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Keywords:

  • area-restricted search;
  • correlated random walk;
  • Cygnus columbianus bewickii;
  • depletion;
  • Potamogeton pectinatus

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix
  • 1
    Tundra swans forage on below-ground pondweed tubers that are heterogeneously distributed in space. The swans have no visual cues to delineate patches. It was tested whether swans employ an area-restricted search tactic. Theory predicts that swans should alternate between an intensive (low-speed, sinuous) search mode in high tuber density areas and an extensive (high-speed, directed) search mode between these areas.
  • 2
    A quantitative analysis of movement paths recorded over short time frames (15 min) revealed that the sequential step lengths were strongly autocorrelated. After partitioning the data in low-speed paths and high-speed paths, this autocorrelation was very much reduced.
  • 3
    Movement paths with low speed were non-directional and could well be described as random walks. In contrast, high-speed paths were directed forward, and were better described as correlated (i.e. directional) random walks.
  • 4
    Movement paths recorded over longer time frames (1–4 h) provided empirical evidence that an alternation of low-speed, sinuous and high-speed, directed searches occurred.
  • 5
    There was a spatial autocorrelation in tuber biomass density, being significantly positively correlated until c. 10 m distance. The scale of the food clump size and step length of high-speed paths matched, suggesting that they were causally linked.
  • 6
    Computer simulations confirmed that swans using the observed search tactic achieved a higher energy gain than swans using only an intensive search mode, provided that the tuber biomass density occurred in clumps. They also achieved a higher gain than swans that alternated between intensive and extensive search mode, but always moved in a random direction.

Introduction

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

Foragers feeding on cryptic prey can be divided in two categories: one in which these hidden prey occur in identifiable patches, and one in which also the patches cannot be instantly identified by the animals. An example of the first category is provided by woodpeckers feeding on hidden larvae in easily recognizable branches. The foraging behaviour of these birds can be understood using Bayesian statistics, in which the birds’ decisions to leave a patch are based on estimates of future prey capture rates in the current patch (Olsson et al. 1999). This reasoning cannot be directly applied to foragers that are not able to delineate a patch because the information about prey captures cannot be evaluated on a patch basis. These foragers might perform well by adjusting their searching behaviour to changes in food capture rates (Bell 1991).

Benhamou (1992) investigated how animals should adjust their searching behaviour in order to forage optimally in environments with hidden patches (which he calls ‘continuous patchy environment’). He showed that in this type of environment, the optimal behaviour is to decrease speed and increase sinuosity when a prey item is detected. In another simulation study, Zollner & Lima (1999) found that the best nonsystematic way to encounter a new patch is a nearly straight search path. Thus, the optimal behaviour should be a combination of low-speed, sinuous searches in high-resource density areas, and high-speed, directed steps between these areas (see also McIntyre & Wiens 1999). Together, these make up an ‘area-restricted search’, consisting of an alternation of intensive and extensive search modes, respectively, aimed at concentrating the feeding effort within high-resource areas (Benhamou 1992).

Such an alternation between extensive and intensive search modes has been recorded in real animals, sometimes as a response to prey capture. Ward & Saltz (1994) found that dorca gazelles feeding on leaves of lilies alternated bouts of concentrated feeding in small areas with long steps to new foraging areas. In carabid beetles, the switch from one movement pattern to the other was probably mediated by the hunger state of the animals, enabling them to concentrate in favourable habitat patches (Grüm 1971; Baars 1979). Zach & Falls (1976a) found that the paths of ovenbirds became less straight when patches with increased mealworm density were encountered. Also, search paths prior to and after locating a patch were straighter than while in the patch (Zach & Falls 1976b). In contrast, thrushes change their search path after a prey capture mainly by increasing the autocorrelation in turning angles, so that they are looping, without changing the absolute turning angles (Smith 1974a,b).

Since the pioneering work of Baars (1979), Kareiva & Shigesada (1983) and Root & Kareiva (1984), several studies have tested model predictions of movement against observations on real animals. Most of this work involved invertebrates. In the last decade these tests have been extended to vertebrates, more specifically mammals (Benhamou 1990; Ward & Saltz 1994; Focardi, Marcellini & Montanaro 1996; Bascompte & Vilà 1997; Bergman, Schaefer & Luttich 2000; Fortin 2000). The majority of these tests compare the observed movement paths with those predicted by a correlated random walk model. In this model, a movement path is represented by straight steps from one measurement point to the next. In a random walk, turning angles are drawn from a uniform distribution. In a correlated random walk directionality of movement is accounted for by assuming a nonuniform distribution of turning angles. The concentration around a certain turning angle (usually 0°, representing forward movement) provides a measure of the degree of directionality (Kareiva & Shigesada 1983). Both the random walk and the correlated random walk model assume that there is no autocorrelation in sequential step lengths or turning angles, so the length and turning angle of a step are independent from those of the previous step (Turchin 1998). Because ‘correlated’ in correlated random walk refers to directionality (and not to autocorrelations in the movement pattern), we prefer to use the term directional random walk here.

