption about the probabilities of detecting a predator in its final dash, given that the prey is scanning (probability of detection assumed one) or is not scanning (probability of detection assumed zero). In interpreting our 'vigilance curves' as showing overall probabilities of detection we make assumptions about the probability of detection before and after % essentially that the probability of detection is zero during the stalk which precedes the final dash. A third type of assumption, which we have not implied in this paper, affords interpretation of the results in terms of the probability of capture. This type of assumption concerns the probability of capture given that the predator is detected. The simple assumption here is that the prey always escapes if the predator is detected, and it ]nay be difficult to avoid implying this in discussing the adaptive value of alternative strategies assessed using 'detection probabilities'. So what is meant by ~-? It is effectively defined by the types of assumptions about detection and capture that we have just discussed. If these assumptions are made in the simple ways outlined above, then ~ is the time taken for the predator to travel from a point where it breaks from perfect cover into perfect observability, to a point where it moves from a range at which alert prey always escape into a range at which a capture is inevitable. This definition of ~- is implied if we state or imply that the vigilance curves measure the probability of capture. It is obviously unrealistic but at the comparative level of analysis this is unlikely to cause serious error. If on the other hand we propose a detailed cost-benefit analysis of the vigilance behaviour of a particular animal, then it becomes necessary to define ~- such that we can measure it and obtain its frequency distribution in the field. Furthermore, if we wish to assess the evolutionary stability of one vigilance strategy against a
1222
ANIMAL
BEHAVIOUR,
hypothetical alternative then we must take account of the changes in the predator population and in predator tactics that could occur if the mutant prey become common. This is a much more difficult problem.
Hunting Tactics Our approach gives detection probabilities for two kinds of predator: those attacking randomly and those using prey behaviour to time an attack. The former is, perhaps, more usually associated with attacks by avian predators. The marsh harrier (Circus aeruginosus), for example, usually quarters flat areas in low flight using ground vegetation and habitat edges to surprise prey. Once the prey is located, it drops with claws outstretched. Attacks are seldom repeated if capture fails (Schipper 1977). Similarly Tinbergen (1946) describes how the sparrowhawk (Accipiter nisus) flies low along hedges, slipping from one side to the other, along wood edges, among bushes and buildings, ready to snatch any unsuspecting bird. There is no evidence that these predators use prey behaviour to time an attack. In contrast, many ground predators are known to use prey behaviour, at least during the stalk if not at the attack. Thus Leyhausen (1973) reports that during a stalk, domestic cats (Felis domesticus) freeze when prey are vigilant. Biben (1979) was unable to confirm this experimentally, but the tests were carried out in small experimental rooms with poor ground cover. Leyhausen reports that in such conditions cats dispense with the preliminary stalk and rush at prey (Leyhausen 1973). Some predators, including lions, are known to wait until a prey animal has its head down feeding before launching an attack (Schaller 1972). Of the two tactics exploiting prey behaviour that we have considered, check-wait predators are marginally less successful than double-check predators. However, double-check predators will tend to incur higher waiting-time costs. That is, they will have to spend more time waiting for a prey animal to put its head down, extra time in which they might be spotted or in which prey might move out of range. Moreover, this cost will increase with group size for as group size increases so the proportion of time one or more birds spend scanning will tend to increase. In fact, lions attacking larger groups of prey switch from a 'stalk' to a 'running crouch' tactic (Schaller 1972; Bertram 1978). Although such a running crouch is more likely
32,
4
to be detected, calculations by Elliott et al. (1977) suggest that such a switch will lead to a significant increase in the probability of a successful hunt since it increases the lion's velocity relative to the prey at the moment of detection. This change in tactics may represent the abandonment of tactics using prey behaviour.
Comparison with Pulliam's Model The assumptions that determine the nature of r are common to our model and that of Pulliam (1973). Where the two differ is in the assumptions made in deriving the probability that the prey will look up within the period r. (1) Scans are assumed to be instantaneous in Pulliam's (1973) model (though not in that of Pulliam et al. 1982). This assumption is generally unrealistic and is not made by our model. (2) Pulliam (1973; Pulliam et al. 1982) assumes that the distribution of inter-scan intervals for individuals follows the negative exponential. This assumption fails for the ostriches observed by Bertram (see his Fig. 3, 1980; the Kolmogorov-Smirnov test given by Pearson & Hartley indicates significant deviations from the negative exponential for ostriches alone and in groups). It also fails for blue tits (Lendrem 1983) and in many cases for house sparrows (Elgar & Catterall 1981; Elcavage & Caraco 1983). The cost of retaining this assumption when it fails is twofold: the reliability of the vigilance measure obtained is reduced, and the investigator may overlook important information about the predator-prey interaction that a more detailed examination of the scanning pattern would reveal. Our model makes no assumption about the distribution of inter-scan intervals. (3) In deriving the probability of detection for groups of prey Pulliam (1973; Pulliam et al. 1982) assumes that the prey scan independently of one another. We have shown how group vigilance can be calculated using our model with or without the assumption of independence. We tried calculating vigilance levels using the independence assumption (method in Appendix) but they were much higher than those in Fig. 4, which suggests important deviations from independence. Bertram (1980), using the same data, shows that ostriches in groups of three and four had their heads up concurrently slightly, but significantly, more of the time than would be expected if they were behaving independently. Lazarus (1979) observed that the probability of quelea in flocks detecting an artificial stimulus was not so high as predicted assuming that they
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR were scanning independently. Given the choice, we prefer to avoid the independence assumption unless it can be shown that it is not violated. (4) Pulliam (1973; Pulliam et al. 1982) assumes a randomly-attacking predator. Extension to predators that cue their attacks using prey behaviour in a simple manner is straightforward and illuminating. The probability of detection for different cueing tactics may depend crucially on the form of the distribution of inter-scan intervals, a fact which emphasizes the value of our non-parametric analysis. The adaptive significance of such differences is difficult to assess unless the cost to the predator of waiting for the moment to attack can be quantified. Nonetheless, many ground predators are known to use prey behaviour and in these cases models for cueing predators will give more realistic detection probabilities. Finally, we reiterate that while our model is more realistic than Pulliam's it does retain simplistic assumptions in the definition of predator approach time. Either Pulliam's model or ours, as appropriate, could be employed as a basis for further development in cases where suitable data are available. Examples of modifications that might be considered are to allow variation in the probability of detection according to the length of scan or the number of group members scanning, and to investigate whether probability of escape depends on the distance from the predator when detected. Another matter for attention is the possibility of a correlation between the lengths of successive inter-scan intervals, which would offer great advantages to predators that used it to cue their attacks. Obviously, data interpretation will be prejudiced if complications such as these exist in a particular case and are ignored. It is up to the investigator to choose the most appropriate vigilance measure for the problem in hand; what we have done is to provide an alternative starting point to Pulliam's model, and indicate the value of detailed inspection of the data.
