Carrying capacity in overwintering birds: when are spatial models needed?

Authors

  • J. D. Goss-Custard,

    Corresponding author
    1. Centre for Ecology and Hydrology (CEH) Dorset, Winfrith Technology Centre, Dorchester, Dorset, DT2 8ZD, UK
      J. D. Goss-Custard, 30 The Strand, Topsham, Exeter EX3 0AY, UK (e-mail j.d.goss-custard@exeter.ac.uk).
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  • R. A. Stillman,

    1. Centre for Ecology and Hydrology (CEH) Dorset, Winfrith Technology Centre, Dorchester, Dorset, DT2 8ZD, UK
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  • R. W. G. Caldow,

    1. Centre for Ecology and Hydrology (CEH) Dorset, Winfrith Technology Centre, Dorchester, Dorset, DT2 8ZD, UK
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  • A. D. West,

    1. Centre for Ecology and Hydrology (CEH) Dorset, Winfrith Technology Centre, Dorchester, Dorset, DT2 8ZD, UK
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  • M. Guillemain

    1. Centre for Ecology and Hydrology (CEH) Dorset, Winfrith Technology Centre, Dorchester, Dorset, DT2 8ZD, UK
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    • *

      Present address: Office National de la Chasse et de la Faune Sauvage, CNERA Avifaune Migratrice, La Tour du Valat, Le Sambuc, 13200 Arles, France.


J. D. Goss-Custard, 30 The Strand, Topsham, Exeter EX3 0AY, UK (e-mail j.d.goss-custard@exeter.ac.uk).

Summary

  • 1We sometimes need to predict the maximum number of bird-days that can be supported by the food supply in a site used by migratory birds outside the breeding season. So defined, carrying capacity is often estimated using the daily ration model (DRM). In this, the total biomass of accessible food, aggregated across all patches of differing food density, is divided by an individual's daily requirement. Carrying capacity can also be estimated using spatial depletion models (SDM), in which patches of differing food density are treated separately. We identify here some of the features of the food supply that enable patches to be amalgamated so that the very simple DRM can be used instead of a more complex SDM.
  • 2We show by theoretical modelling that the predictions of the DRM and SDM are often the same even though initial food density varies between patches. The wide range of conditions over which this is so are specified.
  • 3A DRM and a SDM of wildfowl eating seagrass Zostera spp. in a nature reserve produced similar predictions for the number of bird-days supported intertidally before the birds switched to farmland.
  • 4We conclude that a DRM can often be used instead of a SDM to predict bird-day carrying capacity. We identified two conditions in which the DRM cannot be used: (i) when the rate of additional food loss due to factors other than depletion by the birds themselves differs between patches; and (ii) when the relative profitabilities of patches, and thus the number of birds using each patch, change through the depletion period in non-simple ways that cannot be predicted without a SDM. Examples of such exceptions are described.
  • 5Synthesis and applications. By showing when a DRM can be used instead of a SDM, this study should help nature managers to predict the bird-day carrying capacity of a site in the simplest way yet available. However, predicting the maximum number of bird-days supported is not equivalent to predicting demographic rates and should not be confused with predictions for population size. Rather, it is probably best regarded as a measure of site quality.

Introduction

There is sometimes a need to predict the maximum number of bird-days that can be supported by the food supply in a site used by a migratory bird species outside the breeding season. Such carrying capacity predictions can help select the best policy for managing a site to maintain or increase its conservation value (Sutherland & Allport 1994). The simplest calculation uses the daily ration model (DRM), in which the total amount of consumable food is divided by the daily food requirement of an individual, all of which is assumed to be identical and interference-free. ‘Consumable food’ is the total biomass of accessible food that is above the minimum starvation or leaving ‘threshold food density’ needed for an average bird to feed fast enough to obtain its daily requirements in the time available. Examples in which carrying capacity has been calculated this way are provided by Alonso, Alonso & Bautista (1994), Anderson & Low (1976), Charman (1979), Cornelius (1977), Korschgen, George & Green (1988), Lovvorn & Baldwin (1996) and Michot (1997).

Recently, spatial depletion models (SDM) have been developed to calculate carrying capacity. These models recognize that food supplies usually comprise patches that differ in certain features, such as food density and accessibility and frequency of disturbance by people. The birds are again assumed to be identical and interference-free but are distributed between patches according to simple rules. For example, in Sutherland & Allport's (1994) pioneering model for bean geese Anser fabilis fabilis L. and wigeon Anas penelope L., model birds initially forage in the densest patch until food density there is depleted to that of the next dense patch, whereupon birds exploit both patches in equal numbers. As depletion proceeds, the birds gradually exploit a wider range of patches and spread out over the food gradient until food density everywhere reaches the ‘threshold density’ at which all birds leave or starve simultaneously. Such models are usually applied to a local site, such as a nature reserve, and it is assumed that birds using different patches do not incur significantly different movement costs in time and energy. The models have a spatial dimension but they are not spatially explicit because patch co-ordinates are not specified.

This study expands the point made by Sutherland & Anderson (1993) that, when calculating the carrying capacity of a site, SDM are only required when patches differ in certain significant features unless, of course, the spatial sequence of food depletion is itself of interest. We identify some of the features of food patches that, solely for the purpose of measuring the combined bird-day carrying capacity of multiple patches, allow them to be amalgamated. Our results show that patches of differing initial food density, for example, can often be amalgamated. This is useful to know because it enables nature managers to use a very simple model to measure a site's carrying capacity.

Methods

natural history assumptions

We have in mind the maximum number of bird-days for migratory birds that can be supported in the non-breeding season by a site in which food is little replenished by growth and recruitment and so declines through depletion and other causes, such as inclement weather; these being called ‘additional losses’, L. Sometimes food is replaced by growth, reproduction and immigration and may contribute significantly to carrying capacity (Hockey et al. 1992). Our models can include food gain by changing the sign of L and so can be applied to many migratory animals. However, to make our main point, we need only discuss losses. Food density refers to the food items that are accessible to the birds and therefore excludes food that, for one reason or another, is out of the birds’ reach.

DRM and SDM can only be used when it is appropriate to assume that a single daily energetic requirement applies to all birds in the site so that all have the same movement and food-acquisition costs, irrespective of where they feed. If these costs are regarded as important, and are likely to differ between birds feeding in different patches, a spatially explicit model must be used, such as that developed by R.W.G. Caldow in Pettifor et al. (2000).

spatial depletion model

Description

The formal description of the individual behaviour-based model (IBBM) used here is in Goss-Custard et al. (2002). Used as a SDM, the parameters defining individual competitive ability were set to zero so that all individuals were identical. There was no interference or body reserves. If at any point during a simulation two or more patches provided the same intake rates, the birds were distributed equally between them. This is a slightly different allocation procedure from other SDM in which birds are distributed equally between patches with the same food density and not intake rate. However, this makes no difference. Our functional response is described by a monotonically increasing function so that intake rate increased across the whole range of food density. Intake rates were only the same in two patches if their food densities were also the same.

The model was seeded with the same initial food supply on day 1 in all simulations. The food was distributed between five patches of equal area Aj = 2000 units, with initial food densities fj = 1000, 800, 600, 400 and 200 food items unit area−1 in each patch. The total quantity of food at the start was therefore 6000 000 items. The daily energy requirements of the average individual was, for simplicity, constant throughout each simulation at r= 100 units. The food would therefore support 60 000 bird-days if all could be consumed. The gradient of the functional response can affect carrying capacity (Goss-Custard et al. 2002) so two kinds of functional response were used, as illustrated in fig. 1a in Goss-Custard et al. (2002). Response A had a steeper gradient than response B so the threshold food density was lower in A than in B. The two parameters that describe functional responses A and B, respectively, were: food density at which the consumption rate was 50% of the maximum, f50 = 50 and 500, and the asymptote, Imax= 500 and 500. The number of birds occupying the site in autumn, N, was varied over the range 50–1000. All other parameters in table 1A of Goss-Custard et al. (2002) were set to zero.

Table 1.  The daily ration model, showing the equations used to calculate the number of days for which the population is supported (T), food abundance (F) as a function of time (t) within the range t= 0–T (the time period during which animals are alive), and the number of bird-days supported (S). The remaining parameters are A= patch area, f0= initial food density, f* = threshold food density for survival, F0= initial food abundance, N0= initial number of animals, r= food intake rate required for survival. See Appendix 1 for the derivation of f*, Appendix 2 for the derivation of the model with no additional food losses and Appendix 3 for the derivation of the models with additional food losses, expressed either as a constant or proportionate daily amount
ModelTFS
No additional source of food lossinline image
F0rN0t
inline image
Additional source of food loss at a daily constant rateinline image
F0− (rN0+Lconst)t
inline image
Additional source of food loss at a daily proportionate rateinline imageinline imageinline image

Incorporating additional food losses

Additional losses can be represented as an absolute amount lost each day or as a daily percentage of the current standing crop. Removing an absolute amount daily means that an increasing proportion of the diminishing standing stock is removed on each successive day. Removing a fixed daily percentage means that the absolute amount removed is higher early in the simulation before the standing stock has much decreased and greatest in the patches with the highest standing crop biomass.

