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Keywords:

  • brent geese;
  • depletion model;
  • despotism;
  • green algae;
  • ideal free distribution

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

1. The functional and aggregative responses of dark-bellied brent geese Branta bernicla bernicla feeding on the green algae Enteromorpha intestinalis and Ulva lactuca are described.

2. Bite rate showed maxima at both high and low algal biomass, although the functional response in terms of biomass intake rate was linear (type-1). The differences in the shapes of these responses can be explained by the fact that bite size declines with decreasing algal biomass.

3. There was little interference between the geese; the rate of aggression was only weakly related to goose density, and time spent feeding decreased only slightly with increasing density.

4. Geese fed on all parts of the algal bed, showing weak, but significant aggregation in patches of higher algal biomass in both October and November.

5. The observed pattern of aggregation was weaker than that predicted by the ideal free distribution with a low value of interference.

6. Simulation models were used to explore four possible behavioural factors leading to this deviation from predictions: cost of travel between patches, constraints on the perception of patch quality, disturbance and short-term resource guarding. It is concluded that resource-guarding is likely to be the most important contributory factor.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Numerous theoretical studies have shown that spatial processes can be fundamental to population dynamics (Hassell & May 1973, 1974; Pacala, Hassell & May 1990). In the interaction between prey populations and predators (used here to include herbivores), the central processes are the functional response (the relationship between intake rate and prey abundance) and the aggregative response (the relationship between predator density and prey abundance). These responses combine to affect the spatial pattern of prey mortality. The functional and aggregative responses can, in turn, be understood as the outcome of individual foraging decisions, thus making it possible to relate population biology to behavioural decisions (Hassell & May 1985; Bernstein, Kacelnik & Krebs 1988; Koehl 1989; Goss-Custard & Durell 1990; Sutherland 1996).

The functional response has traditionally been described by models in which intake rate is defined by search efficiency and handling time. The most frequently used model of this sort is Holling's (1959) disc equation, which describes the relationship between food availability and instantaneous intake rate. The random predator equation (Rogers 1972) is obtained by integrating the functional response to give the intake rate over a longer period of time as depletion proceeds. These models describe asymptotic (type-2) functional responses, in which handling time determines the maximum rate of intake, i.e. the asymptote of the functional response is the reciprocal of the handling time, while search efficiency determines the rate at which the asymptote is approached. Linear (type-1) responses are generally interpreted as cases where handling time is negligible relative to the rate of encounter, and the asymptote is never therefore approached over the range of prey availability measured.

The aggregative response can be viewed as the cumulative result of decisions made by individuals and thus may be examined using game theory, based on the interference between predators (Royama 1971; Sutherland 1983) and resource depletion (Comins & Hassell 1979; Sutherland & Anderson 1993). Many attempts to predict the spatial distribution of predators from individual behaviour are based on the ideal free distribution (IFD), which assumes that predators feed where their intake is highest and as a result the mean intake rate is equal in all patches (Fretwell & Lucas 1970). However, most natural distributions do not conform to the predictions of the basic IFD model as intake rate varies between patches (Parker & Sutherland 1986; Kacelnik, Krebs & Bernstein 1992), and it has therefore been modified in a number of ways to incorporate biologically more realistic assumptions. The main modifications have been the incorporation of differences in competitive abilities between individuals (Lomnicki 1978, 1980; Parker & Sutherland 1986; Sutherland & Parker 1992), lack of complete knowledge of resource distribution (Abrahams 1986, 1989; Bernstein et al. 1988; Getty & Pulliam 1992), costs of travel between patches (Bernstein, Kacelnik & Krebs 1991) and sensitivity to predation risk (MacNamara & Houston 1990; Newman 1991).

The aim of this paper is to describe the functional and aggregative responses of a vertebrate herbivore, the dark-bellied brent goose Branta bernicla bernicla L., together with the spatial pattern of depletion of the plants eaten, inter-tidal green algae; and to attempt to identify the behavioural mechanisms underlying these responses. Brent geese spend the winter in the study site, feeding preferentially on green algae when they first arrive in the autumn, moving to salt marsh and pasture when the algae are depleted (Vickery et al. 1995). Aggregative responses of the geese on salt marsh are described in Rowcliffe, Watkinson & Sutherland 1998).

