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Dr R. W. G. Caldow, Centre for Ecology and Hydrology, CEH Dorset, Winfrith Technology Centre, Winfrith Newburgh, Dorchester, Dorset DT2 8ZD. Tel.: 01305 213568, Fax: 01305 213600, E-mail:firstname.lastname@example.org
1Type II functional responses, which can be described by Holling’s disc equation, have been found in many studies of predator/prey and host/parasite interactions. However, an increasing number of studies have shown that the assumptions on which the disc equation is based do not necessarily hold. We examine the functional response of kleptoparasitically feeding Arctic skuas (Stercorarius parasiticus L.) to the abundance of fish-carrying auks and, by examination of the assumptions of the disc equation, test whether it can explain the function.
2The rate at which individual skuas make successful chases is a decelerating function of the abundance of auks. However, it would appear that this is not determined by factors that influence their probability of success, but by the rate at which they initiate chases. This too is a decelerating function of the abundance of auks. Arctic skuas have a Type II functional response.
3Although Arctic skuas exhibited a direct numerical response there was no evidence that components of predation connected to the density of predators (direct prey stealing, or increased host avoidance) had any effect on the rate at which individual skuas made chases or were successful in their chases. We conclude that the observed functional response is free from any effects of interference.
4We suggest that abnormally high levels of foraging effort expended by breeding skuas and their poor breeding success in the years of observation argue against the limit to the observed functional response being determined by skuas’ energetic requirements.
5Several of the assumptions underlying the disc equation do not hold. The duration of chases (handling time) was not a constant; it decreased with increasing host abundance. Moreover, the chase duration predicted by the disc equation, if handling time limited the functional response, was far in excess of that observed. Furthermore, the observed rate of decline in the searching time per victim with increasing host abundance suggested that skuas’ instantaneous rate of discovery was also not constant. Possible reasons for these observations are discussed. The basic disc equation may describe Arctic skuas’ functional response, but it cannot explain it.
In a series of landmark papers Holling (1959a,b, 1966) developed an approach to understanding the process of predation or parasitism which acknowledged that ‘the characteristics of any specific example of a complex process can be determined by the action and interaction of a number of discrete components’. Holling (1966) identified 10 components of the functional response, i.e. the change in the number of prey consumed/hosts parasitized by individual predators or parasites in response to changes of prey/host density (Solomon 1949). Those that are present in all situations he termed ‘basic’ and the others he termed ‘subsidiary’. The three basic components of the response of predators to the density of their prey (D) are: (i) the instantaneous rate of discovery (a); (ii) the time for which predator and prey are exposed (T); and (iii) the handling time per prey captured (h) (Holling 1966). These form the basis of Holling’s disc equation (Holling 1959b):
( eqn 1)
where N= number of prey taken by a predator in time T. This equation describes a decelerating rate of increase of feeding rate to an asymptotic value as prey density increases (Holling 1959a) and is based on the assumptions that a predator: (i) takes all the prey it encounters (i.e. there is no selectivity), (ii) searches randomly, and that (iii) the predator’s instantaneous rate of discovery, and (iv) the handling time required to kill and consume a prey item are constant and independent of prey density or feeding rate. Holling (1959b) found that the disc equation could explain a functional response generated by a very simple experimental predation system and provide an adequate description of a variety of published host-parasite responses. However, Holling noted that it was ‘dangerous to suppose that it completely explains the responses of these parasites’. Subsequently, Hassell, Lawton & Beddington (1976) and Hassell (1978) pointed out that despite the ability of such simple models to describe data, it is unlikely that both the rate of discovery a and handling time h are constant. Both are functions of several subcomponents, some of which have been shown in studies of invertebrate predators or parasites to vary with prey density or predator feeding rate (Hassell et al. 1976; Hassell 1978).
Type II functional responses, like the other two types identified by Holling (1959a), must ultimately reach an upper limit that is set by the subsistence requirements of the predator or parasite (Holling 1959a). However, the form of the disc equation is such that when this ultimate limit does not operate its asymptotic value will be determined by the time required to kill, i.e. the asymptote is the reciprocal of the handing time. Although, in one experiment Holling (1959b) found a close match between measured values of handling time and the estimate derived from the fitted disc equation, this was not the case in another experiment in which the measured handling time was lower than that calculated. Hassell et al. (1976) point out that such discrepancies are almost inevitable because the calculated value of handling time includes any periods of non-searching activity. In agreement with this, Wanink & Zwarts (1985) demonstrated that in several studies across a range of taxa the observed mean handling time was considerably less than that necessary to set the plateau of an observed functional response. Thus, another ‘assumption’ of the basic disc equation is often violated.
Holling (1959a) noted that the functional responses of insect parasites searching out their hosts could be expected to be similar in form to those of predators seeking prey. Kleptoparasitism (Rothschild & Clay 1952; Brockmann & Barnard 1979) is a foraging strategy in which kleptoparasites gather food by stealing it from others. In this paper, we present data on the functional response of a kleptoparasitic bird, the Arctic skua Stercorarius parasiticus (L.) to the abundance of its hosts, fish-carrying auks (family Alcidae), that they pursue in order to steal their fish.