Predicted bird-days might differ according to how additional losses are represented in SDM and spatially dimensioned models might give different, and more realistic, predictions than the DRM in which all the food is aggregated into one patch. The difficulty in exploring these possibilities was to ensure that the threshold food density was reached in all patches of the SDM on the same day, whether the ‘percentage’ or ‘absolute’ methods were used to represent additional loss. Were this not to happen, the total amount of additional food losses would not be equal at the end of each simulation.

Our solution was to run preliminary simulations in which 1% of the current standing crop was removed daily from each patch until all 500 birds starved. We prevented food from being removed from a patch once food density had reached the threshold density, f*, so that all subsequent additional losses occurred in the consumable food, thus ensuring comparability with the DRM. This procedure estimated the total additional losses of food, L, and the number of days elapsed, T, before the birds starved or left. L/T is the daily absolute rate of additional loss required to reduce the food by the same total amount over the same total period as when the loss was expressed as a fixed daily percentage of the current standing crop.

In the simulations themselves, the daily absolute rate of additional loss, L/T, was distributed between patches in proportion to the initial food abundance in each patch on day 1. By this means, the absolute reduction was set at such a level that, on the day the last individual in the ‘percentage’ simulation died or left, the total additional loss in the ‘percentage’ and ‘absolute’ simulations were the same. If the two ways of expressing additional loss made no difference to bird-day predictions, all individuals should starve or leave on the same day whether the daily additional loss was expressed as a percentage or an absolute amount, and the maximum bird-day predictions would be identical.

daily ration model

Description

The total amount of consumable food across all patches combined, Fcons, is the quantity above the threshold density, f*, at which animals cannot feed fast enough to obtain their energy requirements, r, in the feeding time available per 24 h. The maximum number of bird-days (S) that can be supported is the total consumable food divided by the daily ration: S=Fcons/r.

Appendix 1 describes how the threshold density, f*, can be derived from the functional response. Appendix 2 derives the simplest DRM in which no additional loss of food occurs. Appendix 3 derives the DRM that includes additional food losses expressed either as a constant absolute or percentage rate. Table 1 gives a summary of the equations used to calculate S, the number of days for which the bird population is supported (T) and the food abundance (F) as a function of time (t) within the range t= 0–T (i.e. the time period during which birds are alive).

Aggregating patches at different shore levels

Sometimes patches are not available for the same time each day, e.g. wildfowl feeding on intertidal vegetation (Percival, Sutherland & Evans 1998). A patch downshore is available for less time per tidal cycle than is a patch upshore. If a bird were to restrict its foraging to either the top or bottom of the shore, the threshold food density, f*, would be much higher downshore. The different values of f* for each shore level could be used in the DRM.

But this would usually be very unrealistic. Birds unable to obtain their requirements solely by feeding downshore would feed upshore on the ebbing and flooding tides and would therefore have the same time to feed as birds feeding only upshore. Unless individuals are forced to feed only at one shore level, the value of f* should be the same at all shore levels and correspond to the time available for feeding upshore.

Furthermore, the threshold density would be reached more-or-less simultaneously at all shore levels because any disparity in food densities between upshore and downshore would be equalized over the low water period when all shore levels are accessible. Thus a high rate of upshore depletion over one tidal cycle as the tide was ebbing and flowing would cause foragers to concentrate during the next low water period on the now more profitable downshore, tending to equalize food density across shore levels.

That such equalization can be achieved intertidally by unrestrained birds was tested using a modified version of the SDM in which the time available for feeding per tidal cycle differed between the two food patches used. The upshore patch was exposed for 8 h tidal cycle−1; 2 h on the receding tide, 4 h over low tide and 2 h as the tide advanced. The downshore patch was exposed for 4 h over low water only. Both patches had the same size (5000 m2) and food density (272·725 g m−2, or 6000 kJ m−2). There were 1500 birds, each needing 100 kJ day−1. Simulations were run with type A and B functional responses, and either with no additional food loss or with a daily loss rate of 0·020408 day−1. For simplicity, birds fed at the same rate night and day. Birds failing to obtain enough food over low water downshore fed upshore on subsequent advancing and receding tides.