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The study was carried out at Titchwell, Norfolk, UK (National Grid reference TF 753 453), where outcrops of peat allow patchy growth of green algae Enteromorpha intestinalis Link. and Ulva lactuca L. over an area of 5·3 ha of the inter-tidal zone. The analysis in this paper treats the two algal species as a single food resource and reference to algae, therefore, means both species collectively.

Algal sampling

In order to map the distribution of algae on the bed and monitor its abundance through time, the area was divided by eye into 26 blocks of between 500 and 1500 m2, each with roughly uniform algal density and with boundaries discernible from a distance. An accurate map of the algal bed was drawn from a recent aerial photograph, allowing block areas to be calculated. Biomass on the algal bed was estimated using a point quadrat. For each sample, a 30 × 30-cm grid of 100 points was scored for presence/absence of algae at each point, to give an estimate of percentage algal cover. Percentage cover estimates (c) were converted to biomass in g m–2 (b) using a calibration equation (log (b) = 1·919log (c) – 2·059, r2 = 0·97, n = 30, P < 0·0001), which was obtained from 30 samples of known percentage cover by clipping the algae at mud level, washing and drying them at 70°C for 48 h before weighing. Ten randomly placed estimates of algal biomass were made in each block on 5 October, 31 October and 22 November 1990, giving a mean algal biomass for each block at each sample date.

To investigate the direct effects of grazing on the algae, 10 2 × 2-m exclosures were constructed at randomly chosen locations on the algal bed. These were made with two strands of fencing wire, attached to corner posts at 25 and 50 cm height. The exclosures were sufficient to exclude grazing geese, but not to shelter the enclosed algae from wave action. Algal biomass was sampled using the point quadrat as described above to give estimates of percentage cover both inside exclosures and at adjacent points outside each exclosure at 2-week intervals over the autumn, from 3 October to 20 December 1990.

Goose behaviour and aggregation

Geese feeding on the algal bed were observed by telescope from dunes overlooking the site, which were sufficiently far from the bed to avoid influencing the behaviour of the geese. The area was scanned every 15 min during the 3–4 h of low tide, when the algal bed was exposed, and the numbers of geese, their positions and the proportions feeding were recorded. Densities of birds could then be calculated. Geese never roosted on the algal bed, and any birds recorded as not feeding were always engaged in brief periods of vigilance between feeding bouts. The surveys were made at intervals of 1–6 days through October and November 1990, so that seasonal changes in the spatial pattern could be determined.

Peck rates, pace rates and rates of aggressive interaction of focal individuals were recorded. An aggressive interaction was defined as when one or a pair of birds gave an aggressive display with head lowered and beak open, and when one bird visibly moved away as a result. Times taken for 10 pecks and 10 paces were used to record peck and pace rates, respectively, in October, while in November the method of Goss-Custard & Rothery (1976) was used, which yields information on search and handling times, as well as peck and pace rates. With this method, the time taken for 10 paces is recorded, while simultaneously counting pecks. Pecking slows the speed of movement, each peck increasing the time taken for 10 paces by an amount equivalent to the handling time of the peck. Handling time can thus be calculated from the regression of time taken to make 10 paces against the number of pecks observed in that time, the slope of this regression giving the average handling time. A minimum of 30 observations were used for each estimate of handling time. Subtracting the handling time per peck from the total time per peck yields a parameter estimate, which may be seen as proportional to search time and which, for ease of reference, shall be referred to as search time. The mean pace length was obtained by measuring the distance between footprints in soft mud from 20 trails, allowing pace rates to be converted to speeds of movement.