Arctic skuas kleptoparasitize their victims during the breeding season by intercepting a continuous flow of hosts returning to their nests with fish. Thus, their prey is effectively replenished as it is exploited. It should therefore be possible to make accurate estimates of the constant instantaneous coefficients of the functional response from empirical data (Hassell et al. 1976). First, we establish the shape of skuas’ numerical and functional responses, and determine whether the latter can be described by a Type II response. We then seek to establish whether, in this non-experimental study, skuas’ numerical and functional responses are simply spurious, and the result of uncontrolled environmental effects. We then explore whether there is any evidence for confounding predator density effects on one of the key components of the rate at which skuas make successful chases, i.e. the probability that having initiated a chase, they succeed in stealing the fish. Finally, we go on to establish whether the assumptions underlying the disc equation hold true, namely that handling time and rate of discovery are constant, and limiting in the case of the former and, hence, whether Holling’s simple disc equation can adequately explain skuas’ functional response.
Fieldwork was carried out on Foula, an island that lies west of mainland Shetland (60°08′N 02°05′W). Arctic skuas breeding here obtain most of their food by chasing sandeel (Ammodytes spp.)-carrying hosts within 1 km of the east coast of the island (Furness 1978). All observations were made from a cliff top observation point at Heddlicliff on Foula’s east coast. The study area extended approximately 500 m along the cliff and 500 m out to sea. Observations were made between 10·00 and 19·00 BST on alternate days between mid-June and mid-August 1986, and at approximately weekly intervals between mid-May and mid-August 1987. Observation sessions were divided into periods of 1 h. During an hour, either all interactions between Arctic skuas and their victims within the study area were recorded (‘chase watches’), or observations were made of all the auks flying in-front of the cliff-face and the number of fish-carrying auks arriving was counted (‘potential victim watches’). Hereafter, all references to auk numbers in the text refer to those carrying fish. Potential victims comprised puffins Fraterculaarctica (L.), guillemots Uriaaalge (Pontopp.), razorbills Alcatorda (L.) and an occasional black guillemot Cephusgrylle (L.). Generally, hours spent observing chases and counting potential victims were alternated.
For each interaction recorded in 1986 a large number of variables was recorded (see Caldow & Furness 1991 for details). Of relevance here are: (i) the speed of reaction of the victim to the attack (classified following Furness (1978) as ‘fast’, ‘intermediate’ or ‘slow’); (ii) the duration of the chase (from the instant that the skua began accelerated flight towards the victim until the skua broke off (unsuccessful) or secured fish (successful); (iii) the number of attackers; (iv) whether the victim lost its fish; and (v) which (if any) attacker succeeded in securing fish. All variables except the speed of reaction were also recorded in 1987. At the start and end of each hour spent observing chases, and on every quarter hour in-between, the number of Arctic skuas within the study area was recorded. The average of these counts was used as a measure of the density of competing skuas and, in conjunction with the number of chases and the number of successful chases observed per hour, to derive the number of chases skua−1 h−1and the number of successful chases skua−1 h−1.
Periods of time were also spent each day recording the length of time that individual skuas spent searching for a victim. A patrolling skua was selected at random and followed until it made an attack. The length of time from the end of that attack until the start of the next one was then recorded. If a searching skua went out of sight before commencing a second attack, the length of time that it searched before disappearing was recorded. In general at least 10 such records were made in succession on, as far as could be determined, different skuas.
Records for a variety of meteorological variables that could have influenced the arrival patterns of auks or the behaviour of skuas were obtained from the weather station on Foula. Variables recorded were: maximum and minimum daily temperature (dry bulb, °C) between 09·00 and 21·00 BST, rainfall (mm) between 09·00 and 21·00 BST, and visibility scored [on a scale from 0 (< 100 m) to 86 (60 km)] every 3 h between 06·00 and 21·00 BST.
Because records made during ‘chase watches’ and ‘potential victim watches’ were collected alternately, skua and chase observations in 1 h had to be related to the number of potential victims counted either during the preceding or following hour, or if possible to the average numbers during both. A similar approach was applied when relating searching times to the density of skuas and the abundance of potential victims.
The validity of this approach was investigated by statistical analysis of the relationship between the number of auks arriving in any given hour (NT1) and the number arriving in a subsequent observation period up to a maximum of three hours later on the same day (NT2). This revealed a significant relationship with an intercept that did not differ significantly from zero [NT2 = 5·59 + 0·817*NT1r2 = 75·7% d.f. = 29 P < 0·001 (95% CI of intercept = −0·38 − 11·55, 95% CI of slope = 0·64 − 0·99)]. A subsequent model fitted without an intercept had a slope that did not differ significantly from one [NT2 = 0·936*NT1t = −1·02 d.f. = 30, NS (95% CI of slope = 0·81 − 1·06)] (Fig. 1). Interpolation of auk arrivals from preceding and subsequent observation periods on a given day is therefore unlikely to result in the use of inappropriate values in the analyses.