Appendix: Corporate Vigilance Assuming Independent Scanning If ostriches scan independently of each other then the probability of a group detecting an approaching predator (their 'corporate vigilance') is one minus the probability of all birds failing to detect that lion. Once again, the probability of an individual failing to detect a lion will depend on the tactics employed by that lion.
1223
Random Attacks on Groups of Prey The probability of at least one member of a group detecting an attack that begins at a random point during a period when all heads are down is given by n ( . ) : 1 -- (1 -- G(-c))N (10) where N is the group size, and G(.) is given by equation (2). This assumes that group members are scanning independently of one another. So for a randomly-timed attack, the probability of detection for attacks that start when no ostrich is scanning will be H(.), but attacks that start when one or more ostriches are scanning will always be detected. Thus the overall detection probability PI(T) = ps + (1 -- ps)n(7)
(11)
where ps is now the proportion of time one or more ostriches spend scanning if birds are behaving independently of each other. For example, the proportion of time one or more birds spend scanning, ps, in a pair of birds is ps = x + y - - x Y , where x and y are the proportions of time for which each bird has its head up. That is, ps is the sum of the time each bird has its head up minus the time that both have their heads up concurrently (see also Bertram 1980).
Cued Attacks on Groups of Prey The probability of one or more ostriches detecting a lion that cues its attack on ostrich behaviour will again depend on the precise tactics employed by the lion. The last bird to put its head down to feed will detect a double-check tactician with a probability F(~-). The remaining birds will detect that lion each with a probability G(~-). Thus their corporate vigilance will be Pa(~') = 1 -- (1 -- F(~-)). (1 -- G(~-))N-1 (12) again assuming that birds are scanning independently. For check-wait tacticians, on a proportion ps of the time, one or more birds will be scanning and the lion will wait until the start of the next period when all heads are down. On this ps of occasions the probability of detection will be the same as that for a double-check tactician (equation 12). However, for (1--ps) of the time all birds will be feeding and the lion will begin its attack immediately. On these occasions the probability of detection will be given simply by H(~-). Thus the overall probability of detection for a check-wait predator will be Pz(~') = p s P a + (1 --ps)H(r) (13)
t224
ANIMAL
BEHAVIOUR,
where H ( , ) is given by equation (10) a n d ps is the p r o p o r t i o n of time when one or more ostriches are scanning, assuming independence (as outlined above). Finally, if a lion waits for two or more ostriches to p u t their heads down simultaneously before b e g i n n i n g an attack then the probability of predator detection is given b y P(~-) = 1 - - (1 - - F(~-))~v' . (1 - - a(~-)) iv-N' (14) where N is the group size a n d N ' the n u m b e r of individuals o n whose behaviour the attack is cued. I n the extreme case where all birds lower their heads simultaneously then P(~-) = 1 - - (1 - - F(T)) u
(15)
I n reality the cost to the predator of waiting for such a rare event is likely to be prohibitive.
Acknowledgments W e are immensely indebted to m a n y of our colleagues. Brian Bertram kindly loaned us his data to illustrate the model. The model itself emerged from discussions with Dave Stephens, Alasdair H o u s t o n , A l a n G r a f e n a n d R o b Kirkwood. Stimulating discussion a n d constractive criticism came from Chris Barnard, John Bithell, Les H u s o n , J o h n Lazarus a n d T o n y Pettit. M a r k Elgar gave us access to his data a n d discussed his analysis. R o n Pulliam, Des T h o m p s o n , R o n Ydenberg a n d a n a n o n y m o u s referee criticized earlier drafts. Carol M u n r o redrew the figures. D. W. L. a n d A. H. were supported by studentships from the Medical Research Council a n d the Science a n d Engineering Research Council respectively. REFERENCES Barnard, C. J., 1980. Flock feeding and time budgets in the house sparrow (Passer domesticus L.). Anim. Behav., 28, 295-309. Bertram, B. C. R. 1978. Pride of Lions. London: Dent. Bertram, B. C. R. 1980. Vigilance and group size in ostriches. Anita. Behav., 28, 278-286.