We tested whether the movements of swans, which exploit below-ground pondweed tubers that occur aggregated in hidden patches, can be understood as an area-restricted search as predicted by Benhamou (1992), and whether the observed search tactic was the optimal one in this particular continuous patchy environment. Two types of field data were collected. First, swans were tracked for 15 min, and their search paths compared with pseudopaths generated with the random and directional random walk models. We made a distinction between searches with a low and a high speed. Second, a small number of swans was tracked for >1 h to check whether an alternation between the two postulated search modes occurs. Based on the observed scale of resource aggregations, we subsequently used a game-theoretical, individual-based model to check whether the observed searching behaviour was the optimal tactic in this continuous patchy environment. We evaluated five search tactics, consisting of different combinations of the direction and speed of movement. As the currency for comparison we used the rate of energy gain, which is the difference between the metabolizable energy intake rate and the energetic activity costs.

Methods

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

STUDY SYSTEM

Bewick’s swans (Cygnus columbianus bewickii Yarrell), a subspecies of tundra swans, were foraging on below-ground tubers of fennel pondweed (Potamogeton pectinatus L.) in monospecific stands in 30–70 cm deep water (Nolet et al. 2001b). No above-ground plant material was present during the swan exploitation. The tubers are hidden in the first 30 cm of the sediment (Santamaría & Rodríguez-Gironés 2002). Swans dig craters by trampling their feet, and obtain the below-ground tubers with their heads submerged. Crater size is typically c. 1 m2 (Beekman, van Eerden & Dirksen 1991). Partly due to the swan activity, the water is very turbid with Secchi depths of typically around 20 cm.

The study was performed at two sites that are both visited by the same population of Bewick’s swans on their migration (Beekman et al. 1991; Nolet et al. 2001a). One site was situated in the Dvina Bay (64°51′N, 40°17′E), a part of the White Sea in Russia. The size of the pondweed bed was 600 × 60 m2. The other site was located in the Lauwersmeer (53°22′N, 06°13′E) in the Netherlands, in a creek that is closed for the public. The segment of the creek that was studied was 800 m long and 300 m wide, containing two parallel, c. 100-m broad beds of pondweed separated by a c. 100 m wide gully without vegetation. After depletion of the pondweed bed, the swans switch to harvest remains of sugarbeets on the fields surrounding the Lauwersmeer (Beekman et al. 1991).

MEASUREMENTS OF MOVEMENT PATHS

Swans were observed from a 3-m high hide in the Netherlands in October 1995 and 1996, and from a 3-m high sand dune in Russia in May 1995, both at fixed locations. The position of the swans was determined by measuring the distance (±1 m) and azimuth (±1°) to the observer using a laser range finder (Geovid 7 × 42 BDA, Leica, Solms, Germany). The distance to the birds ranged usually between 50 m and 300 m (but see below) in the Netherlands and between 250 m and 300 m in Russia.

In the Netherlands, a foraging bird was arbitrarily chosen from a flock and tracked for 15 min, taking distance and azimuth every 100 s (i.e. 10 measurements in total). The bird was kept in sight constantly in order to avoid confusion with other birds.

In Russia, birds were tracked for 1–4 h, taking measurements every 5 min. Here birds were chosen that were marked with neck-collar or leg-band. One unmarked bird was followed at a time that few other birds were nearby, hence confusion with other birds was unlikely in this case. These paths were collected to check whether an alternation between localized search and directed search could be detected when swans were tracked for longer time periods than 15 min.

ANALYSIS OF MOVEMENT DATA

The 15-min paths collected in the Netherlands were analysed quantitatively. The length and the turning angle (ranging from –180° to 180°) between subsequent positions were calculated, defining right turns as negative and left turns as positive by convention (0° being straight ahead). We selected the paths with a maximum distance between observer and bird of less than 230 m, at which distance a 1° difference in azimuth is equal to 4 m, and defined a step as a change in position of ≥4 m. The effect of the height of the observation point on a calculated step length was so small (0·2–0·001% for birds at 50–230 m) that it was ignored. Given the approximate crater size, an analysis with a resolution of 1 m would in principle be desirable. However, an analysis based on a finer scale than 4 m is biased, since a displacement is more often apparent from the distance measurement than from the azimuth measurement, in which case the turning angle is calculated to be 0° or ±180°. Apart from this bias, the analysis based on ≥1-m steps gave rather similar outcomes as the analysis of ≥4-m steps, yielding confidence in the validity of the presented approach.