Bite size

Functional responses are usually either expressed in terms of items captured per unit time or biomass intake rate. In this study, both approaches were explored so that the results could be compared. The intake of items per unit time was measured directly in terms of peck rates (see above) and, in order to arrive at an estimate of biomass intake rate, it was therefore also necessary to measure bite size. The shape of the bite size response to spatial variation in algal biomass was obtained by removing artificial bites of algae with a brent goose bill from a range of biomass levels. The bill was held closed with elastic to give a constant bite pressure and bite samples were taken by fully opening the bill, placing it vertically onto a patch of algae, releasing the beak and removing it from the substrate. This was repeated five times in each of 50 30 × 30-cm quadrats of known algal cover in both October and November 1990; the samples of algae were then dried and weighed, so that the amount of algae removed per bite could be related to the amount available.

A direct estimate of average bite size over the whole bed was calculated from defecation rates (R), dropping weights (W) and digestibility (d, defined as the proportion of ingested biomass assimilated, calculated from the relative proportions of an indigestible marker in the food and droppings) obtained from the same site by Lane (1994), together with behavioural data. Since intake rate (I) is given by I = RW/(1 –d), mean bite size (B) can be calculated using peck rate (P) and proportion of time feeding (f) estimates by B = I/Pf. Separate estimates of mean bite size were obtained in October and November. The log-linear regression equations for artificial bite size against biomass were used to predict expected mean bite sizes for the mean biomasses sampled in October and November. It was assumed that the shape of the response of artificial bite size to biomass represented the true response shape, but that the absolute values of artificial bite size could be biased. The position of the bite size response was therefore adjusted by multiplying artificial bite size values by the ratio of observed to predicted mean bite size for each month. The relationship between adjusted bite size and biomass was used to predict values of bite size for the biomasses at which peck rates were recorded. Peck rates could then be multiplied by the appropriate bite sizes to give estimates of intake rate.

Simulation models

In order to explore the implications of goose behaviour for spatial patterns of aggregation and algal depletion, a computer model was constructed to simulate the depletion of algae by geese (see Fig. 1). The model was based on the 26 algal patches defined in the field study; mean algal biomass in each patch was initially taken from the measurements at the beginning of October. Within the simulations, the biomass of each patch was changed according to the daily rate of change recorded in the exclosures (Fig. 2a). This method ignores any possible feedback between grazing and growth; however, little growth occurs over the autumn and early winter season, and any feedback between growth and grazing is therefore likely to be unimportant in this system. The total number of geese feeding on the algal bed was taken from the seasonal and diurnal patterns recorded during the study (unpublished data). The intake rates of individual geese in each patch were calculated using the observed intake (Fig. 3) and percentage feeding responses recorded in the field study (unpublished data). The model was iterated every 10 min, calculating the total depletion of algae by geese in each patch for each iteration.

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Figure 1. Summary of the structures of grazing simulation models using (a) free-movement, and (b) viscous-movement rules for aggregation of geese.

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Figure 2. (a) Seasonal changes in mean algal biomass inside (●) and outside (m) grazing exclosures; error bars are standard errors. (b) Percentage change in algal biomass due to goose grazing (m) and other processes, such as productivity and wave action (●).

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Figure 3. (a) Peck rate in relation to algal biomass. y = 80–2·4x + 0·087x2; r2 = 0·26, n = 35, P < 0·01. Data from different periods are indicated: 13–20 Oct (●); 25 Oct–4 Nov (m); 14–18 Nov (o); 22–23 Nov (n). (b) The change in artificial bite size with algal biomass. Each point is the mean of five bites removed from a single quadrat of known algal cover on 18 October (●) and 16 November (m); y = 0·627x + 0·029, r2 = 0·67, n = 95, P < 0·0001. See methods for details of bite size calculation and adjustment. (c) The intake rate, calculated as the product of peck rates (Fig. 3a) and bite size (Fig. 3c), in relation to algal biomass at different dates. Symbols are as in Fig. 3(a). 13–20 Oct: y = 0·025x, r2 = 0·99, n = 13, P < 0·0001; 25 Oct – 4 Nov: y = 0·023x, r2 = 0·96, n = 7, P < 0·0001; 14–18 Nov: y = 0·026x, r2 = 0·99, n = 8, P < 0·0001; 22–23 Nov: y = 0·035x, r2 = 0·99, n = 7, P < 0·0001.