Analyses of skuas’ numerical and functional responses involved fitting a variety of non-linear functions to the raw data. This was done using the NLIN procedure of SAS (SAS 1989). In analyses of the influence of competitor density and the abundance of potential victims on various aspects of chases, the mean number of skuas in the study area during an hour was used as the measure of competitor density and the number of auks arriving per hour was used as the measure of victim abundance. In the case of categorical variables, either the binary (BLOGISTIC) or ordinal (OLOGISTIC) logistic regression procedures of minitab were used (minitab 1996) to examine the proportions of chases in each category in each hour. In other cases, ordinary least squares regressions were calculated. To avoid problems of pseudo-replication, analyses were conducted on mean chase durations (averaged over all chases within an hour) and mean searching times (averaged over all records within a session). Proportions and means used as values of the response variables were included in the analyses only if they were based on 10 or more records.
In this field study, the variation in the abundance of auks arose naturally. Accordingly, skuas’ functional response to the abundance of auks was not derived under constant conditions. Moreover, there was of course, no way in which to ensure a constant density of ‘predators’, a condition essential to the application of the basic disc equation to natural situations (Holling 1959b). We therefore examined the factors that might explain the variation in the abundance of auks and then explored whether skuas’ numerical and functional responses to the abundance of auks might be influenced by these same factors, independently of the abundance of auks. The relationships of diurnal, seasonal and weather covariates with each of auk numbers, skua numbers and skua chase rates were assessed. The partial relationships between skua numbers and chase rates, and auk numbers, allowing for correlations with the other factors, was assessed with multiple regressions involving the double square root transformation of auk numbers (auks0·25). This approximated the non-linear relationship between auks and skuas (see below). The covariates examined were: date (days since 1st June), time of day (minutes since midnight), maximum daytime temperature, daytime rainfall and visibility.
Describing skuas’ numerical response
The number of Arctic skuas was positively related to the abundance of auks. This relationship was better described, as judged by r2 values, by a decelerating function than by a linear relationship (Fig. 2a, Table 1). Because of this direct numerical response, any plateau to the functional response may reflect the effects of interference between skuas (Holling 1966). Applying the basic functional response equation may be inappropriate, unless the density of competitors has no measurable effect on the key components of the functional response.
Table 1. Results from fitting a linear and three non-linear regression models to hourly records (n = 75) of; the number of Arctic skuas (numerical response), the total number of chases h−1, the number of chases skua−1 h−1 and the number of successful chases skua−1 h−1 (functional response) in relation to the number of fish-carrying auks arriving per hour. The form of each model is presented. The Hyperbolic and Disc equation models are of identical format and, hence, generate the same r2 values. The former is fitted in all cases and the latter only in those in which it is appropriate to consider parameters a and b as being equivalent to the rate of discovery and handling time, respectively. For each model, the parameter estimates are given with their standard error in parentheses. Figures in bold denote parameters that differed significantly from zero at the 5% level
The total number of chases per hour was better described, as judged by r2 values, by a decelerating function of the abundance of auks than by a linear relationship (Fig. 2b, Table 1). There was a positive linear relationship between the total number of chases per hour and the average number of skuas in the study area (r2 = 0·627, d.f. = 74 P < 0·001). However, the number of chases skua−1 h−1, derived from the two preceding response variables, did not increase linearly with the number of potential victims, the relationship again being better described, as judged by r2 values, by any one of three decelerating functions (Fig. 3a, Table 1). An equation of the same form as Holling’s disc equation provided as good a fit to the data as the other two similar functions.
The number of occasions per hour on which an Arctic skua, having initiated a chase, succeeded in stealing fish, was also better described, as judged by r2 values, by decelerating functions of the abundance of auks than by a simple linear relationship (Fig. 3b, Table 1). In this case it was not, however, possible to fit a non-linear model in which both parameter estimates differed significantly from zero. Nonetheless, Arctic skuas clearly exhibit a Type II functional response to the abundance of auks, the rate at which they make successful chases being approximately constant across most of the observed range of auk abundance values.
Exploring the effects of environmental covariates on skuas’ numerical and functional responses
The most important explanatory factor for the variation in the abundance of auks was seasonal. There was a pronounced quadratic effect (Table 2), auk numbers reaching a peak in mid-season. In conjunction with this seasonal effect there was no significant diurnal effect, a very marginal effect of visibility, but highly significant linear effects of both temperature and rainfall (Table 2). Auks tended to be more numerous on cooler, drier days. In excess of 60% of the variation in the abundance of auks was explained by a model incorporating seasonal, temperature and rainfall effects (Table 2).