The average net squared displacement calculated for the observed paths after each nth step ( inline image) was compared with the inline image predicted by directional random walk or random walk models using the bootstrap procedure described by Turchin (1998). It has become conventional to use the squared displacement rather than the linear displacement because its expectation has been analytically derived, and because it tends to show a linear relationship with time (Kareiva & Shigesada 1983). Only complete 15-min paths (30 in 1995 and 49 in 1996) were used. Empirical distributions of step lengths and of turning angles were constructed by pooling the data for all individuals in the two years. Subsequently, starting from (x0, y0) = (0, 0) for convenience, 1000 pseudopaths were generated. For each step, a step length and turning angle were drawn. In the case of a random walk, a step length was randomly drawn (with replacement) from the empirical distribution, but the turning angle was drawn from a uniform distribution (ranging from –180° to 180°). In the case of a directional random walk, both the step length and turning angle were drawn from the empirical distributions. The net squared displacement from the starting point was calculated after each nth step as:

  • Rn2 = xn2 + yn2.

Subsequently, we randomly drew as many Rn2 values as there were observed paths with at least n steps, and calculated inline image for this set. This was repeated 1000 times. Finally, these inline image were sorted in an ascending order, and the mean and 95% confidence interval calculated by taking the average, 26th and 975th value of these. inline image was also calculated using Kareiva & Shigesada’s (1983) equation 2 (that assumes turning angles symmetrical around 0°).

The main assumption of the (directional) random walk model (i.e. a lack of autocorrelation of both step lengths and turning angles) was tested by calculating the correlation between sequential (log-transformed) step lengths (using Pearson correlation coefficient) and the correlation between sequential turning angles (using Jupp-Mardia circular correlation coefficient). In addition, Stephens circular correlation coefficient was calculated to determine the sign of the autocorrelation of turning angles (Stephens r = max(r+, r), where r+ and r indicate positive and negative coefficients, respectively). Whether the turning angles were uniformly distributed on the 360° circle (i.e. no directionality) was tested by calculating the angular concentration (or mean vector length) and using the Rayleigh test (Batschelet 1981).

FOOD DISPERSION

Below-ground tubers were sampled in the Lauwersmeer by taking 40-cm deep cores (diameter 10 cm) and filtering the sediment through a 3-mm sieve. Tubers were dried for 72 h at 70 °C.

In order to establish the number of cores to be taken per square metre, a 100 × 100 × 60-cm high frame, which was subdivided into 8 × 8 squares, was pushed 40 cm deep into the sediment. Hereby it was possible to take cores close together, one core per square, extracting half of the sediment in the square. The tubers were weighed per core. This yielded a grand average of 116·8 g m−2 ± 97·3 SD (N = 64) fresh mass, equivalent to about 35 g m−2 dry mass. It was subsequently determined how the estimate of the tuber biomass density at a given square metre was dependent on the number of cores. A specific core result was randomly drawn (without replacement), and after each draw the percent deviation of the current average from the grand average (i.e. of the 64 cores) was calculated. This procedure was repeated 100 times. The median percentage deviation decreased from 20% after six cores, 15% after nine cores, 10% after 18 cores, to 5% after 36 cores. It was decided to take six cores m−2 in a sampling scheme designed to describe the tuber biomass density distribution of the pondweed bed, and nine cores m−2 in a sampling scheme designed to measure the spatial autocorrelation in tuber biomass density within that bed.

The distribution of tuber biomass density was measured by taking six cores in 1-m2 plots that were 10 m apart. In early October 1995, a total of 78 plots were sampled on sandy sediment (see Nolet et al. 2001b) and in less than 60 cm deep water.

The spatial autocorrelation in food dispersion was determined using the method of Lovvorn & Gillingham (1996). In the last week of September 1998, tuber biomass density was measured at 17 stations 100 m apart. Now nine cores were taken per 1-m2 plot. Plots were situated at the centre of the station, and 1, 3, 9 and 27 m from the centre in the direction of the centre of the next station (at 100 m). Pearson correlation coefficients between respective tuber biomass density estimates were calculated using a one-tailed test, because a positive correlation was expected.

SIMULATIONS

The rates of energy gain and the tuber depletion by swans using different search tactics were compared with an individual-based model that was developed within the framework of OSIRIS (Mooij & Boersma 1996). The swans were foraging on a virtual pondweed bed until they left the bed and flew to a sugarbeet field. The pondweed bed was modelled as a grid, consisting of 40 × 40 cells of 1 m2 each. The cell size of the grid was equal to the size of a foraging pit (Beekman et al. 1991), and taken to be equal to the grain of the swans; in other words, the swans were assumed not to respond to variation in biomass density within cells (Kotliar & Wiens 1990). The grid had wrap-around margins so that a swan swimming out of the grid on one side reappeared on the opposite side. This represent foraging on a large bed and in a large flock. The tuber biomass in the bed was modelled with and without spatial autocorrelation. In the case of spatial autocorrelation clump size was 10 × 10 m2, so 16 clumps fitted in the grid (Fig. 1, see Appendix). The model was parameterized using field and laboratory data (Beekman et al. 1991; Nolet & Drent 1998; Nolet et al. 2001b).

image

Figure 1. Example of the modelled dispersion of tuber biomass density (g m−2) in a 40 × 40 m2 bed (a) without and (b) with spatial autocorrelation over a distance of 10 m.