32,
4
Biben, M. 1979. Predation and predatory play behaviour of domestic cats. Anita. Behav., 27, 81-94. Caraco, T. 1982. Flock size and the organisation of behavioural sequences in juncos. Condor, 84, 101-105. Caraco, T., Martindale, S. & Pulliam, H. R. 1980. Avian flocking in the presence of a predator. Nature, Lond., 285, 400-401. Diaconis, P. & Efron, B. 1983. Computer-intensive methods in statistics. Scient. Am., 248 (5), 96-108. Elcavage, P. & Caraco, T. 1983. Vigilance behaviour in house sparrow flocks. Anita. Behav., 31, 303-304. Elgar, M. A. & Catterall, C. P. 1981. Flocking and predator surveillance in house sparrows: test of an hypothesis. Anita. Behav., 29, 868-872. Elliot, J. P., Cowan, I. M. & Holling, C. S. 1977. Prey capture by the African lion. Can. J. Zool., 55, 1811-1828. Jennings, T. & Evans, S. M. 1980. Influence of position in the flock and flock size on vigilance in the starling, Sturnus vulgaris. Anita. Behav., 28, 634635. Lazarus, J. 1979. The early warning function of flocking in birds: an experimental study with captive quelea. Anim. Behav., 27, 855-865. Lendrem, D. W. 1982. Vigilance in birds. D.PhiI. thesis, University of Oxford. Lendrem, D. W. 1983. Predation risk and vigilance in the blue tit (Parus caeruleus). Behav. Ecol. Sociobiol., 13, 9-13. Lendrem, D. W. 1984. Sleeping and vigilance in birds, II. An experimental study of the Barbary dove (Steptopelia risoria). Anim. Behav., 32, 243-248. Leyhausen, P. 1973. Verhaltensstudien an Katzen 3. Z. Tierpsychol. Beiheft, 2, 1-232. Pearson, E. S. & Hartley, H. O. 1972. Biometrika Tables for ,Statisticians, Vol. 2. Cambridge: Cambridge UniversityPress. Pulliam, H. R. 1973. On the advantages of flocking. J. theor. Biol., 38, 419-422. Pulliam, H. R., Pyke, G. H. & Caraco, T. 1982. The scanning behavior of juncos: a game-theoretical approach. J. theor. Biol., 95, 89-103. Schaller, G. B. 1972. The Serengeti Lion. Chicago: Chicago University Press. Schipper, W. J. A. 1977. Hunting in three European harriers (Circus) during the breeding season. Ardea, 65, 53-72. Siegel, S. 1956. Non-parametric Statistics for the Behavioural Sciences. New York: McGraw-Hill. Tinbergen, L. 1946. Die sperwer als roofvij an van zangvogels. Ardea, 34, 1-213. (Received 25 March 1983; revised 16 January 1984; MS. number: 2357)
VIGILANCE AND SCANNING PATTERNS IN BIRDS A N D R E W HART* & DENNIS WILLIAM L E N D R E M ~
Animal Behaviour Research Group, Department of Zoology, South Parks Road, Oxford, OX1 3PS, U.K. Abstract. We present a new approach to the analysis of scanning patterns in feeding birds. We estimate the probability of a bird detecting a predator from the frequency distribution of inter-scan intervals and the proportion of time spent scanning. This method avoids several unrealistic assumptions implied in earlier analyses of vigilance data, and can accommodate predators attacking randomly or using prey behaviour to time an attack. The practical application of this approach is illustrated using data for feeding and vigilance in the ostrich (Struthio camelus). The analysis is discussed with reference to the hunting tactics of predators and Pulliam's (1973) model of feeding and vigilance in birds. intervals is then wholly described by the rate. Unfortunately there is growing evidence that these assumptions are frequently violated. In a direct test of Pulliam's model Elgar & Catterall (1981) found departures from predicted scanning rates in flocks of house sparrows, which they attributed to the fact that scans are not instantaneous but of finite duration. Moreover Elcavage & Caraco (1983) and Lendrem (1982, 1983) report that birds rarely scan randomly, and Lazarus (t979) reports that birds scan synchronously rather than independently of each other. The method which we present here avoids all three of these unrealistic assumptions and encourages a more detailed examination of scanning patterns. We illustrate its application using data on feeding and vigilance in the ostrich kindly made available by Dr B. C. R. Bertram.