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The effects of a number of aggregation rules on the distribution of geese between patches and the resulting pattern of algal depletion were investigated. In free-movement versions of the model (Fig. 1a), geese were assumed to distribute themselves instantaneously according to a fixed aggregation function. Two versions of this model were created. In the first, the aggregation function used was that recorded in the field study, while in the second the observed relationship between goose density and distance from the sea was used. In viscous movement versions of the model (Fig. 1b), geese arrived on the bed at a random block, choosing new blocks to enter from those adjacent on the basis of five different decision functions: random choice, preference for patches nearer the sea, preference for lowest goose density, preference for highest algal biomass and preference for the maximum ratio of biomass to goose density. Time spent in each patch was calculated on the basis of patch size and the observed speed of movement in response to algal biomass. Grazing units randomly chosen between two and 10 birds were used, reflecting the loose flock structure in which the geese fed on algae. The aggregation and decision functions used in the two free-movement and five viscous-movement versions of the model are summarized in Table 1.

Table 1.  The methods used to aggregate geese in the seven versions of the simulation model tested
Model typeVersion 
Free movement 1 2Aggregation function Observed biomass aggregation pattern Observed sea-distance aggregation pattern
Viscous movement 1 2 3Decision function Random patch choice Preference for distance from beach Preference for lowest goose density
 4 5Preference for highest biomass/goose density ratio Preference for highest biomass

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Temporal changes in algal biomass and the effects of grazing

Mean algal biomass outside exclosures initially increased in early October and then rapidly decreased to negligible levels in early December (Fig. 2a). Biomass in ungrazed plots inside exclosures followed the same pattern, but at a consistently higher level after mid-October. Subtracting proportional rates of change inside exclosures from those outside gives the depletion due to geese, which can be compared with rates of change in the absence of grazing (Fig. 2b). The rate of change in the absence of grazing was highly variable (from 3·4 to –3·6% day–1), showing a trend towards increasing depletion over the season. In contrast, grazing depletion was relatively constant at between –0·6 and –2·2% day–1.

Goose behaviour

Peck rate was significantly higher at both high and low algal biomass than at intermediate biomass, although the increase at high biomass was more marked (Fig. 3a). The quadratic term in this relationship is significant (P < 0·05), and analysing the data above and below the biomass mid-point of 15 g m–2 indicates a significant negative linear fit below (P < 0·05) and a significant positive linear fit above (P < 0·01).

The mean values of bite size calculated from egestion rates and digestibility fell well within the range derived artificially, giving confidence in the bite size estimates. The value of bite size calculated from egestion rates in October was 1·1 greater than that predicted from the artificial relationship, and 1·5 times greater in November. Log-transformed values of bite size, adjusted by these ratios, showed significant linear relationships with log biomass in October and November (Fig. 3b). The positions of these two regressions were not significantly different (F1/91 = 3·681, NS), and a common regression for October and November was therefore fitted (Fig. 3b).

Multiplying the peck rate observations in Fig. 3(a) by bite sizes predicted by the relationship in Fig. 3(b) gives values of intake rate, which can, in turn, be related to algal biomass (Fig. 3c). Since biomass and date are strongly correlated in this sample, assessing the independent effects of biomass and date of intake requires that the data be divided into time periods within which there are no changes in biomass over time. Four date categories were used for this purpose: 13–20 October, 25 October–4 November, 14–18 November and 22–23 November. This division separates the data into groups of similar size while minimizing the total time spanned within any one group. Curves fitted using the disc equation (Holling 1959) did not fit the data for any of the four time periods significantly better than linear equations and linear regressions through the origin are therefore shown in Fig. 3(c). The slopes of these four lines were not significantly different (F3/30 = 2·9, NS), although there is a tendency for the slope to be steeper later in the season.

Neither handling time nor search time were significantly related to algal biomass (r2 = 0·01 and 0·06, respectively, both NS). However, although handling time remained constant with time in November (r2 = 0·01, NS), search time declined significantly over this period (r2 = 0·46, P = 0·003, y = 0·485–0·013x).

The speed of movement of birds was negatively related to algal biomass in both October and November (Fig. 4a). The slopes of the linear regressions for October and November were not significantly different (t(27) = 0·5, NS), but the elevation was lower in November (t = (28)2·1, P < 0·05). This indicates that geese moved more slowly in November on a given biomass than in October.