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The model swans foraged in bouts consisting of a fixed sequence of trampling, feeding and resting. They had perfect knowledge of the cell quality after one foraging bout, but learned the quality of the whole bed, which constantly changed due to the depletion by the swans, using a linear operator model (Bernstein, Krebs & Kacelnik 1988). The linear operator model, in which outdated information is devaluated, is in theory a powerful way to track habitat changes (McNamara & Houston 1987; Giraldeau 1997). Previous theoretical work on memory factors has shown that the weight that is given to past events should depend on the rate at which the environment changes. If the environment changes slowly or infrequently, then relatively more weight should be given to the past (McNamara & Houston 1985). We calibrated the linear operator model (the memory factor µ and the giving-up gain rate γGU; see Appendix) against field data in such a way that about the same average tuber biomass reduction was obtained in the simulations as observed in the field (c. 20% in 1995) (B. A. Nolet & W. M. Mooij, unpublished data). In the model the swan swam to another cell when the gain rate that it expected to obtain in the next foraging bout in the current cell was less than the expected gain rate for the bed as a whole. The swans flew away as soon as the expected gain rate for the bed as a whole was less than γGU.

Five search tactics, consisting of different combinations of the speed and direction of movement, were tested (iR, iD, iReR, iDeD and iReD search; Table 1). Swans swam either randomly (‘R’) to one of the eight adjacent cells or directionally (‘D’) to one of the four adjacent cells NE, N, NW or W of the current cell. This directional movement prevents backtracking. If a target cell was already occupied by another swan, the focal swan swam to one of the other optional cells. The model swans could use an intensive (‘i’) search mode, or alternate between intensive (‘i’) and extensive (‘e’) search modes. In the intensive search mode a swan made one move to an adjacent cell before resuming trampling. In the extensive search mode it made 12 movements before resuming trampling. In this way, the swan on average moved 10 m when moving directionally. As mentioned above, a swan made a decision to swim to another cell based on a comparison of the expected gain rate in the current cell and that for the bed as a whole. If the swan decided to swim after the first foraging bout in the current cell, swans subsequently used the extensive search mode to move to another cell. Otherwise (i.e. if more than one foraging bout was performed in the current cell), they used the intensive search mode.

Table 1.  Modelled search tactics of swans foraging on below-ground pondweed tubers
TacticModeDirection
  • *

    Between intensive and extensive mode.

iRintensiverandom
iDintensivedirectional
iReRalternating*random in both modes
iDeDalternatingdirectional in both modes
iReDalternatingrandom in intensive and directional in extensive mode

We investigated whether a swan with the observed iReD search tactic could invade a population of swans with one of the other search tactics and vice versa by comparing the rates of energy gain of the mutant with that of the wild type while simultaneously foraging on a virtual pondweed bed with and without spatial autocorrelation in tuber biomass density. We performed 16 simulations of each scenario.

Results

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

15-MIN PATHS

The increase in the average net squared displacement inline image with the number of steps in a path was similar in 1995 and 1996 (Fig. 2a). These years were subsequently pooled in the rest of the analysis.

image

Figure 2. Average net squared displacement against the number of consecutive steps for swan movement paths. (a) Observed average net squared displacement (N ≥ 3) in 1995 and 1996. (b)–(d) Comparison of the observed average net squared displacement (95 + 96) with that predicted for a straight walk (SW), random walk (RW) or directional random walk (DRW) with turning angles symmetrical around 0°. The solid lines represent the average and the dashed lines the 95% confidence intervals of the net squared displacement calculated by bootstrapping (black: random walk, grey: directional random walk). (b) All paths, (c) low-speed paths, (d) high-speed paths. Inset in (c) shows low-speed paths at different scale.

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Using all paths, the random walk and directional random walk models predicted virtually the same net squared displacement, and observed inline image were somewhere between a straight walk and a (directional) random walk (Fig. 2b). Sequential step lengths were strongly autocorrelated, violating one of the basic assumptions of the (directional) random walk model (Table 2). The autocorrelation between sequential turning angles was not significant, but unlike a random walk, there was a significant directionality in turning angles (Table 2). The turning angles had a bias in forward direction (Fig. 3a).

Table 2.  Tests of the main assumptions of random walk and directional random walk models. Stephens circular concentration coefficient indicates the sign of the autocorrelation, but can give rise to false significance whenever the angular concentration indicates directionality
 Step lengthsTurning angles
AutocorrelationNDirectionalityAutocorrelationN
Pearson correlation coefficientAngular concentrationJupp-Mardia circular correlation coefficientStephens circular correlation coefficient
  • *

    P < 0·05;

  • **

    P < 0·01;

  • ***

    P < 0·001.

All paths0·46***1570·18*0·26r = 0·14109
Low-speed paths0·30** 790·060·31r+ = 0·22 49
High-speed paths0·23* 780·33***0·29r = 0·28* 60
image

Figure 3. Directions of turning angles (1995 and 1996 pooled) of steps for (a) all paths combined (N = 158), (b) low-speed paths (N = 78) and (c) high-speed paths (N = 80). Turning angles were classified into octants: (1) –22·5–22·5°, (2) 22·5–67·5° ... (8) –67·5 to –22·5°.

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Step lengths averaged per path i (&#x006c;̄i) were not normally distributed (Kolmogorov–Smirnov D = 0·125, N = 79, P < 0·01), but the distribution was skewed to the right (skewness 0·88) with a heavy tail (kurtosis 0·66). Steps were always measured over 100 s, so step length was a measure of displacement speed. In order to distinguish between low- and high-speed paths, the data were split in two groups: paths with short average steps (&#x006c;̄i < &#x006c;̄) and paths with long average steps (&#x006c;̄i > &#x006c;̄), where &#x006c;̄ is the grand average step length (15·2 m, N = 218). Sequential step lengths within these groups were to a much lesser extent (but still significantly) autocorrelated (Table 2).

15-MIN LOW-SPEED PATHS

The observed inline image of low-speed paths did not differ significantly from both random walk and directional random walk models (again giving very similar predictions; Fig. 2c). Sequential turning angles were not significantly autocorrelated (Table 2), satisfying this underlying assumption of these models. There was also no significant directionality in turning angles in accordance with a random walk model (Table 2 and Fig. 3b). Thus, apart from the autocorrelation in step lengths, the low-speed paths were well described as a random walk.

15-MIN HIGH-SPEED PATHS

For high-speed paths, the random walk and directional random walk predictions partly deviated (Fig. 2d). From five steps onwards, the observed inline image was significantly different from the random walk model, but remained within the 95% confidence interval of the prediction of the directional random walk model. The observed inline image tended to be greater than that predicted by the directional random walk model. The Jupp-Mardia r did not reveal significant autocorrelation between subsequent turning angles, but Stephens r indicated that subsequent turning angles were on average negatively autocorrelated (Table 2). This leads to greater net displacements than without autocorrelation. There was statistical evidence for directionality in turning angles (in forward direction; Table 2 and Fig. 3c). Hence, high-speed paths mainly deviated from a random walk through the directionality in the turning angles, in accordance with a directional random walk. Apart from, again, the (weak) autocorrelation in step lengths, the high-speed paths had the characteristics of a directional random walk.

MORE THAN 1-H PATHS

Recording movement paths of >1-h duration in Russia revealed that swans could forage for long periods on a small part of a pondweed bed. Swan 221A foraged 240 min in an area of 25 × 25 m2. Swan 557P foraged for 105 min on an area of 15 × 15 m2 before flying 160 m to the next area of 10 × 25 m2 where it foraged for another 90 min. The tracks of the other three swans showed alternation between nondirectional movements on a small area and directed movements with large displacements (Fig. 4).

image

Figure 4. Movement paths of five swans foraging in Russia. Values along the axes are in metres. The position of the swans was determined every 5 min for 70 min (202P), 85 min (TJY), 195 min (557P), 240 min (221A) or 255 min (Unmarked). Solid lines indicate displacements by swimming and the dotted line a displacement by flight. The edges of the pondweed bed are indicated with thick grey lines.

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FOOD DISPERSION

The tuber biomass density in one square metre was significantly positively correlated (one-tailed P < 0·05) with that in the adjacent square metre. Combining the Pearson correlation coefficients for different distances between plots, it was found that the running average stayed at the significance level until the plots were >10 m apart, after which it gradually decreased to zero for plots 100 m apart (Fig. 5). We conclude that the spatial autocorrelation in tuber biomass density did generally not extend beyond 10 m.

image

Figure 5. Pearson correlation coefficients of tuber biomass densities in relation to the distance between sampled square metres. The dashed line indicates the critical value for a one-sided test (P < 0·05). The grey line is the three-point running average. Note that the dots are not independent.

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Based on the above argument, the 1-m2 plots sampled in the Lauwersmeer in 1995 may be assumed to give independent estimates of tuber biomass density because they were ≥10 m apart. The tuber biomass density (g m−2) was ln-normally distributed with mean 2·27 and standard deviation 0·89 (i.e. not significantly different from normal distribution after ln-transformation of tuber biomass density: Kolmogorov D = 0·073, N = 78, P > 0·15).

SIMULATIONS

The observed search tactic was modelled as an iReD search, i.e. an alternation of random movements in intensive search mode and strong directional movement in extensive search mode. While in the pondweed bed, the rate of energy gain of swans with different search tactics was in all simulated games remarkably similar (Figs 6 and 7). However, large differences between search tactics were found in the moment that swans decided to leave the pondweed bed (i.e. the patch residence time). The error bars in Figs 6 and 7 may seem large, but these are SD above and below the average; in this case, the 95% confidence intervals are only about half the SD. If we assume that the departing swans achieved a gain rate in the alternative habitat (i.e. the sugarbeet fields) equal to the giving-up gain rate γGU, the cumulative gain of early and late departing swans can be compared. This shows that in the end late departing swans gained more energy than early departing swans (Figs 6 and 7).

image

Figure 6. Cumulative gain of swans with different search tactics at the moment of pondweed bed departure after foraging on (a)–(d) randomly distributed or (e)–(h) spatially autocorrelated tuber biomass densities in computer simulations. Presented are the average ± SD of one simulation of 16 wild types and the average ± SD of 16 simulations of one mutant. The wild type (•) consists of swans with (a,e) iR search, (b,f) iD search, (c,g) iReR search and (d,h) iDeD search; the mutants (○) are swans with the observed iReD search (for the meaning of the codes see Table 1). The solid grey line indicates the gain while in the patch that increased with the same rate for the two search tactics involved per simulation. The dashed lines are projected cumulative gains after the swans have left the pondweed bed, assuming that the gain rate at the alternative habitat equals γGU.

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image

Figure 7. As Fig. 6, but now with swans with the observed iReD search as the wild type (○). The mutants (•) were swans with (a,e) iR search, (b,f) iD search, (c,g) iReR search and (d,h) iDeD search.

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In a bed without spatial autocorrelation in tuber biomass density, swans with the observed iReD search left the bed earlier than swans that continuously were in intensive search mode (iR or iD search). This held irrespective of whether the swans with iReD search were the mutant (Fig. 6a,b) or the wild type (Fig. 7a,b). In contrast, in a bed with spatial autocorrelation the reverse was found, swans with the observed iReD search staying considerably longer than swans with iR or iD search (Fig. 6e,f and Fig. 7e,f). Swans with iReD search also left later than swans that alternated between intensive and extensive search mode but always moved in a random direction (iReR search), in particular when the tuber biomass density was spatially autocorrelated (Figs 6g and 7g). In a bed with spatial autocorrelation swans with the observed iReD search did not differ from swans that alternated search modes but had a directional movement in both modes (iDeD search) (Figs 6h and 7h).

Hence, swans with the observed search tactic (iReD search) left a spatially autocorrelated pondweed bed later than swans with only a single search mode or swans alternating modes but moving randomly, and by doing so achieved a higher energy gain over the same time period.

Discussion

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

The random walk and directional random walk models did not provide an adequate description of the 15-min movement paths of the swans recorded in the Netherlands. The presence of a strong autocorrelation in step lengths suggested that more than one type of movement pattern was involved. The skewed distribution of the step lengths averaged per path i (i.e. &#x006c;̄i) provided further evidence for the existence of two types of searching. This was further confirmed by five >1-h paths recorded in Russia, which showed that swans exhibit localized searches with short steps and more directional searches with long steps, and can alternate between the two. We distinguished between these two types by splitting the 15-min paths on the basis of &#x006c;̄i. Thereafter the observed net square displacements when displacing at low speed fitted the predictions of a random walk, whereas, the displacements fitted the predictions of a directional random walk when displacing at high speed.

The food resources occurred in clumps with a diameter of roughly 10 m. The displacements of low-speed paths were generally shorter than this clump size, whereas those of high-speed paths were greater, in line with the supposition that the search modes and the patchy food distribution were causally linked.

By computer simulations we investigated whether the area-restricted search is superior over alternative search tactics in terms of the rate of energy gain. The memory factor used in these simulations results in a fairly rapid adjustment of the expected gain rate after a change in actual gain rate (see Appendix). This is in line with empirical evidence. Studies in starlings and pigeons have suggested that these birds in fact only react to the most recent experience (Cuthill et al. 1990; Shettleworth & Plowright 1992). Cuthill, Haccou & Kacelnik (1994) experimentally switched travel times from long to short or vice versa, and found clear evidence for transient behaviour in the number of prey items taken by starlings. This indicates that the starlings at least had some form of reference memory and took more than just the last experience into account, but adjusted their behaviour rapidly.

In the simulations swans with different search tactics obtained very similar gain rates during the depletion; they differed in the moment they decided to leave the pondweed bed. The late departing swans were more effective foragers because the actual gain rates of departing swans were generally still above the giving-up gain rate γGU at the moment of bed departure. For the comparison between early and late departing swans we assumed that the actual gain rate achieved at the alternative habitat (sugarbeet fields) equals γGU. We think that is a good approximation since birds are thought to switch habitat as soon as they expect higher rewards elsewhere (Sutherland 1996). Which search tactic was most effective in this sense was dependent on whether tubers were or were not aggregated in larger spatial units than the grain of the swans.

In a hypothetical environment with randomly distributed tubers, swans that used only intensive search mode would be more effective than swans that alternated between intensive and extensive search modes. This is because the latter spent time swimming longer distances during which their expectation of the quality of the bed as a whole decreased (and might fall below γGU). In contrast, in a patchy continuous environment, swans with alternating search modes more effectively exploited the tubers than swans that used intensive search mode only. The swans using only intensive mode experienced large decreases in their quality assessment of the bed as a whole while in a 10 × 10 m2 area with a low tuber density. Swans with alternating modes were able to escape such areas and were able to obtain an average gain above γGU long after the swans that used intensive mode only had flown away.

The extensive search mode with high-speed, directed steps is apparently a successful tactic to search for new food clumps. This was show previously by Zollner & Lima (1999), who simulated search paths with different sinuosities when dispersing through a patchy landscape. They found that the best nonsystematic search paths to a new patch were nearly straight, the optimal angular concentration being >0·9. We found a much lower value (0·3), in accordance with the values between 0·1 and 0·5 reported by Bergman et al. (2000) in long-distance migrating caribou. However, Zollner & Lima (1999) show that the probability of successful search is a nearly flat function of angular concentration when the patches are clumped. In other words, a new food patch is readily found in a clumped habitat as long as the path is neither random nor completely straight.

Considerable theoretical progress has been made towards understanding the optimal foraging behaviour on clumped, discrete prey that occurs in easily recognizable patches (Green 1988; Olsson & Holmgren 1998, 2000). For the continuous patchy environment considered here, we studied two important aspects of this behaviour, namely the search speed and direction, but the optimal decision rules of other aspects have yet to be determined. The swans’ memory factor and giving-up gain rate were calibrated against data of a particular year, and did not differ depending on the search tactic or tuber biomass aggregation. The optimal parameter values may well be different and this may explain why the remaining tuber biomass density in the virtual pondweed bed was generally a little (0·2–1·4 g m−2) greater than the giving-up density of 7·0 g m−2. Our model swans used the extensive mode when they decided to swim after only one foraging bout in the current cell, but again the optimal rule for switching between search modes may depend on the circumstances (Benhamou 1992). The next step would be to establish the optimal rules for the different search tactics under various conditions and to investigate whether the outcome of this study is sensitive to these changes or not.

We conclude that the observed search pattern was in accordance with the predicted area-restricted search, which consists of an alternation of intensive search mode with low speed, sinuous moves in high-resource areas, and extensive search mode with high-speed, directed moves between these areas. The scale of the food clumps and movement patterns were found to match, so we suggest that swans travel small, random distances within food clumps, but great, more directional distances between food clumps. In this way, they conceivably achieve a greater energy gain on the spatially autocorrelated pondweed bed than they would with a single search mode or by always moving randomly.

Acknowledgements

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

We would like to thank Harold Claassen, Kirsten Engelaar, Marcel Klaassen, Oscar Langevoord, Roef Mulder and, last but not least, Erik Wessel for their assistance in collecting the movement path data. The food dispersion was measured together with Thijs de Boer, Ten Dekkers, Helen Hangelbroek, Oscar Langevoord, Koos Swart and Erik Wessel. We acknowledge Staatsbosbeheer for giving permission to work in their nature reserve in the Lauwersmeer. We would like to thank Valery Andreev for his invaluable support in Russia. Daniel Fortin, John Fryxell, Marcel Klaassen, Ola Olsson and one anonymous referee helped to improve this paper with their comments. The work in Russia was funded by the Netherlands Organization for Scientific Research (NWO grant 047–002–008). BAN is also grateful to John Fryxell for hosting his sabbatical at the University of Guelph. This is publication 2886 of the Netherlands Institute of Ecology.

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  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix
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Appendix

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

INDIVIDUAL-BASED MODEL OF SWAN FORAGING ON TUBERS

Time step

The time step x during the simulations was taken to represent 1 s.

Pondweed bed

In the bed without spatial autocorrelation in tuber biomass, the initial tuber biomass in each cell of the 40 × 40 m2 grid was assigned by a draw from the ln-normal distribution established for 1995. A bed with spatial autocorrelation was created by similarly drawing an average initial tuber biomass for each 10 × 10 m2 subplot. Subsequently, an initial tuber biomass was assigned to each cell within a given subplot by a draw from a normal distribution with a standard deviation of 5 around the back-transformed mean for that subplot. This standard deviation of 5 gave a Pearson correlation coefficient of c. 0·72 (N = 15, P2t < 0·005) between the tuber biomass in an arbitrarily chosen focal cell and that in other cells in the same subplot. The average and standard deviation of the overall tuber biomass density D (g m−2) in the bed with spatial autocorrelation was 2·29 ± 0·84 after transformation ln(D + 1).

Forager density

We used a swan density of c. 0·01 birds m−2, equivalent to the peak density observed in the field (B. A. Nolet et al., unpublished data), by introducing 16 virtual swans of the wild type and one mutant on the 40 × 40 m2 grid. The initial position of these swans was randomly assigned.

Foraging behaviour

A foraging bout consisted of a fixed sequence of trampling, feeding and resting. At the beginning of the simulations the swans started with trampling. The trampling phase had a fixed duration of 1 s in accordance with field data (Nolet et al. 2001b). The duration of the feeding phase was calculated using a modified version of the model of Houston & Carbone (1992). The details and parameter values are given in Nolet et al. (2001b). In the model, oxygen reserves are used during feeding and replenished during surfacing. At the end of a foraging bout, the expected rate of energy gain during feeding in the next foraging bout is calculated from the tuber biomass density in the current cell, the functional response, and the duration of the last (optimal) feeding phase (see Forager departure rules). The optimal surfacing duration s* and the associated optimal feeding duration t* that maximize the rate of energy gain while maintaining the oxygen balance over a foraging bout, are then found iteratively. The maximum t was set at 12 s. s* varied little at the scale of the time-step of the simulation model and a fixed duration (2 s) was used for the surfacing phase based on field observations (Nolet et al. 2001b).

Forager gain

The gross intake rate i while feeding was calculated according to the type II functional response: inline image (Holling’s disc equation), where D is the resource bio- mass density, a is the area of attack (0·001 m2 s−1), and th is the handling time (2·0 s g−1) (Nolet et al. 2001b). This was multiplied with the energy concentration of 17·3 kJ g−1 and a metabolisability of 0·90 (Nolet & Drent 1998) to obtain the metabolisable energy intake rate.

The activity costs were measured using heart rate following the methodology of Nolet et al. (1992). The rate of energy expenditure was estimated to be 140 J s−1 during trampling, 40 J s−1 during feeding, and 20 J s−1 during surfacing (Nolet et al. 2001b). Energy expenditure during swimming costs was measured to be 32 J s−1 (B. A. Nolet, R. M. Bevan and M. Klaassen, unpublished data). These activity costs were subtracted from the rate of metabolisable energy intake rate to obtain the net intake rate or gain rate g.

Forager learning

The quality of the bed constantly changed due to food depletion by the swan itself and by its flock members. Each swan tracked these changes by updating its expected gain rate γ with the actual gain rate g, weighted by the memory factor µ at every time step x:

  • γx+1 = µ · gx + (1 – µ)γx

(Bernstein et al. 1988). As the memory factor we used µ = 0·005. The memory factor determines the rate of learning: after a sudden 10% change in actual gain rate, it will take a swan with this value of µ 10 min before the swan’s expected gain rate is within 1% of the new actual gain rate.

The prior expected gain rate γ was set at 200 J s−1 at the start of a simulation. This is the expected average gain rate in a foraging bout at a tuber biomass density of 25 g m−2, common at the start of swan foraging (Nolet et al. 2001b). Under these conditions, the optimal duration of the feeding phase t* is 6 s. γ was updated at every time step x (i.e. not only during feeding but also during trampling, surfacing and swimming when g is negative).

Forager departure rules

The swan swam to another cell when the expected gain rate in the next foraging bout in the current cell was lower than its current expected gain rate γx for the bed as a whole. The expected gain rate in the next foraging bout was based on the functional response. The final resource biomass density Df at the end of the next foraging bout can be calculated from the initial tuber biomass density Di and the current optimal duration of the feeding phase t*, since:

  • image

(Case 2000: 253). Df, which is found by iteration, then gives the cumulative gross intake (i.e. A[Di – Df], with A, the area of a cell, being 1 m2), which can be converted to rate of gain g (see Forager gain).

The swans flew away when γx < γGU, with γGU = 35·3 J s−1. This was calculated as the gain rate that a swan achieved in a foraging bout at the giving-up tuber density (7·0 g m−2) found in the field (Nolet et al. 2001b).

Forager movement speed

The observed swimming speed was 0·7 m s−1 (B. A. Nolet, unpublished data), so the duration of the swimming phase in the intensive search mode (one move to an adjacent cell with an average distance of 1·2 m) was set at 2 s. In the extensive search mode (12 movements) the swimming phase lasted 14 s in total.