Scanning rates are widely used as indices of vigilance in feeding birds (e.g. Lazarus 1979; Barnard 1980; Bertram 1980; Caraco et al. 1980; Jennings & Evans 1980; Elgar & Catterall 1981 ; Caraco 1982). It is assumed that the probability of detecting an approaching predator increases as scanning rate (or time spent scanning) increases. However this ignores the pattern of scanning behaviour. Two birds may have identical scanning rates but very different scanning patterns, and the probability of predator detection will depend on those patterns. This is because what determines whether an attack at a particular time is detected is not average scanning behaviour, but whether the prey looks up before it is too late; that is, detection depends on the length of the particular inter-scan interval in which the attack occurs. This in turn depends on both the scanning pattern of the prey and the predator's choice of when to attack. In this paper we present a method for the analysis of scanning patterns in birds. The method estimates the probability of predator detection from the proportion of time spent scanning and the frequency distribution of interscan intervals, and it takes account of the tactics used by the predator. Like Pulliam (1973) we consider the probability of predator detection in an interval, % during which the predator breaks cover to come within striking distance of its prey. Pulliam's model assumes that birds scan randomly, instantaneously and, when in flocks, independently of one another. If these assumptions hold, mean scanning rate is indeed an adequate measure of vigilance, since the distribution of inter-scan
The Model Consider a lion hunting an ostrich. It stalks the ostrich unseen. Suddenly it breaks cover and makes a final uncovered dash to capture the ostrich. Let this dash take ~-s. The probability of our ostrich detecting the lion is the probability that it will make a scan before the lion completes its final uncovered dash. In order to calculate this probability we need to know the frequency distribution of inter-scan intervals, the proportion of time spent scanning and the tactics employed by the lion. In the approach outlined below we consider lions using three different kinds of hunting tactic. 'Random' lions do not use ostrich behaviour to time an attack; an attack is initiated regardless of whether an ostrich is feeding or scanning. In contrast, both 'check-wait' and 'double-check' lions delay their attack if an ostrich is scanning. The former (check-wait) attack if ostriches are feeding but otherwise delay until the start of a new feeding bout. The double-check lion always
Present addresses: *Mammals and Birds Department, Worplesdon Laboratory, Tangley Place, Guildford GU3 3LQ, U.K. ]'Department of Psychology, The Ridley Building, Claremont Place, Newcastle upon Tyne, NE1 7RU, U.K. 1216
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR waits until the start of the next feeding bout whether or not the ostrich is currently feeding. We first consider attacks against solitary ostriches. Solitary Prey Let the proportion of time spent scanning be ps and the probability density of inter-scan intervals bef(t). Both ps and the discrete form of f(t) may easily be obtained empirically (see later). If the lion attacks randomly then on a proportion (l-ps) of occasions the ostrich's head will be down, and the start of the attack will be undetected. How long will it be before the ostrich raises its head ? A randomly-timed attack is proportionately more likely to fall into a long inter-scan interval than into a short one, and more likely to fall into a frequent length of interval than an infrequent length. In fact, the proportion of all attacks which will fall into intervals of a certain length, t s, is given by the total time for all intervals of that length, as a proportion of all time. The total time in intervals of length t is proportional to tf(t), and thus the probability density of all attacks falling in intervals of this length is
tf(t)
g(t) -
(1) 0f ~176 tJ(t)dt
The function g(t) describes the frequency distribution of the lengths of intervals which are interrupted by a randomly attacking lion. The probability of detection depends on the length of the interrupted interval. If it is less than % the time taken for the final uncovered dash, then the probability of detection is one. If the interval is greater than z then detection depends on the point at which it has been interrupted; on .r/t occasions the attack will begin less than -r s before the next scan and the predator will be detected. Thus the probability of detecting an attack started at random within an inter-scan interval is given by + oo
G(-r) =o
g(t) dt
~-
tg(t) dt
(2)
However, on a proportion ps of occasions an ostrich will be scanning when an attack begins. The overall probability of detecting an attack by a 'random' lion is therefore pl(~) -- ps + (1 - ps) a(~)
(cf. Lendrem 1984).
(3)
1217
Cued Attacks on Solitary Prey We now derive the probability of detection if a lion uses the ostrich's behaviour to time its attack. As outlined above, two tactics of this sort are considered. In the first, check-wait, the lion does not attack randomly but delays if the ostrich is scanning. If the ostrich is not scanning it begins its attack immediately. The detection probability on the proportion (1-ps) of occasions when the ostrich is not scanning is given by G(~-), as in equation (2). On theps occasions when the attack is delayed until the end of a scan the probability of detection is given not by G(z), but by f(~-),
F(~') = 0
f
,r
f(t) dt
(4)
F(~') is the probability that the next scan will start within a period r when the attack begins at the start of an inter-scan interval rather than at a random point within the interval. Thus for this kind of tactic the overall detection probability is given by pz(~') = ps F(~-) q- (1 -- ps) G(~')
(5)
The second type of delaying tactic (doublecheck) is for a lion always to wait until the beginning of an inter-scan interval. For this kind of predator the overall detection probability is simply p~(z) = F(~-)
(6)
Attacks on Groups of Prey Calculating levels of corporate vigilance (the probability that at least one member of a group will detect an approaching predator) is fairly straightforward if we assume that individual birds scan independently of each other (see Appendix). However, it is even easier if we sidestep the question of independence altogether by treating the group as a unit and asking directly what is the probability of the group detecting an approach predator? Levels of corporate vigilance can then be calculated using equations (3), (5) and (6) above, provided we derive F(~-) and G(~-) from the distribution of intervals when all birds have their heads down and ps is the observed proportion of time when one or more birds are scanning. The raw data must therefore consist at least of timed records of the beginning and end of each period for which no member of the group is scanning.