Habitat loss and disturbance

Percival, Sutherland & Evans (1998) used a SDM to predict the effects on intertidal herbivorous wildfowl of food loss downshore due to sea level rise and upshore due to encroachment by cord grass Spartina anglica C.E. Hubbard. Disturbance from hunting upshore forced birds initially to feed downshore until the food there was so depleted that they had to risk feeding further upshore, closer to the hunters.

To model this with the DRM, calculations were made in two stages. In stage 1, all 1500 birds only fed downshore until depletion and additional food losses (if any) had reduced food density to the point at which the birds could no longer meet their requirements in the 4 h tidal cycle−1 the zone was exposed. T was calculated as in Table 1 using the appropriate value of f*. The stage 1 values of T varied between 102 and 161 days. The number of bird-days accumulated by the end of stage 1 was T × 1500.

In stage 2, birds were assumed to feed both upshore and downshore. In simulations with no additional losses, the initial upshore food supply was simply added to the quantity remaining in the downshore zone at the end of stage 1. In simulations with additional losses, the initial food supply upshore was first reduced by the losses that had occurred during stage 1, using the equation to calculate F in Table 1. The DRM in Table 1 was then used to calculate how many bird-days the food supply present in the downshore and upshore zones combined at the start of the second stage would support until it had been reduced to the threshold density, f*, appropriate for an 8-h tidal cycle. In simulations in which the rate of additional loss differed between zones, an average value was used weighted by the total amount of food present in each zone at the start of stage 2. Summing the predicted bird-days for stages 1 (downshore only) and 2 (downshore and upshore combined) estimated the maximum number of bird-days supported throughout.

brent goose model

Most comparisons here between the predictions of SDM and DRM are made using hypothetical parameter values chosen to cover a wide range of natural situations. Although the conclusions should therefore have widespread relevance, some readers may prefer to have the findings replicated in a real case. A comparison was therefore made between the predictions of a DRM and SDM for brent geese Branta bernicla bernicla L. spending the non-breeding season on the Exe estuary, UK, the only system for which we had the data. The IBBM used had been built by RWGC to predict the effect of changes in the quality of one or more wintering sites on the equilibrium population size of the entire brent goose population in north-west Europe; model details and parameter values are in Pettifor et al. (2000). It was used here to calculate the maximum number of goose-days supported by the Exe's eight intertidal beds of Zostera spp. before the birds switched to grassland. The IBBM was run until the last bird switched to the model grassland, and the number of goose-days on the Zostera patches up until the average bird made the switch was calculated.

Some data generated by this model were used to parameterize the DRM in order to ensure that both models were parameterized identically and so differed only in their spatial dimension. The average daily ration and daily rate of additional food loss were calculated from day 1 until the day the average bird abandoned Zostera. These, together with the biomass density of Zostera on day 1, were used to define in the DRM the daily ration, the daily additional loss and the threshold biomass density. Using the output of the SDM to parameterize the DRM is legitimate because we were not comparing the predictive abilities of two independent models. Rather it is a real-world illustration that, using the same parameters, a SDM and DRM predict the same carrying capacity.

Results

initial food density and additional losses

In the simplest simulations, patches differed only in their initial food density. With no additional losses, bird-days were constant across the whole range of N in both models and exactly the same in each, whichever functional response was used, being 58 755 with functional response A and 47 505 with functional response B. Bird-days were greater with response A because the threshold occurs at a lower food density than in response B, allowing a greater proportion of the food to be used before birds starved. When patches differed only in their initial food density, the spatial component in the SDM did not produce any differences in predicted bird-days from those of the DRM.

Further simulations were run to test whether the DRM and SDM still gave the same bird-day predictions when additional food losses occurred at the same daily rate in all patches. The predictions of the SDM were essentially the same as those of the DRM irrespective of the manner in which additional food losses were included and of the functional response used (cf. rows 1 and 2 in Table 2 with the values given in the legend). [Note too that the method used to include additional losses did not affect the predictions either with higher and lower bird numbers (rows 3–6; Table 2). But carrying capacity did depend on the initial number of animals, regardless of the shape of the functional response, because it determines how long it takes the birds to deplete the food and therefore for how long the food is also reduced by additional losses.]

Table 2.  Bird-days supported according to the means by which additional food losses due to causes other than depletion are represented in the spatial depletion model with either 500, 50 or 1000 birds (N) occupying the food supply on day 1 of the simulation. The prediction with only one patch is identical to the daily ration model predictions of 37 998 and 28 030 bird-days for functional responses A and B, respectively
RowManner in which food was removedNNumber of food patchesBird-days supported with functional responses A and B
AB
1By a constant daily amount 500537 99928 086
2By a fixed proportion each day 500538 00028 276
3By a constant daily amount  50511 841 7 050
4By a fixed proportion each day  50511 650 7 168
5By a constant daily amount1000545 52535 000
6By a fixed proportion each day1000545 52435 000

We conclude that the DRM and SDM give the same bird-day predictions when the initial food density differs between patches and either the absolute or proportionate rates of additional losses, along with the threshold food density, f*, are the same in all patches.

spatial variation in the time available for feeding on the shore

To apply the DRM intertidally where patches are accessible for different amounts of time, it was first necessary to test that the threshold food density, f*, would be the same in the downshore and upshore zones and would be similar to that calculated from the functional response using a foraging period equivalent to the duration of the upshore zone (see the Methods section). In accordance with the prediction, the upshore and downshore threshold food densities were very similar to each other and to the values calculated from the functional response, irrespective of whether additional losses occurred (Table 3). The values from the SDM were a little lower because the model tracks whether birds are alive at the end of each 24 h, allowing birds to continue consuming food on the day they starve after f* has been passed.

Table 3.  The density of food remaining in the upshore and downshore patch at the end of the day on which all the birds starved, as predicted by the spatial depletion model (top two rows). The initial food density in all patches was 272·7 g m−2. Lprop= the instantaneous rate of additional food loss. f *, the threshold density at which birds cannot feed fast enough to obtain their energy requirements in an 8-h tidal cycle, are shown in the third row
LpropFunctional response A Upshore (g m−2)Downshore (g m−2)Functional response B Upshore (g m−2)Downshore (g m−2)
00·310·263·973·80
0·020408 24 h−10·340·284·023·84
 0·400·404·014·0

Depletion by model birds therefore equalized the densities of the food in the two zones down to the density predicted from the functional response. The maximum bird-days supported by both zones combined can therefore be calculated by summing their initial food and using the value of f* appropriate for the period for which the upshore feeding grounds are exposed.

habitat loss and disturbance

Simulations with a SDM of wildfowl by Percival, Sutherland & Evans (1998) suggested that removing upshore food would reduce carrying capacity by much more than would removing equivalent amounts downshore. Disturbance from shooting upshore forced birds initially to concentrate downshore until depletion downshore forced the birds to feed upshore, closer to the hunters.

This was modelled by first ignoring the disturbance so birds fed in both zones from the start. The DRM again gave identical results to our SDM, irrespective of the functional response used and whether there were additional food losses (Fig. 1). Predicted bird-days were identical whether up to 90% of the food was removed from upshore or downshore except with functional response B and when > 90% of the food had been removed. Bird-days were then reduced by 6000 (2%) more when food was removed from upshore than from downshore.

Figure 1.

The maximum number of bird-days supported by an upshore and a downshore intertidal patch combined, as predicted by (a) the spatial depletion model and (b) the daily ration model, using functional response A or B. Top pairs of symbols = no additional food losses due to factors other than bird depletion; bottom pairs of symbols = additional losses occurred at a daily rate of 0·0020408, both upshore and downshore. Circles = food removed from upshore on day 0 of the simulation, before depletion by the birds had begun; crosses = food removed from downshore on day 0. The initial biomass is the total quantity of food present upshore and downshore combined at the start of day 1 of the simulation.

These results may contrast with those of Percival, Sutherland & Evans (1998) because we excluded upshore hunting disturbance. But repeating the calculations with birds being forced first to feed only downshore made no difference; the result was the same whether food was removed from upshore or downshore (Fig. 2). Further simulations across a wide range of parameter values for f* and Lprop were run, including putting 75% of the total food supply either upshore or downshore. The results were always the same, except when Lprop differed between upshore and downshore (Fig. 3). With a lower rate of additional loss upshore than downshore, food loss upshore now had a greater effect on carrying capacity than an equivalent food loss downshore, as found by Percival, Sutherland & Evans (1998).

Figure 2.

The maximum number of bird-days supported by an upshore and a downshore intertidal patch combined, as predicted by the daily ration model, using functional response A or B. Hunting upshore prevented birds from feeding in the upshore zone until they had depleted the food in the downshore zone to the threshold density for that zone. Top pairs of symbols = no additional food losses due to factors other than bird depletion; bottom pairs of symbols = additional food losses occurred at a daily rate of 0·0020408, both upshore and downshore. Circles = food removed from upshore on day 0 of the simulation, before depletion by the birds had begun; crosses = food removed from downshore on day 0. The initial biomass is the total quantity of food present upshore and downshore combined at the start of day 1 of the simulation.

Figure 3.

The maximum number of bird-days supported by an upshore and a downshore intertidal patch combined, as predicted by the daily ration model, using functional response A or B. Hunting upshore prevented birds from feeding in the upshore zone until they had depleted the food in the downshore zone to the threshold density for that zone. Additional losses due to factors other than bird depletion occurred initially at a daily rate of 0·0020408 upshore and 0·010204 downshore, but at a weighted-average rate after birds started feeding upshore. Circles = food removed from upshore on day 0 of the simulation, before depletion by the birds had begun; crosses = food removed from downshore on day 0. The initial biomass is the total quantity of food present upshore and downshore combined at the start of day 1 of the simulation.

brent geese and zostera

The prediction of the DRM for goose-days supported by Zostera before geese switched to grassland was 70 791, almost identical to that of 70 568 of the SDM model. [The actual number of bird-days recorded, 65 815, was quite close, bearing in mind that the value of Lprop used had not been measured on the Exe, and is likely to vary between winters and sites.]

Discussion

initial food density and additional losses

The simulations showed that the DRM and SDM give the same bird-day predictions when the initial food density differed between patches, additional losses occurred at the same absolute or proportionate rates in each patch and the threshold food density, f*, was also everywhere the same. This happens because all the identical birds starve or leave on the same day. A given quantity of food, aggregated across patches, therefore lasts a certain number of days, T, irrespective of the patch from which it is taken and thus independently of the distribution of birds across patches. The total amount of food remaining at the end of each day of the depletion period does not depend on the birds’ distribution across patches either because they remove the same amount wherever they feed. With the daily rate of additional loss also the same in all patches, the total amount additionally lost each day, or over the whole of time T, is always the same, irrespective of the patches from which it is lost and of the way in which bird depletion is distributed across patches. In these very simple circumstances, the spatial dimension is not required to predict the maximum number of bird-days a site can support.

spatial variations in the threshold food density, F*.

Sometimes the time and energy costs of exploiting food differ between patches. For example, Bewick's swans Cygnus columbianus bewickii L. foraging on submerged tubers of sago pondweed Potamogeton pectinatus L. stop feeding at different tuber densities according to water depth and sediment silt content (Nolet et al. 2001). By the time every patch has been depleted to the threshold food density, f*, food density will not have been equalized across all patches. However, before f* has been reached at its different levels in every patch, birds would be expected to congregate in the currently most profitable patch so that the net rates of food intake would be equalized across patches as depletion proceeds. If this reasoning is correct, f* should be reached more-or-less simultaneously in all patches, even though the values of f* differ between patches.

If the differences in energy costs between patches are small compared with the total daily energy expenditure, the daily requirements, r, can be considered the same in all patches. The DRM can then be used by simply calculating the quantity of food in each patch above the patch-specific value of f* and summing the consumable food across all patches. But if the energy costs differ significantly between patches, the DRM cannot be used because r, and the time available for acquiring it, would differ between patches. A spatially explicit model would be needed to calculate the daily energy requirements of birds that move between patches and incur patch-specific time and energy costs.

spatial variation in the time available for feeding on the shore

Simulations with the SDM showed that the threshold food densities, f*, in two intertidal zones that were accessible to birds for different amounts of time each day were the same and equivalent to that predicted from the functional response using the longer exposure period of the upshore zone. Equalization might sometimes not occur when there is insufficient time, or too few birds, to deplete the downshore sites over low water by an amount sufficient to reach the current food density upshore (see below). Otherwise, carrying capacity can again be calculated by aggregating the food across the shore and using a value of f* appropriate for the exposure period upshore.

habitat loss and disturbance

Percival, Sutherland & Evans (1998) suggest that the greater impact on a shore's carrying capacity for wildfowl of removing upshore food (through Spartina encroachment) compared with removing downshore food (through sea-level rise) is caused by the longer period for which upshore sites are accessible each tidal cycle. Although this is difficult to evaluate because the way exposure period affected their model birds is unclear, our results suggest that the different exposure time upshore and downshore zones was not the reason. Our SDM showed that the birds generally starve when the food density in both zones reaches the value of f* appropriate to the upshore 8-h tidal cycle foraging period because birds feeding at low water downshore can also feed upshore as the tide ebbs and flows. Upshore and downshore zones are therefore part of the same food supply so that habitat loss reduces the amount of consumable food and thus the bird-days supported by the same amount, irrespective of its location.

We found a difference between removing upshore and downshore food with functional response B but only after > 90% of the food had been removed. Bird-days were then reduced by 6000 (2%) more when food was removed from upshore than from downshore. This exceptional case arose because the threshold density was reached in the upshore zone 4 days before it would have been reached downshore and, during those 4 days, there was not enough foraging time for birds to survive by feeding downshore alone. All birds starved 4 days before all the food on the shore reached the 8-h tidal cycle threshold density, and so not all of it was used. This explanation might also account for the different giving-up food densities reported in intertidal Bewick swans (Nolet & Drent 1998). How often this occurs in nature is likely to depend on the gradient of the functional responses because, in our simulations, it did not happen with response A as, with its steep gradient, food density upshore and downshore reached f* at almost the same time.

In our simulations, habitat loss upshore reduced bird-days by more than did habitat loss downshore only when the rate of additional food loss, L, was greater upshore than downshore. This probably explains the difference between our results and those of Percival, Sutherland & Evans (1998). In their study, the upshore zones were dominated by Z. noltii, which had a slower rate of additional loss than Z. angustifolia, which dominated downshore. A given biomass of food upshore supports more bird-days than an equivalent amount downshore because a greater proportion of the upshore food is used by the foragers before it is lost in other ways. Removing 1000 kg of upshore food therefore removes more consumable food than does the loss of an equivalent amount downshore, and therefore has a greater impact on bird-day carrying capacity.

when can a drm be used instead of a sdm?

Our main finding is that, with identical and interference-free individuals, carrying capacity measured as bird-days by SDM and DRM is often the same even though the initial density of food varies between patches and irrespective of the functional response shape, the initial number of animals occupying the food supply, the time for which different patches are accessible in intertidal systems or the precise manner (percentage or absolute rates) in which food is removed by causes other than depletion by the birds themselves. A SDM is therefore not required for predicting carrying capacity over a wide range of conditions. The much simpler DRM gives essentially the same result by using the equations in Table 1 and (i) the total quantity of accessible food that occurs at densities above the food threshold density, aggregated across all patches; (ii) the daily ration required by the average individual to survive in the site; (iii) the additional food losses (or gains) due to sources other than depletion; and (iv) the number of animals initially occupying the site. Carrying capacity has often been measured this way and our results suggest that the absence of a spatial dimension will often not have affected reliability.

It may sometimes be necessary to carry out some initial calculations. For example, when the birds use patches in a sequence dictated by factors other than depletion, such as disturbance, and if additional losses (or gains) also occur at different rates in different patches, a separate DRM must be used sequentially for each patch or each group of similar patches within the site, as illustrated above with intertidal wildfowl. Although rather more cumbersome, this is still much simpler than building a SDM.

There are circumstances, however, when the DRM cannot be used in place of a SDM. The over-riding conditions for this seem to be (i) when the rate of additional loss differs between patches and (ii) when the relative profitabilities of patches, and thus the number of birds using each patch, change through the depletion period in non-simple ways that cannot be predicted without a SDM. One example of this is when the rate of additional loss, L, differs between patches and birds are not constrained to use patches in a specified sequence, as the wildfowl at Lindisfarne were forced to do by disturbance (Percival et al. 1998). A SDM is then needed to track the changing relative profitabilities of different patches, and thus bird distribution across patches, arising from the simultaneous processes of depletion and additional losses. Similarly, Sutherland & Allport (1994) showed how patch value to one species depends on the amount of prior exploitation by another. As the relative profitabilities of the patches at every stage of the depletion process are the result of an interaction between the rates of additional loss and gain, they cannot be forecast without a SDM. Another example is Nolet et al.'s (2001) Bewick swans in which the energy costs of foraging, and so net profitability, differs between patches. If this difference is small it can be ignored without too much loss of precision and a site-value of r (which of course includes foraging energy costs) used across all patches in the DRM. But if the difference is large the SDM would have to be used to track the continually changing net profitabilites, and thus bird numbers, in the different patches.

application to nature management

Despite these important exceptions, in many situations the DRM provides a simpler alternative to the more complex SDM for calculating carrying capacity measured as the maximum supportable bird-days. However, whether provided by DRM or SDM, carrying capacity predictions provide only limited information for nature managers. Because they assume all individuals are identical, such models cannot predict mortality rate or body condition at spring migration, the two most important pieces of information on fitness required when the demographic consequences of alternative policies are being evaluated. Predicting the maximum number of bird-days supported is not equivalent to predicting demographic rates and should not be confused with predictions for population size, the quantity that nature managers usually aim to maintain or increase. To do so may be very damaging to birds because many may starve or lose condition, and so fail to migrate successfully, well before a site's capacity has been reached (Goss-Custard et al. 2002). For this reason, the maximum supportable number of bird-days is probably best regarded as a measure of site quality in systems in which birds exhaust the food supply in one site before moving to the next, whether large numbers stay for a short time or smaller numbers stay for longer.

Acknowledgements

We are very grateful to Mike Bell, Bruno Ens, Jan van Gils, Ian Johnstone, Romke Kats, Jaap van der Meer, Theunis Piersma, Anne Rutten, Bill Sutherland and Wouter Vahl for helpful comments. M. Guillemain was supported by a Marie Curie Fellowship from the European Community programme ‘Human Potential’ under contract number HPMF-CT-2000 00945.

Appendices

Appendix 1

calculating the threshold food density for survival from the functional response

We assume that a population of identical animals feeds within a single patch in which food is uniformly distributed, and that intake rate increases with increased food density up to an asymptotic rate. As a specific case, we assume the following functional response:

image(eqn 1)

where I= intake rate, Imax= maximum intake rate achieved when food is superabundant, f= food density within patch and f50= food density at which intake is 50% of maximum.

We assume that animals need to consume food at a rate r to meet their requirements and that animals do not store reserves and so die of starvation if their intake rate falls below r. All animals are identical and so all survive if intake rate is greater than or equal to requirements. The threshold food density (f*) below which all animals starve, and above which they all survive, is found by substituting f* for f and r for I in equation 1:

image(eqn 2)

which can be rearranged to find the threshold food density for survival:

image(eqn 3)

Appendix 2

derivation of the daily ration model with no additional food loss

This simple ration model assumes that (i) all the food is in a single patch; (ii) depletion by animals is the only factor affecting food density; (iii) all animals are identical; (iv) interference is absent; (v) animals do not store energy reserves; and (vi) animals must consume food at a fixed rate in order to avoid starvation.

The model considers a population of N animals foraging in a patch of area A in which food occurs at a density f. The total amount of food in the patch (F) is therefore:

F=Af.(eqn 4)

Each animal consumes food at a rate r in order to meet its requirements. The rate at which food abundance is reduced by the animal population is therefore:

image(eqn 5)

The intake rate of animals is a function of food density and so the number surviving at any time is also a function of food density. As intake rate decreases with decreased food density and all animals are identical, all survive if the density of food is above a threshold value (f*). The number of surviving animals is therefore:

image(eqn 6)

where N0= initial number of animals. The model is only concerned with the period of time during which animals are alive (i.e. when f ≥ f*). Substituting equation 6 into equation 5 when f ≥ f* gives:

image(eqn 7)

Integration then gives:

F=arN0t(eqn 8)

where a= integration constant. We assume that F=F0 when t= 0 and so a=F0. Substituting F0 for a gives:

F=F0rN0t,(eqn 9)

where F0= initial food abundance. Food abundance is converted to food density by substituting equation 4 for F and F0:

Af=Af0rN0t,(eqn 10)

where f0= initial food density. Rearranging and simplifying gives:

image(eqn 11)

The time for which the animal population is supported (T) (i.e. the time taken for food density to be reduced from f0 to f*) is found by setting f to f* and t to T and rearranging:

image(eqn 12)

The number of animals alive from t= 0 to T is found from equation 6 when f ