The functional response
Functional responses in herbivores have generally been found to be either linear (Batzli, Jung & Guntenspergen 1981; Trundell & White 1981; Anderson & Sæther 1992) or, more commonly, asymptotic (Murton, Isaacson & Westwood 1966; Allden & Whittaker 1970; Wickstrom et al. 1984; Hudson & Frank 1987; Spalinger, Hanley & Robbins 1988; Gross et al. 1993). However, there is little agreement over how intake is defined in the case of herbivore functional responses (e.g. see Crawley 1983) and the results given here highlight the potential confusion arising from this lack of agreement on the definition. Originally, the functional response was defined in terms of the number of hosts attacked by an enemy (Solomon 1949), which is clearly inappropriate to herbivores. Since then, the functional responses of herbivores have been given in terms of either bites, biomass or plant parts consumed per unit time. Studies also differ as to whether intake is measured per unit time spent feeding or per day, which includes time spent on other activities. In this study, the functional response is examined in terms of both bite rate and biomass intake rate per unit time feeding in relation to plant biomass available (Fig. 3a,c), and the forms of the two responses are quite different. Bite rate was high at both high and low algal biomass, with a low point at intermediate biomass, a function which does not correspond to any previously described functional response types, while biomass intake increases approximately linearly with algal biomass (a type-1 response).
The unusual shape of the bite rate response can be largely explained by the fact that bite size gets smaller with decreasing algal biomass, with the result that a higher bite rate does not necessarily result in a higher intake rate, and the intake response is thus close to the more familiar linear type. However, a change in behaviour over time may also provide part of the explanation. It has been suggested that optimal diet choice decisions (Charnov 1976; Krebs et al. 1977) may affect the form of the functional response (Krebs, Stephens & Sutherland 1983; Abrams 1990), and an example of this process taking place in the field has been reported (Wanink & Zwarts 1985). Although the slopes of the intake responses for different periods were not significantly different, they did show a tendency to increase later in the season (Fig. 3c). If this is a real effect, the available evidence suggests that it may be the result of a change in foraging behaviour in accordance with the optimal diet choice model, with successively smaller bites being included in those selected as depletion progresses.
The evidence for this hypothesis lies in the changes in search time and pace rate responses over time. First, search time per bite was found to decrease with date through November, but there was no effect of biomass on this parameter. This fits with the expectation of the diet choice model, that inclusion of new, smaller bite-size classes late in the season provides an effectively greater density of bites and, hence, a lower search time. Secondly, for any given biomass, the speed of movement was significantly lower later in the season (Fig. 4a), suggesting that more time is spent on a given biomass later in the season. This is the opposite of the expectation for a constant search strategy, suggesting a higher degree of selectivity when food is more abundant overall. The available evidence is thus consistent with an optimal diet choice model, however, more experimental data would be required to test this idea more thoroughly. In particular, measurements of actual bite selectivity at different overall biomass levels would enable the hypothesis to be tested directly.
The aggregative response
Geese aggregated in areas of higher algal biomass, the response being slightly, but not significantly, stronger than linear (Fig. 4b). How can the behaviour of individual geese account for this response?
An extensive body of theory under the heading of the ideal free distribution (Fretwell & Lucas 1970) forms the basis of most studies of the processes underlying patterns of distribution in animals (Milinski & Parker 1991; Kacelnik et al. 1992; Sutherland 1996). This assumes that animals move in order to maximize their rate of food intake, leading to a precise set of predictions of aggregation patterns under given conditions. A central variable influencing aggregation is the degree of interference, causing a decline in intake with increasing predator density. Assuming an ideal free distribution, Sutherland (1983) showed that the distribution of predators can be related to the degree of interference by: P = cN1/m, where P is the proportion of predators in a given patch, N the proportion of prey, c a constant and m the degree of interference. The reciprocal of m is thus equivalent to the aggregation constant µ in the previous aggregation equation; estimates of the interference necessary to produce a given aggregation pattern can therefore be calculated as the inverse of the fitted values of µ. This gives estimates of m = 0·84 ± 0·34 (standard error) in October, and m = 0·69 ± 0·14 in November. Both values are high in relation to field estimates of interference from other studies (Hassell 1978; Sutherland & Koene 1982). Sutherland & Anderson (1993) suggest that interference is likely to be negligible for gregarious herbivores feeding on superabundant resources, leading to the prediction that all animals should feed in the most profitable patch, only dispersing as this becomes depleted. This is clearly not the case for geese feeding on algae, but can interference alone account for the relatively weak aggregation observed?
Hassell & Varley (1969) showed that searching efficiency (a) can be related to predator density (P) by the linear equation: log a = log Q – m log P, where log Q is the search efficiency in the absence of interference, and the slope m is the degree of interference. Some interference was detectable in this study in the form of a lower proportion of birds feeding at high density (Fig. 5), and this can therefore be quantified as the slope of the relationship between log(proportion feeding) and log(goose density). This gives a value of interference, m, of 0·12 ± 0·03 (standard error). Since there was no relationship between peck rate in feeding birds and bird density (F1,37 = 1·99, P > 0·15), this value of m can reasonably be assumed to account for all the interference in this case. The values of m inferred from the degree of aggregation (0·84 and 0·69) are significantly and substantially higher than this, and it is therefore clear that factors other than interference must be acting to reduce the degree of aggregation.
Travel costs, perceptual constraints, predation risk, and short-term resource guarding can be suggested as reasons for weaker aggregation than expected under the assumptions of a basic ideal free distribution. The following paragraphs examine each of these, in turn, assessing the likelihood that each is important in this study by comparing the patterns of goose distribution and algal depletion predicted by different versions of the simulation model (Figs 7 and 8, Tables 2 and 3) with the patterns observed.
Travel costs may, in theory, have a significant effect on distribution if the scale of patchiness is large in relation to the area, which a predator can sample in a short space of time (Bernstein et al. 1991; Zhang & Sanderson 1993), and it has been suggested that they may affect aggregation in some experimental situations (Korona 1990). Viscous movement versions of the model effectively simulate the case of aggregation with travel costs, while free movement versions assume no travel costs. All viscous movement versions of the model incorporate slower movement in richer patches (after Fig. 4a), which Stillman & Sutherland (1990) have shown can in theory lead to aggregation. Although this process alone was not sufficient to give rise to aggregation here (Fig. 7b), the predicted aggregation and density-dependence of algal depletion are both much stronger than observed when geese can choose the richest patch from those available (Figs 7f and 8g). Travel costs alone are thus clearly not constraining aggregation in this case.
Perceptual constraints can prevent ideal aggregation if predators learn about the distribution of resources slowly in relation to the rate of depletion (Bernstein et al. 1988), or if predators cannot accurately detect differences in resource availability when encountered (Abrahams 1986), and they have also been suggested to affect some experimental distributions (Sutherland, Townsend & Patmore 1988; Gotceitas & Colgan 1991). We consider this by modelling the two extremes of random patch choice and a preference for higher biomass. The former gives the case of complete perceptual constraint (whether through an inability to learn or an inability to perceive differences is not addressed), while the latter gives the case of perfect knowledge, i.e. no perceptual constraint.
The viscous movement model with random patch choice predicts no aggregation (Fig. 7b), while the viscous model with preference for highest biomass predicts excessively strong aggregation (Fig. .7f). The nature of the density-dependence of algal depletion reflects this, with no density-dependence under random patch choice (Fig. 8c), but very strong density dependence when there is a preference for higher biomass (Fig. 8g). It is possible that an intermediate ability to choose the best patch leads to the observed intermediate degrees of aggregation and algal depletion. However, geese are able to sample the available area rapidly, they frequently return to the same patches, and it is likely that the biomass of neighbouring patches is visible to the geese as they feed. While this does not prove that the geese do learn to distinguish between patches on the basis of biomass, it is clear from other studies that geese are well able to perceive differences in food availability (Williams & Forbes 1980; Sedinger & Raveling 1984; Summers, Stansfield & Perry 1993; Rowcliffe et al. 1998), and readily learn about its spatial distribution (Cooke & Abraham 1980). Thus, although some degree of perceptual constraint must operate in this case, the circumstances of this study would suggest that perceptual constraints do not have an over-riding influence on the pattern of aggregation.