1218
ANIMAL
BEHAVIOUR,
Calcuiating Detection Probabilities from Real Data Our analysis provides a measure of vigilance behaviour which takes account of both the amount and the .pattern of scanning behaviour. This measure is the detection probability, calculated from the data for one or more predator types and over a range of values for -r. Basic procedures for maldng comparisons are applied to data on ostriches in the next section. Here we deal with the use of the formulae derived above in calculating detection probabilities and plotting vigilance curves from real data. The first step is to construct the frequency distribution f(d) (the discrete analogue of f(t) in the model) by sorting the observed inter-scan intervals into classes. These classes must be arranged such that the values of ~- for which detection probabilities are to be calculated fall on the class boundaries. The choice of the number of classes to use is critical, and the usual guidelines for constructing histograms apply. If the classes are too narrow then the number of observations in each class will be small, and the general form of the distribution is more likely to be obscured by sampling fluctuations. If, on the other hand, classes are too wide, then resolution is lost and detailed features of the distribution may be overlooked. If it is only possible to obtain a reasonably 'smooth' histogram by using a very small number of classes then more data are required. Suppose that the number of classes is c, denoted by i-= 1, 2, 3. . . . . c. If the xth interval class is the one whose upper limit is -r then the discrete form of F(-r) is obtained as X
F(,)
=
Xf(d~) i=1
(7)
where di is an inter-scan interval length representative of the ith class. In this case, classes are best represented by the median of the observed values which they contain. If the distribution is not strongly skewed or uneven, and if the classes are not too wide, then the midpoint will be an adequate measure. Using the same notation, the discrete form of G(~-)is obtained as x
G(T)=
Z g(dl) + i=1
C
~-
X --g(dO i~-x+l d~
(8)
32,
4
where g(dQ is obtained as
d, f(di) g(&) =
(9) r
Z d, f(d~) i--1 Equations (7), (8) and (9) are used to calculate G(~-) and F(~-) for successive values o f - r and substituted into the appropriate equation (from 3, 5, and 6) to obtain the respective detection probabilities. These are best presented graphically, plotted against ~-.
An Example: Vigilance in Ostriches Bertram (1980) observed ostriches (Struthio camelus) feeding in Tsavo West National Park, Kenya between July and October, 1977. The habitat in the park consists of open, undulating grasslands with many scattered bushes and a few trees. Observations were made of adults without young, and birds were considered to be part of a group if they were within 65 m of one another. Birds were usually 30-40 m apart. Further details of the study area and data collection methods are published in Bertram (1980). Bertram observed that, in pairs of ostriches, males seemed more vigilant than females. Accordingly we have analysed male-female and female-female pairs separately. In fact, further significant variation exists within group types for Bertram's data, but for the purposes of illustration we have ignored it. Frequency distributions of intervals when all heads were down in four different types of group are illustrated in Fig. 1. There are sufficient data in each group type for fairly narrow class intervals, of 5 s throughout. Levels of corporate vigilance were calculated without assuming independence using equations (3), (5) and (6). F(~-) and G(~-)were derived from the frequency distributions shown in Fig. 1 according to equations (7) and (8) as described above. The proportion of time for which one or more birds were scanning, ps, was obtained directly from the pooled data for each group type. The median values of ps over instances of each group type are given alongside the respective histograms in Fig. 1. Vigilance curves (the probability of predator detection as a function of predator approach time, ~-) were plotted for each type of group. Randomly attacking lions were at a serious disadvantage compared with their more patient counterparts (Fig. 2). In fact, unless scans are instantaneous (so ps = zero), the detection
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR probabilities for check-wait and double-check lions must always be lower than for randomly attacking lions, other things being equal. In order to show a difference between check-wait and double-check tactics it is sufficient to show that the birds are not raising their heads at random. Only if head-raising is random (a stationary Poisson process) will the expected waiting time to the end of an inter-scan interval be constant and independent of the time since the interval began. It is easy to see that if this is true then the probability of detecting an attack starting at any point during an inter-scan interval is the same, and therefore check-wait and double-check predators will be detected with equal probability. We tested the fit of the observed distributions in Fig. 1 to the negative exponential using the Kolmogorov-Smirnov one-sample test, as given by Pearson & Hartley (1972). This test does not involve grouping into interval classes and so is more sensitive to deviations. Using this test we found significant deviations from the negative exponential for all groups. This means that for all groups the differences (shown in Fig. 2) between
80 i60140L
solitary birds p. =0 343 s
2ot
N=253
OH
_.1
I
pairs
60
ps= 0.520
2
N=249
40
>,
o
the vigilance curves for double-check and checkwait attacks are significant. The nature of deviations from the negative exponential may not be clear from a simple histogram of the data. If the sample size is not large (np to 120) then a graphical method described by Pearson & Hartley (1972) may be useful. This method involves plotting the ranked observations against the expected values of the order statistic. If the distribution conforms to the negative exponential then the points should fall on a straight line with a slope that is estimated by the mean of the observations. Figure 3 shows the plot obtained using the data collected by Bertram for one instance of a solitary male ostrich. The Kolmogorov-Smirnov one-sample test shows that the distribution of inter-scan intervals for this ostrich deviates significantly from the negative exponential (Dmax = 1.44, N = 25, P < 0.01). The points in the plot against the expected values of the order statistic suggest that the shortest intervals are larger than expected, and the longest intervals shorter than expected on the basis of the negative exponential distribution. This result is typical of the majority of those for instances of solitary male ostriches, and indeed for solitary females and for groups.
1.0 0.8I 0.7 O.E ?, 0.~= 0.4 0.3 o~. 0.1
0 . 9 ~
P..~-iP--i.~-~.
81~male-female
solitary .
=9
80[
female-female
u.
Ps= 0.475 2 0 ~ = 1 6 8
pairs
80 r
three & four birds
600~---]
ps= 0 . 4 6 4
2oLE]:: ":!62 5
15
25
35 45
1219
.
.
.
.
.
.
pairs
male-female '
. . . . .
~
i
I
i
~' 1.o
0 10 20 30 40 50 0 10 20 30 40 50 55
Inter-scan intervals (s) Fig., 1. Frequency distributions of intervals when all birds were feeding, and the median proportion of time spent scanning (ps) in groups of ostriches.
Predator approach time (s) Fig. 2. Corporate vigilance in groups of ostriches. Randomly attacking lions (P1) were more likely to be detected than either check-wait (P2) or double-check (Pa) lions. Double-check lions were slightly but consistently more likely to pass undetected than check-wait lions.
ANIMAL
1220
BEHAVIOUR,
The vigilance curves are replotted for betweengroup comparison in Fig. 4. The probability of detecting a double-check predator does not depend on ps (see equation 6), and so a test for differences among the f(t) is a sufficient test for differences between the vigilance curves in Fig. 4C. The Kruskal-Wallis one-way non-parametric analysis of variance (Siegel 1956) revealed significant variation between the group types in the distribution of inter-scan intervals ( H = 28.11, c.lf= 3, P<0.0001; see Fig. 1 for histograms). The Kolmogorov-Smirnov two-sample test could be used for pairwise comparisons of the f(t), but it is in fact often less sensitive (Pearson & Hartley 1972). To compare the probability of detecting random or check-wait predators for different .group types it is necessary to test for differences m ps as well as f(t). Again the Kruskal-Wallis test indicates significant variation between grouptypes ( H = 9.21,df = 3, P < 0 . 0 5 ; medians forps are shown in Fig. 1). Figures 4A and 4B show how these significant differences in ps and f(t) combine to give differences in corporate vi.gilance. Unfortunately contradictions may arise. If the between-group-types variation in ps a n d f ( t ) have contrary implications for vigilance then the reliability of the between-groups differences in vigilance for random and checkwait predators becomes questionable. This is because the effects ofps andf(t), though individually significant, may effectively cancel each other out. In the present case this complication arises in the comparisons of pairs of females with solitary ostriches and with groups of three and
32,
4
four. The remaining comparisons are not affected, as may be seen by comparing the trends inf(t) andps (Fig. 1). Examination of Fig. 4 indicates that corporate vigilance does not simply increase with group size, but rather that male-female pairs were the most vigilant groups, and solitary birds and pairs of females least vigilant. Clearly it would be very useful if confidence limits could be attached to the vigilance curves, both for visual pairwise comparisons and to resolve the difficulties caused by contradictory differences i n f ( t ) and ps.
1.0A 0.9 0.8 0.7 0.6
0.5 0.4 0.,3
0.2 0.1 O.G
1.0
.
.
.
.
.
i
i
i
i
i
B
o.9 0.8
0.7 0.6
g
0.5 0.4 0.:3 0.9-
0.1 O.C
"-- 20
1.0c
>
0.9 O.E
~- 10 0
r
L
5
0
0
0
S
=1.44,N=25 ; P < 0 . 0 1
0.7 0.E 0.~
1
f
I
2
3
Expected order statistic
Fig. 3. Ranked inter-scan intervals for a solitary male ostrich plotted against the expected values of the order statistic. If the distribution of inter-scan intervals conforms to the negative exponential then the points should fall along the line whose slope (6.84) is given by the mean of the observations. The deviations shown by this individual are typical of the majority of Bertram's ostriches.
0.4 0.3 0.2 0.1 0,0
i
,
,
,
, . ,
=
,
i
,
0 10 20 30 4(3 50 Predator approach time (s)
Fig. 4. Corporate vigilance as a function of group size given (A) random, (B) check-wait and (C) double-check attacks. 1: solitary; 2 ~ : male-female pairs; 2~9: female-female pairs; 3-t-4: three and four birds.
HART & L E N D R E M : V I G I L A N C E A N D S C A N N I N G BEHAVIOUR
In fact it seems that confidence limits might be obtained by 'bootstrapping', a technique lucidly explained and demonstrated by Diaconis & Efron (1983), though some thought would have to be given to the way in which variation in ps andf(t) was combined. We have explored the use of the 'bootstrap' for double-check predators, which involves only fit). The results for a predator approach time of 5 s are presented in Fig. 5, which shows for each group type the distribution of detection probabilities obtained from 1000 bootstrap samples. Each bootstrap sample is of the same size as its parent dataset, and is computergenerated by sampling at random with replacement from the parent dataset. The results we obtained correspond broadly to the result of the Kruskal-Wallis analysis of variance: the confidence intervals shown in Fig. 5 indicate that at least some of the group types differ significantly in detection probability.
400-
~
200-
~_~
3 and 4 birds
I
--
~o 400- hale-female* 200.
pairs
400
* female-female
400.
*
7 "1
_~
200-
i
solitary
[L
0.0
015
1.0
Detection probability Fig. 5. Distribution of double-check predator detection probabilities (corresponding to ~ = 5 s in Fig. 4C) obtained from t000 'bootstrap' samples for each grouptype (see text). The arrows indicate the detection probabilities obtained from the original data. The 95 ~ confidence interval is given by the unshaded portion under each histogram, drawn to exclude the 25 most extreme bootstrap samples at each end.
1221
Discussion Interpretation Both Pulliam's (1973; Pulliam et al. 1982) model and our own employ the concept of 'predator approach time'. The interpretation of the models, and of the results of calculations based on them, depends upon what is meant by ~-. What we actually calculate is the probability that the ostriches will look up before the end of a period -r, given various assumptions about when ~- begins (corresponding to various predator tactics). In interpreting this as the probability of detection during ~- we make an assumption about the probabilities of detecting a predator in its final dash, given that the prey is scanning (probability of detection assumed one) or is not scanning (probability of detection assumed zero). In interpreting our 'vigilance curves' as showing overall probabilities of detection we make assumptions about the probability of detection before and after % essentially that the probability of detection is zero during the stalk which precedes the final dash. A third type of assumption, which we have not implied in this paper, affords interpretation of the results in terms of the probability of capture. This type of assumption concerns the probability of capture given that the predator is detected. The simple assumption here is that the prey always escapes if the predator is detected, and it ]nay be difficult to avoid implying this in discussing the adaptive value of alternative strategies assessed using 'detection probabilities'. So what is meant by ~-? It is effectively defined by the types of assumptions about detection and capture that we have just discussed. If these assumptions are made in the simple ways outlined above, then ~ is the time taken for the predator to travel from a point where it breaks from perfect cover into perfect observability, to a point where it moves from a range at which alert prey always escape into a range at which a capture is inevitable. This definition of ~- is implied if we state or imply that the vigilance curves measure the probability of capture. It is obviously unrealistic but at the comparative level of analysis this is unlikely to cause serious error. If on the other hand we propose a detailed cost-benefit analysis of the vigilance behaviour of a particular animal, then it becomes necessary to define ~- such that we can measure it and obtain its frequency distribution in the field. Furthermore, if we wish to assess the evolutionary stability of one vigilance strategy against a
1222
ANIMAL
BEHAVIOUR,
hypothetical alternative then we must take account of the changes in the predator population and in predator tactics that could occur if the mutant prey become common. This is a much more difficult problem.
Hunting Tactics Our approach gives detection probabilities for two kinds of predator: those attacking randomly and those using prey behaviour to time an attack. The former is, perhaps, more usually associated with attacks by avian predators. The marsh harrier (Circus aeruginosus), for example, usually quarters flat areas in low flight using ground vegetation and habitat edges to surprise prey. Once the prey is located, it drops with claws outstretched. Attacks are seldom repeated if capture fails (Schipper 1977). Similarly Tinbergen (1946) describes how the sparrowhawk (Accipiter nisus) flies low along hedges, slipping from one side to the other, along wood edges, among bushes and buildings, ready to snatch any unsuspecting bird. There is no evidence that these predators use prey behaviour to time an attack. In contrast, many ground predators are known to use prey behaviour, at least during the stalk if not at the attack. Thus Leyhausen (1973) reports that during a stalk, domestic cats (Felis domesticus) freeze when prey are vigilant. Biben (1979) was unable to confirm this experimentally, but the tests were carried out in small experimental rooms with poor ground cover. Leyhausen reports that in such conditions cats dispense with the preliminary stalk and rush at prey (Leyhausen 1973). Some predators, including lions, are known to wait until a prey animal has its head down feeding before launching an attack (Schaller 1972). Of the two tactics exploiting prey behaviour that we have considered, check-wait predators are marginally less successful than double-check predators. However, double-check predators will tend to incur higher waiting-time costs. That is, they will have to spend more time waiting for a prey animal to put its head down, extra time in which they might be spotted or in which prey might move out of range. Moreover, this cost will increase with group size for as group size increases so the proportion of time one or more birds spend scanning will tend to increase. In fact, lions attacking larger groups of prey switch from a 'stalk' to a 'running crouch' tactic (Schaller 1972; Bertram 1978). Although such a running crouch is more likely
32,
4
to be detected, calculations by Elliott et al. (1977) suggest that such a switch will lead to a significant increase in the probability of a successful hunt since it increases the lion's velocity relative to the prey at the moment of detection. This change in tactics may represent the abandonment of tactics using prey behaviour.
Comparison with Pulliam's Model The assumptions that determine the nature of r are common to our model and that of Pulliam (1973). Where the two differ is in the assumptions made in deriving the probability that the prey will look up within the period r. (1) Scans are assumed to be instantaneous in Pulliam's (1973) model (though not in that of Pulliam et al. 1982). This assumption is generally unrealistic and is not made by our model. (2) Pulliam (1973; Pulliam et al. 1982) assumes that the distribution of inter-scan intervals for individuals follows the negative exponential. This assumption fails for the ostriches observed by Bertram (see his Fig. 3, 1980; the Kolmogorov-Smirnov test given by Pearson & Hartley indicates significant deviations from the negative exponential for ostriches alone and in groups). It also fails for blue tits (Lendrem 1983) and in many cases for house sparrows (Elgar & Catterall 1981; Elcavage & Caraco 1983). The cost of retaining this assumption when it fails is twofold: the reliability of the vigilance measure obtained is reduced, and the investigator may overlook important information about the predator-prey interaction that a more detailed examination of the scanning pattern would reveal. Our model makes no assumption about the distribution of inter-scan intervals. (3) In deriving the probability of detection for groups of prey Pulliam (1973; Pulliam et al. 1982) assumes that the prey scan independently of one another. We have shown how group vigilance can be calculated using our model with or without the assumption of independence. We tried calculating vigilance levels using the independence assumption (method in Appendix) but they were much higher than those in Fig. 4, which suggests important deviations from independence. Bertram (1980), using the same data, shows that ostriches in groups of three and four had their heads up concurrently slightly, but significantly, more of the time than would be expected if they were behaving independently. Lazarus (1979) observed that the probability of quelea in flocks detecting an artificial stimulus was not so high as predicted assuming that they
HART & LENDREM: VIGILANCE AND SCANNING BEHAVIOUR were scanning independently. Given the choice, we prefer to avoid the independence assumption unless it can be shown that it is not violated. (4) Pulliam (1973; Pulliam et al. 1982) assumes a randomly-attacking predator. Extension to predators that cue their attacks using prey behaviour in a simple manner is straightforward and illuminating. The probability of detection for different cueing tactics may depend crucially on the form of the distribution of inter-scan intervals, a fact which emphasizes the value of our non-parametric analysis. The adaptive significance of such differences is difficult to assess unless the cost to the predator of waiting for the moment to attack can be quantified. Nonetheless, many ground predators are known to use prey behaviour and in these cases models for cueing predators will give more realistic detection probabilities. Finally, we reiterate that while our model is more realistic than Pulliam's it does retain simplistic assumptions in the definition of predator approach time. Either Pulliam's model or ours, as appropriate, could be employed as a basis for further development in cases where suitable data are available. Examples of modifications that might be considered are to allow variation in the probability of detection according to the length of scan or the number of group members scanning, and to investigate whether probability of escape depends on the distance from the predator when detected. Another matter for attention is the possibility of a correlation between the lengths of successive inter-scan intervals, which would offer great advantages to predators that used it to cue their attacks. Obviously, data interpretation will be prejudiced if complications such as these exist in a particular case and are ignored. It is up to the investigator to choose the most appropriate vigilance measure for the problem in hand; what we have done is to provide an alternative starting point to Pulliam's model, and indicate the value of detailed inspection of the data.