Table 2. Multiple regression models relating the number of fish-carrying auks per hour to a variety of environmental variables: days since 1 June (days) and its square (days2), time of day (time), visibility (vis), maximum daytime temperature (maxt), minimum daytime temperature (mint) and daytime rainfall (rain). For each model the significance of the constant term and of the partial regression coefficient of each variable is given (* P < 0·05, ** P < 0·01, *** P < 0·001), as well as the overall adjusted r2 value and model significance. Var1, var 2, etc., refer to the order in which each of the variables is listed under each model
The four response variables presented in Figs 2 and 3 were all positively related to the abundance of auks (Table 3), but with the exception of the number of successful chases skua−1 h−1, the variation in these was better explained (as assessed by comparison of adjusted r2 values) by a quadratic seasonal effect than by the number of auks (Table 3). Controlling for the effect of either stage of the season or the abundance of auks (or both), neither temperature nor rainfall had a significant effect in any case. When models containing both the quadratic seasonal effect and the effect of auk numbers were examined, the former had a highly significant partial relationship in all cases (with the exception of the model of successful chases skua−1 h−1 in which all terms were non-significant), whereas the latter term did not (Table 3). It would seem that the rate at which auks return to their breeding colonies with fish varies primarily as a function of the stage of the season, and that the number of skuas in the study area and the rate at which they chased auks may also simply vary seasonally rather than in direct response to the abundance of auks.
Table 3. Multiple regression models relating: (i) the mean number of skuas in the study area, (ii) the total number of chases h−1, (iii) the number of chases skua−1 h−1, and (iv) the number of successful chases skua−1 h−1 to the abundance of fish-carrying auks (double square root transformed; auks0·25) and/or the number of days since 1 June (days) and its square (days2). For each model the significance of the constant term and of the partial regression coefficient of each variable is given (* P < 0·05, ** P < 0·01, *** P < 0·001), as well as the overall adjusted r2 value and model significance
Although these results might suggest that the positive relationships between skua numbers, and chasing activity and auk numbers are spurious, they cannot explain the observed asymptotic nature of these relationships. The asymptotes to these functions might, however, arise if environmental conditions when auks were particularly abundant differed in some way from other records and acted to depress skua activity. To test this possibility we compared conditions on the occasions on which auk numbers were in excess of 40 h−1 (n = 7) with those when auks were less abundant (n = 12) during the same time window (20th June–18th July). There were no significant differences in any of the variables examined between the two groups of records (Table 4). There is therefore no evidence that when auks were particularly abundant, skua numbers or activity might have been depressed due to particular environmental conditions. Thus, in the following sections we seek alternative explanations for the observed asymptotic shape of the functional response.
Table 4. Comparison, by means of two-sample Wilcoxon rank sum tests, of the environmental variables recorded on those occasions (n = 7), when the abundance of fish-carrying auks h−1 was in excess of 40 (sample 2) and the other records within the same time window (20 June–18 July) when auks were less numerous (n = 12; sample 1). The values shown are the medians of each sample, the value of the test statistic and its significance
Time (minutes after midnight)
Maximum daytime temperature
Minimum daytime temperature
Explaining the functional response
Probability of success
One of the key components of the rate at which skuas make successful chases is the probability that having initiated a chase, they succeed in stealing the fish. Is there any evidence that the abundance of potential victims or competitors influences this, and explains the observed shape of the functional response?
The probability that a skua secured fish was closely associated with the speed with which victims reacted (χ2= 19·47, d.f. = 2 P < 0·001). Arctic skuas secured fish in 6, 8 and 18% of chases when the victim reacted quickly, intermediately and slowly. Because victims vary in their response times and, hence, the probability of being chased successfully, optimal foraging theory would predict that skuas should become increasingly selective of individuals that can be chased successfully as the availability of potential victims increases. They should avoid chasing victims that are likely to respond to them when they are still far away. Skuas’ direct numerical response may mean that through avoidance learning by prey (Holling 1966) victims are more vigilant and, hence, react faster when skuas, and hence victims themselves are more abundant. This could lead to a density-dependent decline in skuas’ success rate that may counteract any increasing selectivity for suitable victims as their abundance increases. However, the speed with which victims reacted to being attacked was not significantly related to either the number of potential victims or to the number of Arctic skuas in the study area (Table 5). There is no evidence that as victims become more abundant, skuas might achieve an increasing success rate by increasingly avoiding those that are likely to react to them while still far away, or that indirect interference due to increasing vigilance of the victims may limit skuas’ success rate.
Table 5. The effect of the average number of Arctic skuas foraging in the study area (competitor density) and the abundance of fish-carrying auks (encounter rate) on: victims’ speeds of reaction; the probability of a chase being joined by one or more skuas; the probability of a skua, having initiated a chase, being successful in stealing fish; chase duration (handling time); and search time per victim. Linear regression models were constructed in all five analyses. In the case of categorical response variables, ordinal or binary logistic regression was performed. In the case of searching time, the abundance of fish-carrying auks was log-transformed as this yielded a considerably better linear model fit. The number of hourly chase watches/searching time sessions (n) that generated proportions or means based on 10 or more records is given in the last row. * P < 0·05, ** P < 0·01, *** P < 0·001
Skuas may also interfere with one another and reduce the probability of success for chase initiators by joining one another’s chases and competing directly for the victim’s fish. If, as might be expected, such contests occur more frequently when the density of competitors is high, skuas’ direct numerical response could again, through the effects of direct interference, limit the rate at which chase initiators make successful chases. The proportion of chases during an hour that involved more than one Arctic skua was not significantly related to the abundance of auks but was positively related to the number of skuas in the study area (Table 5). However, the probability that a victim would drop its fish was significantly related to the number of Arctic skuas in pursuit (0·17, 0·26 and 0·64 when chased by 1, 2 and 3 skuas, respectively; χ2= 27·68, d.f. = 2 P < 0·001). So, too, was the probability that a chase would result in at least one of the attackers securing a fish (0·11, 0·20 and 0·57 when chased by 1, 2 and 3 skuas, respectively; χ2= 33·89, d.f. = 2, P < 0·001). However, the probability that the skua that initiated a chase, succeeded in securing fish was not associated with the number of other skuas that joined it (0·11, 0·11 and 0·09, when on its own, and when joined by one or two skuas, respectively; χ2= 0·07, d.f. = 2 NS). The increased incidence of contested chases at high skua densities cannot explain the plateau in the rate at which chase initiators succeeded in securing fish.
Consistent with all of the above, the probability that an Arctic skua, having initiated a chase, secured fish was not related to either the number of potential victims or to the number of Arctic skuas in the study area (Table 5). All of these analyses suggest that factors that might influence the probability that a skua having initiated a chase will secure fish cannot explain the shape of their functional response. This conclusion is supported by the fact that the relationship between the rate at which skuas initiated chases and auk abundance (Fig. 3a) has the same basic form as the functional response (Fig. 3b). An explanation for the shape of the functional response may lie in the factors determining the rate at which skuas initiate chases.
The rate at which chases are initiated
Is handling time constant and limiting? The duration of all chases was negatively related to the number of potential victims, but not to the numbers of Arctic skuas in the study area (Fig. 4, Table 5). This decline in handling time with increasing abundance of victims should, as pointed out by Hassell et al. (1976), act to oppose the tendency for the number of chases initiated skua−1 h−1 to reach an asymptote as the abundance of victims increased. In addition, this variation in handling time violates one of the assumptions of the disc equation. Moreover, simple arithmetic indicates that handling time cannot limit skuas’ functional response. The average chase rate of skuas was 13·7 chases skua−1 h−1. The average duration of all chases was 7·5 s (95% CI 7·20–7·74 s n = 1408). This yields a total time spent chasing of a mere 103 s h−1. Even the highest derived value of chase rate of 55 chases skua−1 h−1 yields only 413 s chasing h−1. Moreover, the value of the handling time parameter necessary to explain the asymptotic value of 21 chases skua−1 h−1[0·048 h or 173 s (Table 1)] is far in excess of that observed (Fig. 4). The amount of time that skuas spend pursuing their victims does not limit the number of chases that they can make.
Is the rate of discovery constant? When a model of the same form as the disc equation was fitted to data relating the rate at which individual skuas initiate chases and the abundance of auks, the resulting estimate for the rate of discovery a was 2·504 (Table 1). Because we could not make a direct measure of a, we cannot directly verify this value. However, because a equals λD−1 (Wanink & Zwarts 1985), where λ = encounter rate: the inverse of searching time per prey item and D = prey density, it is possible to derive the relationship between the search time per victim and the abundance of potential victims that is required to generate a parameter a that is constant at 2·504, and to compare this relationship with that observed.
As expected, the length of time that a skua searched for a victim declined significantly with an increasing number of potential victims. There was, however, no detectable effect of the number of Arctic skuas in the study area (Table 5). A better fit was achieved if the number of auks was log transformed. Searching time per victim declined at a decelerating rate as the abundance of victims increased (Fig. 5). However, in order for a to be constant at 2·504, skuas would have had to find victims more slowly than they did at low auk abundances and more quickly than they did at high auk abundances (Fig. 5). When the observed values of both mean searching time per victim and auk abundance were logged, the resulting linear relationship had a slope of only − 0·23 (log10 search time per victim = 2·17 − 0·23*log10 number of fish carrying auks r2 = 31·0% d.f. = 31 P < 0·001). Thus, the search time per victim did not decline n−1 times as density increased n times, and hence a cannot have been constant. Indeed, from the observed relationship between searching time per victim and auk abundance it is possible to calculate that the value of a declined from 27·7 to 0·7 across the range of observed auk abundance (Fig. 6). A further assumption of the disc equation is violated.
Can the observed rate of chase initiation be explained on the basis of the observed density-dependence of handling time and searching time per victim? A function of the same form as the disc equation in which the observed density-dependence of both a and h was incorporated yielded a visibly poorer fit to the observed chase initiation rate data than that achieved when the disc equation with constant values for a and h was used to predict the functional response (Fig. 7). Given the observed density-dependent values of a and h skuas should have initiated considerably more chases per hour than they did. It would seem that the observed values of handling time, and the values of discovery rate derived from observed search times per victim preclude an explanation of the observed functional response solely on the basis of the disc equation and its three basic components.
Reliability of estimates of auk availability
The shapes of the functional and numerical responses presented here are dependent upon the counts of auks providing a good estimate of the availability of potential victims. Estimates will be in error if a fraction of the auks was unavailable. This will not seriously affect the calculations provided that the available fraction of auks was independent of their own density and that of the skuas (Wanink & Zwarts 1985). As the number of skuas present in an area increases, an increasing proportion of incoming auks may avoid the area, so reducing the fraction available. However, because auks feeding chicks are constrained to return to their nests, their scope for avoiding places (and times) where skuas are numerous is limited. Alternatively, returning auks may clump their arrivals in time, effectively reducing the fraction that can be chased, and may do so to differing degrees depending upon their own density and that of their attackers. We have no quantitative data to examine this idea, but pronounced clumping of auk arrivals in time was not particularly noticeable (R. Caldow personal observations). However, until further work is done, the discussion that follows is subject to the caveat that the records of the number of auks arriving per hour are reasonably good and unbiased estimates of the availability of potential victims.
Environmental explanations for skua responses
Because the various stages of the breeding season of skuas coincide roughly with those of auks there is necessarily, over the course of a season, a positive correlation between their numbers on Foula. However, such a correlation, which might be due simply to coincidence in the seasonal abundance of two independent groups of organisms, cannot explain the asymptotic nature of the numerical response. Moreover, the other three response variables (Figs 2b and 3a,b) represent the rate at which direct interactions occurred between kleptoparasites and their hosts. It is unlikely that the positive relationships between these variables and the abundance of auks are merely coincidental. Furthermore, the fact that, for example, the number of chases skua−1 h−1 exhibits a quadratic pattern over the course of the season in tandem with auk abundance cannot explain the asymptotic nature of these responses to auk abundance. The lack of any differences in environmental conditions between those occasions on which auks were most abundant and other records made during the same period of the season also refutes the idea that the asymptotic nature of the various responses was due to skuas’ foraging activity being depressed. We conclude that the asymptotic nature of the relationships depicted in Figs 2 and 3 are genuine responses to the abundance of auks.
Predator density effects
In order to apply the basic functional response equation to natural situations, the density of ‘predators’ must be constant (Holling 1959b). This was not the case here. However, there was no indication that the density of competing skuas influenced any of the parameters that might influence the rate at which skuas make successful chases. We conclude that the functional response derived here is effectively free from any effects of competitor density and that it is therefore appropriate to consider the applicability of the basic disc equation to this system.
Limitation due to requirements
Given sufficient resources, all functional responses eventually reach a plateau set by the subsistence requirements or satiation of the predator or parasite (Holling 1959a). Kenward (1985) suggested that the plateau of the kill-rate of pheasants Phasianus colchicus (L.) by goshawks Accipiter gentilis (L.) (c. 0·4 pheasants day−1) may be because a pheasant provides 3–4 days food for a goshawk. Redpath & Thirgood (1999) argued that the fact that the provisioning rates of chicks must level out at some stage provides an explanation for the plateaux of the functional responses of breeding hen harriers Circus cyaneus (L.) and peregrine falcons Falco peregrinus (Tunst.) to the density of red grouse Lagopus lagopus scoticus (Lath.). Is it possible that initiating c. 20 chases h−1 enabled Arctic skuas to gather food at the rate required to meet their energy demands and those of their chicks? In comparison with many other years, Arctic skuas foraged for considerably longer periods of time each day and achieved low chick-growth rates and fledging success in 1986 and 1987 (Phillips, Caldow & Furness 1996; Phillips, Furness & Caldow 1996). Accordingly, we feel that it is unlikely that the plateau in the rate at which skuas made chases was determined simply by their energetic requirements. We proceed, therefore, to explore the mechanistic interpretation of the parameters in Holling’s disc equation based on instantaneous rates.
Testing the assumptions of the disc equation
Constant handling time
One of the key assumptions of the disc equation is that handling time is constant and independent of prey density and feeding rate. Holling (1966) identified three subcomponents that completely define handling time: the time spent: (i) pursuing and subduing each prey, (ii) eating each prey, and (iii) in a ‘digestive pause’ after a prey is eaten and during which the predator is not hungry enough to attack. Each of these subcomponents can vary and, in some cases, have been shown to do so (Hassell et al. 1976). The consumption of fish by skuas occurs in a fraction of a second at the end of a chase (R. Caldow & R. Furness, personal observations) and only in a fraction of chases. Because these birds are collecting food for their young, their own hunger level is also not a factor. Thus, skuas’ handling time is determined almost entirely by the first of these subcomponents. None of the examples illustrated by Hassell et al. (1976) demonstrate variability in this subcomponent of handling time. However, Wanink & Zwarts (1985) and Hulscher (1976) noted that the handling time of an oystercatcher feeding on bivalves declined with increasing prey density. This was attributed to a decline in time spent lifting the shells out of the sediment and cutting the flesh out of the shells. We cannot yet provide a similarly detailed explanation for the observed decline in chase duration with increasing auk abundance but whatever the explanation, the assumption of a constant handling time clearly does not hold.
Another assumption of the disc equation is that a predator takes every prey that it encounters. Two studies of oystercatchers have shown that this assumption does not hold (Wanink & Zwarts 1985; Norris & Johnstone 1998). Faced with prey of varying profitability due either to variation in their depth in the sediment (Wanink & Zwarts 1985) or their size (Norris & Johnstone 1998), oystercatchers showed pronounced selectivity that varied with prey density. This was suggested to be the reason for the decline in handling time noted in the studies of Wanink & Zwarts (1985) and Hulscher (1976). Perhaps the decline in handling time with increasing auk abundance noted here, although slight, could have arisen by skuas becoming increasingly selective of victims that could be chased most quickly and, hence, with the least possible expenditure of energy, even if not necessarily with any more success.
Asymptote set by handling time
The formulation of the disc equation is such that the time required to handle each item of food determines the asymptote of the function. Simple arithmetic and the finding that the observed handling time (7·5 s) was considerably shorter than the calculated value (173 s) indicates that this is not the case here. This finding is, however, consistent with the expectations of Hassell et al. (1976). Such discrepancies arise because the calculated values of handling time always include any unidentified, time-consuming periods of non-searching activity, perhaps associated with prey identification (Holling 1959b) or an internal handling time (Visser & Reinders 1981). For reasons discussed above, an internal handling time or digestive pause is unlikely to be a factor here. We cannot, however, discount the possibility that before skuas begin accelerated flight towards an auk (which is when timing was started) they spend time examining it to establish whether or not it has fish. An observer may not record such an ‘identification time’ that may be associated with every chase. It would seem unlikely, however, that this alone can account for the discrepancy between observed and calculated handling times. There is, however, one other potentially time-consuming component associated with chasing auks that may be missed by an observer. Chases, although short, are conducted at very high speeds and in 90% of cases involve continuous flapping flight (R. Caldow unpublished data). It is likely that such sprint performances are energetically expensive. If so, some fraction of the time that has been defined as searching time may in reality be an additional component of handling time that comprises a recovery period after the end of each chase (or perhaps only after a series of chases). During this time, skuas may be unable or unwilling to initiate another chase. At high auk abundances the ‘real’ search time per victim may continue to decline [such that a is in fact constant (see below)], but this additional time-consuming component of the foraging process may reduce the time available for searching and limit the rate at which skuas can conduct chases.
Constant rate of discovery
Another assumption underlying the disc equation is that the instantaneous rate of discovery (searching effort or efficiency) is constant. This was shown to be the case in Holling’s ‘touch’ experiment. However, the existence of an additional time-consuming component associated with handling a prey item was associated with a density-dependent decline in a in Holling’s ‘sound’ experiment (Holling 1959b). Wanink & Zwarts (1985) found that an oystercatcher’s encounter rate (bivalves attacked per minute) increased with increasing prey density, but not in a linear fashion, and calculated that the bird’s instantaneous rate of discovery declined markedly with increasing prey density. The same would seem to be true here. Wanink & Zwarts (1985) suggested a number of explanations why a might not be constant. A decrease of a at higher prey densities may be due to the existence of some density-dependent, but unidentified time-consuming non-searching behaviour (see above). Alternatively, at higher prey densities the predator may spend a decreasing proportion of its time in actively searching. The values of a derived in this study are, however, based upon records of the time that skuas searched between successive victims separated by on average only some 75 s. This suggests that these were records of actively foraging birds. Two further problems in estimating a and its density-dependence were discussed by Wanink & Zwarts (1985); that of estimating the effective density of prey when the fraction of prey that are available may vary with density, and that of estimating the true encounter rate when the predator exhibits selectivity, which is itself density-dependent. As discussed above, both of these issues are relevant to the current study and require further attention.
Holling (1966) identified four subcomponents that completely define the rate of discovery: the reactive distance of the predator for prey, the speed of movement of the predator and prey and the capture success. Each of these subcomponents can also vary and in some cases have been shown to do so (Hassell et al. 1976). We have demonstrated that the last of these is constant. Unfortunately, we have no data on the others that would enable exploration of their role in generating a density-dependent decline in the rate of discovery.