Predation risk may influence the distribution of animals (Milinski & Heller 1978; Newman 1991), and disturbance may be seen as analogous to predation risk in its effects on aggregation. Several studies have shown that human disturbance can affect the distribution of geese (Owen 1972; Madsen 1988; Black, Deerenberg & Owen 1991; Keller 1991; Gill, Sutherland & Watkinson 1996). The algal bed was frequently approached by people from the landward side and disturbance might therefore be expected to cause geese to avoid this side. This, indeed, was found to be the case, although it is not obvious that disturbance was the driving force in this relationship since algal biomass is also generally lower on the landward side. This makes the importance of disturbance difficult to assess, although model simulations in which patch choice is made entirely on the basis of distance from sea (free movement Figs 7a and 8b; viscous movement Figs 7c and 8d) are clearly insufficient to give rise to the observed aggregation or algal depletion patterns. This arises, despite the initial correlation between shore distance and biomass, because individuals continue to select the seaward side of the bed, even when it has become depleted. Thus, if disturbance was the over-riding factor in patch choice we would expect reduced overall aggregation in relation to biomass as a result. However, the fact that no independent effect of distance from sea can be found suggests that this process probably does not have a great effect on the aggregative response.
Field evaluations of the importance of interference are frequently underestimates because animals avoid densities at which interference starts to occur (Arditi & Akçakaya 1990). This results in a dispersed distribution, but little or no apparent reduction in intake over the observed range of predator densities. A mechanism by which this might occur is aggressive guarding of resources by individuals, and distributions driven by this process are known as despotic (Fretwell 1972). The economics of resource defence have generally been discussed in relation to territorial species (Gill & Wolf 1975; Davies & Houston 1984) and there are a number of models of territorial behaviour (e.g. Davies & Houston 1983). However, these do not apply here since the geese are simply aggressive to those nearby and do not have a restricted territory. Despite this, an element of despotism can also affect the distribution of animals that are not strictly territorial (e.g. Whitham 1980). A possible interpretation of the sudden increase in the rate of aggression above a threshold algal biomass in this study (Fig. 6b) is that it is due to a form of short-term resource defence, a behaviour which has previously been recorded in brent geese feeding on a patchy resource (Prop & Loonen 1986; Prop & Deerenberg 1991; Black et al. 1992). If this is the case, there is a significant cost to feeding with other individuals on high biomass patches, and avoidance of this cost could explain the lower than expected degree of aggregation. We therefore model this as the avoidance of patches with a high goose density in viscous model versions 3 and 4. When the avoidance of high goose density is the overriding factor in patch choice, no aggregation is predicted (Fig. 7d) and algal depletion is density-independent (Fig. 8e). However, when it is combined with an active preference for high biomass (i.e. patch choice maximizes algal biomass/goose density ratio), the predicted aggregation patterns come closer to those observed (Fig. 7e), particularly in October. The decision rule based on this ratio is equivalent to aggregation with an interference constant of one, and it might therefore be expected that values close to this would be extracted from the model predictions. However, the model is individual-based and spatially explicit, and it is not therefore a foregone conclusion that this result would be obtained in the specific case of the algal bed at Titchwell. The fact that it is supports the case for despotism as a relevant factor in the distribution of the geese.
The variability in the aggregative response predicted by the simulation for November (Fig. 7e) is much lower than observed (Fig. 7g) because predicted algal depletion is much more even than expected (Fig. 8f). This general tendency of all the simulations arises because they are deterministic, whereas various stochastic factors (not least error in the sampling of algal and goose densities) are likely to be important in the field. Morrison (1986a,b) found that significant aggregation does not necessarily give rise to significant density-dependent prey mortality because scatter in the relationship may obscure the process, and this is likely to explain why the observed algal depletion patterns are so much more variable than those predicted by the simulations. Nevertheless, the deterministic models described here support the hypothesis that the aggregation pattern is determined largely by resource guarding behaviour (despotism), modifying an essentially ideal (no perceptual constraints), free (no travel costs) distribution.