Appendix: Corporate Vigilance Assuming Independent Scanning If ostriches scan independently of each other then the probability of a group detecting an approaching predator (their 'corporate vigilance') is one minus the probability of all birds failing to detect that lion. Once again, the probability of an individual failing to detect a lion will depend on the tactics employed by that lion.
1223
Random Attacks on Groups of Prey The probability of at least one member of a group detecting an attack that begins at a random point during a period when all heads are down is given by n ( . ) : 1 -- (1 -- G(-c))N (10) where N is the group size, and G(.) is given by equation (2). This assumes that group members are scanning independently of one another. So for a randomly-timed attack, the probability of detection for attacks that start when no ostrich is scanning will be H(.), but attacks that start when one or more ostriches are scanning will always be detected. Thus the overall detection probability PI(T) = ps + (1 -- ps)n(7)
(11)
where ps is now the proportion of time one or more ostriches spend scanning if birds are behaving independently of each other. For example, the proportion of time one or more birds spend scanning, ps, in a pair of birds is ps = x + y - - x Y , where x and y are the proportions of time for which each bird has its head up. That is, ps is the sum of the time each bird has its head up minus the time that both have their heads up concurrently (see also Bertram 1980).
Cued Attacks on Groups of Prey The probability of one or more ostriches detecting a lion that cues its attack on ostrich behaviour will again depend on the precise tactics employed by the lion. The last bird to put its head down to feed will detect a double-check tactician with a probability F(~-). The remaining birds will detect that lion each with a probability G(~-). Thus their corporate vigilance will be Pa(~') = 1 -- (1 -- F(~-)). (1 -- G(~-))N-1 (12) again assuming that birds are scanning independently. For check-wait tacticians, on a proportion ps of the time, one or more birds will be scanning and the lion will wait until the start of the next period when all heads are down. On this ps of occasions the probability of detection will be the same as that for a double-check tactician (equation 12). However, for (1--ps) of the time all birds will be feeding and the lion will begin its attack immediately. On these occasions the probability of detection will be given simply by H(~-). Thus the overall probability of detection for a check-wait predator will be Pz(~') = p s P a + (1 --ps)H(r) (13)
t224
ANIMAL
BEHAVIOUR,
where H ( , ) is given by equation (10) a n d ps is the p r o p o r t i o n of time when one or more ostriches are scanning, assuming independence (as outlined above). Finally, if a lion waits for two or more ostriches to p u t their heads down simultaneously before b e g i n n i n g an attack then the probability of predator detection is given b y P(~-) = 1 - - (1 - - F(~-))~v' . (1 - - a(~-)) iv-N' (14) where N is the group size a n d N ' the n u m b e r of individuals o n whose behaviour the attack is cued. I n the extreme case where all birds lower their heads simultaneously then P(~-) = 1 - - (1 - - F(T)) u
(15)
I n reality the cost to the predator of waiting for such a rare event is likely to be prohibitive.
Acknowledgments W e are immensely indebted to m a n y of our colleagues. Brian Bertram kindly loaned us his data to illustrate the model. The model itself emerged from discussions with Dave Stephens, Alasdair H o u s t o n , A l a n G r a f e n a n d R o b Kirkwood. Stimulating discussion a n d constractive criticism came from Chris Barnard, John Bithell, Les H u s o n , J o h n Lazarus a n d T o n y Pettit. M a r k Elgar gave us access to his data a n d discussed his analysis. R o n Pulliam, Des T h o m p s o n , R o n Ydenberg a n d a n a n o n y m o u s referee criticized earlier drafts. Carol M u n r o redrew the figures. D. W. L. a n d A. H. were supported by studentships from the Medical Research Council a n d the Science a n d Engineering Research Council respectively. REFERENCES Barnard, C. J., 1980. Flock feeding and time budgets in the house sparrow (Passer domesticus L.). Anim. Behav., 28, 295-309. Bertram, B. C. R. 1978. Pride of Lions. London: Dent. Bertram, B. C. R. 1980. Vigilance and group size in ostriches. Anita. Behav., 28, 278-286.
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