There are two spurious reasons why the observed search times per victim did not decline as steeply as required to result in a constant value of a. First, auks may clump their arrivals in time, and skuas may search and chase mostly during such periods. If so, records of searching time at apparently low auk abundances may have been concentrated over short periods when the frequency of auk arrivals was, in fact, greater than suggested by tallies made over the course of complete hours. Furthermore, an inability to record search times when no skuas were present and the truncation of long searching times when skuas went out of sight may mean that average searching times were under-estimated, particularly at low victim abundances. This is almost certainly true. However, average inter-chase intervals derived from the values of chases skua−1 h−1 (on the assumption that chases occurred regularly) do not decline with increasing abundance of auks in the way necessary to generate a constant value for the rate of discovery. This method of estimating inter-chase intervals does not suffer from the problems associated with measuring search times directly, and suggests that the discrepancy between directly measured search times and those expected cannot have arisen entirely for spurious reasons.
Can the observed functional response be explained solely on the basis of observed searching and handling times?
Wanink & Zwarts (1985) showed that even when both handling time and rate of discovery are density-dependent, a Type II functional response can occur. By generating a family of functional responses based on observed density-dependent values of a and hWanink & Zwarts (1985) managed to get a better fit to the observed data than when the disc equation was used to predict the functional response. The relationships shown in Fig. 7 indicate that this is not true here. The reason for the poor fit is largely because, like most of the studies listed in Table 3 of Wanink & Zwarts (1985), but unlike their results, the observed handling times were too short to generate an asymptote at the observed level. When a function in which the observed density-dependence of the rate of discovery was again included but handling time took the fitted constant value of 0·048 h the fit to the observed asymptotic values of chase rate was good. Moreover, inclusion of the observed density-dependent decline in handling time acts to reduce the asymptotic tendency of the function (Hassell et al. 1976). In addition, the more rapid initial rise in the rate of chase initiation predicted on the basis of the observed density-dependent values of rate of discovery reflects the latter’s initially high values (c. 27 at 1 auk/ h) compared to the constant value of 2·5 generated in the disc equation. Holling (1959b) pointed out that when handling time varies with density the basic disc equation will not describe data well unless the rate of discovery varies in such a way as to mask the variability of the former. Only if the rate of discovery were to vary in a very particular fashion would the equation still fit (Holling 1959b). This does not seem to be the case in this study.
The elevated position of the second curve in Fig. 7 in comparison with the observed data arises almost entirely because it is based on the observed (short), handling times. As pointed out above, Hassell et al. (1976) and Hassell (1978) noted that direct observations of the time that ‘the predator or parasitoid spends visibly dealing with each prey’ almost inevitably provide lower estimates of handling time than values calculated by fitting equations to the data. This arises because in order to fit asymptotic data, the calculated value will also include any periods of non-searching activity that are not identified by an observer. In Holling’s ‘sound’ experiment, handling time was measured as 0·0495 min and calculated to be 0·075 (Holling 1959b). This discrepancy was attributed to an additional time-consuming (non-searching) behaviour, which reduced the time available for searching by a fixed amount for each item handled. Close observation revealed that this component was an ‘identification time’. Once distinguished, this component was included in a modified, more complex version of the basic disc equation. Two time-consuming, non-searching behaviours associated with chasing auks that an observer may have missed have been discussed above. If either (or both) of these components exist, and the discrepancies between observed and calculated handling times certainly suggest something else is involved, then clearly a simple two parameter model is an inadequate model of skuas’ functional response. A more complex model may be necessary in this, and probably many other systems.
In conclusion, in line with a number of studies of invertebrates [Hassell et al. 1976 (and references therein); Hassell 1978] and of birds (Hulscher 1976; Goss-Custard 1977b; Wanink & Zwarts 1985), the results of the current study indicate that several of the assumptions underlying Holling’s basic disc equation do not, in fact, hold. Accordingly, although a simple two-parameter model of the same form as the disc equation seems to provide as good a description of Arctic skuas’ functional response as a variety of other mathematical functions, the disc equation cannot provide an accurate explanation of the data in appropriate biological terms. The basic disc equation is, as pointed out by Holling (1959b), the most basic form of functional response that contains only those components that must be present in all situations. This study, in line with others clearly indicates that, as detailed in Holling’s papers (Holling 1959a,b, 1966), natural responses often involve other components that require somewhat more complicated explanatory models. We hope that this study will prompt others to adopt a more rigorous approach to fitting functional responses to field data. One should not simply fit curves by eye or fit the basic disc equation without testing its underlying assumptions, and checking whether the calculated values are both constant and the same as those observed.
We would like to thank the Holbourn family, for granting permission to conduct our fieldwork on Foula, and John Holbourn, for allowing access to meteorological records. We are also indebted to John Goss-Custard, for suggesting a complete revision of an early draft of this paper, and to Richard Stillman, for commenting on various drafts. We would also like to thank Marcus Rowcliffe and an anonymous referee for making helpful comments on our manuscript. This work was funded by The Carnegie Trust for the Universities of Scotland. RWF was funded by the International Fishmeal and Oil Manufacturers Association (IFOMA).
Received 5 October 2000; revision received 5 March 2001