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Figure 4. (a) The relationships between walking speed and biomass in October (●) and November (m). Linear regressions: October, y = 5·48–0·10x, r2 = 0·5, n = 20, P < 0·001; November, y = 4·97–0·12x, r2 = 0·29, n = 15, P < 0·05. (b) The spatial patterns of goose aggregation in October (●) and November (m). The curves are least-squares best fits of the aggregation equation: October, y = 0·76 x1·19, r2 = 0·26, P < 0·01; November, y = 1·03 x1·45, r2 = 0·51, P < 0·001.

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Geese showed a significant tendency to aggregate in patches of higher algal biomass in both October and November (Fig. 4b). This relationship can be described by the aggregation equation (Hassell & May 1973), PicNiµ, where Pi is the proportion of predators in the ith patch, Ni the proportion of prey, c a constant and µ the aggregative constant, which defines the strength of aggregation. When µ = 0, predators are evenly dispersed; when µ = 1, predator and prey numbers are proportional; and when µ > 1 predators are proportionally more abundant at high prey density. Estimates, with standard errors, of µ from the data were 1·19 ± 0·47 in October and 1·45 ± 0·29 in November. Neither estimate was therefore significantly greater than one. Areas of equivalent biomass held higher goose densities in November than in October as the overall biomass of the bed declined.

Both goose density and algal biomass also showed significant negative correlations with distance from the sea, although the strengths of these relationships were lower in November than in October (Goose density: October, r2 = 0·42, P < 0·0005; November, r2 = 0·16, P < 0·05. Algal biomass: October, r2 = 0·51, P < 0·0001; November, r2 = 0·36, P < 0·005. n = 26 in all cases.) To test whether there was a clear effect of shore distance on aggregation independent of algal biomass, the residuals of the density/biomass relationships in Fig. 4(b) were regressed against distance from the sea. In neither month were they significantly correlated (October: F = 3·33, P > 0·05; November: F = 0·02, P > 0·8).

The percentage of geese feeding was related to both goose density and algal biomass (Fig. 5); goose density and algal biomass were not significantly correlated in this sample (F1,16 = 0·4, P > 0·1). In this analysis, goose density was logged and percentage of geese feeding was arcsine-root transformed in order to normalize their distributions. Using a least-squares procedure to fit a multiple polynomial regression, the best fit model included a linear term for algal biomass, a quadratic term for goose density, and a term for the interaction between biomass and density. All terms in this model were significant at the 5% level or lower, and removing the interaction term reduced the amount of variance explained by 19%. The surface fitted in Fig. 5 indicates that the proportion of geese feeding is highest at intermediate goose densities, declining at both higher and lower densities. It also shows a strong linear decline with increasing algal biomass at low goose density, but relatively little change with biomass at high goose density. A decline in feeding with increasing goose density may be indicative of interference between geese, and the strength of any such interference may therefore be quantified by the slope of the feeding-density response. It is therefore important to determine the significance and strength of this decline. The peak in percentage feeding at the mean algal biomass (20 g m–2) is at 225 geese ha–1. Regressing log proportion feeding against log goose density for all data above 225 geese ha–1 gives a significant decrease with a slope of 0·123 (r2 = 0·76, n = 10, P < 0·001).

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Figure 5. The percentage of geese feeding (F) in relation to goose density (D) and algal biomass (B). Multiple regression: arcsine√F = 0·464–0·012B + 0·381logD – 0·098logD2 + 0·004BlogD; all terms are significant at the 5% level or lower; r2 = 0·72, n = 18, P < 0·0001.

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A probable mechanism by which increased goose density may cause a reduction in time spent feeding is through an increase in the occurrence of aggressive interactions, causing individuals to stop feeding more frequently. The rate of aggressive interactions showed significant increases with both goose density and algal biomass (Fig. 6), partial correlations indicating that both relationships were independent of the other variable. However, the strengths of the two relationships were very different, with goose density having only a weak effect, but algal biomass having a very strong effect on the rate of aggression. Aggression showed a threshold response to increasing algal biomass, the vast majority of interactions taking place on areas with a biomass of greater than 25 g m–2 (Fig. 6b).

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Figure 6. The rate of aggressive interactions (see methods for definition) in relation to (a) goose density and (b) algal biomass. Partial Kendall correlations (relating the rate of aggression to each x variable, while holding the other constant): (a) goose density: T = 0·30, z = 2·50, P < 0·01; (b) algal biomass: T = 0·54, z = 4·48, P < 0·000005.

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Simulated spatial patterns

Apart from the free movement model using the observed aggregation function, which self-evidently gives rise to aggregation and is not therefore shown in Fig. 7, only viscous movement models incorporating preference for high biomass patches predicted significant aggregation of geese (Fig. 7e,f, Table 2). When patch choice is on the basis of biomass only (Fig. 7f), aggregation is much stronger than observed, while tempering this with avoidance of high goose density (Fig. 7e) gives predicted patterns of aggregation closer to those observed. There is a tendency in November for viscous models incorporating either high biomass preference (Fig. 7f), avoidance of high goose density (Fig. 7d) or both (Fig. 7e), to result in lower variability than observed in both algal biomass and mean goose density.

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Figure 7. A comparison of aggregation patterns (a–f) predicted by simulation models and (g) observed for October and November. See Methods and Table 1 for details of the models. The first free-movement model in Table 1 uses the observed aggregation function and does not therefore appear in this figure.

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Table 2.  The goodness of least-squares fits of the aggregation equation (Hassell & May 1973) to mean goose densities predicted by simulation models and for the observed aggregation patterns (Fig. 7). See Table 1 and methods for details of models. µ defines the strength of aggregation
Model versionFigure October r2PµNovember r2Pµ
Free 27a0·11NS– 0·01NS
Viscous 17b0·01NS– 0·002NS
Viscous 27c0·004NS– 0·03NS
Viscous 37d0·001NS– 0·07NS
Viscous 47e0·28<0·011·030·38<0·0010·58
Viscous 57f0·16<0·055·000·39<0·0013·60
Real data7g0·26<0·011·190·51<0·0011·45

The spatial patterns of algal depletion predicted by the simulation models and observed in the field are shown in Fig. 8, and the statistics of linear regressions of these data are summarized in Table 3. In October, change in the algal biomass of grazed areas was spatially density-independent, although the lowest biomass blocks tended to show growth, rather than die-back. Weak density-dependent mortality occurred in November and blocks with initially low biomass again tended to show some growth (Fig. 8h). Density-dependent depletion of algae was predicted only by the free-movement model using the observed aggregation function (Fig. 8a) and the two viscous-movement models using an element of preference for high biomass (Fig. 8f,g). All other models predicted density-independent depletion and all models predicted less variability in depletion than observed (Fig. 8).

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Figure 8. A comparison of algal survival patterns (a–g) predicted by simulation models and (h) observed for October and November. See methods and Table 1 for details of the models.

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Table 3.  Summary of the strengths of density dependent survival of algae predicted by models and from real data. The statistics are from linear regressions of data in Fig. 8. See Table 1 and methods for details of models
Model versionFigure October r2PslopeNovember r2Pslope
Free 18a0·230·01–0·0380·220·02–0·028
Free 28b0NS0·10NS
Viscous 18c0NS0·02NS
Viscous 28d0·07NS0NS
Viscous 38e0NS0·05NS
Viscous 48f0·53<0·0001–0·0450·82<0·0001–0·027
Viscous 58g0·78<0·0001–0·1290·95<0·0001–0·077
Real data8 h0·12NS0·280·005–0·076

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

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.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

We are grateful to Juliet Vickery, Simon Lane and John Davies for assistance with field work, and to the Royal Society for the Protection of Birds for permission to work at Titchwell. The manuscript was greatly improved by comments on earlier drafts by John Goss-Custard and two referees. The work was support by a grant from the Agriculture and Food Research Council under the Joint Agriculture and the Environment Programme.

References

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  6. Discussion
  7. Acknowledgements
  8. References
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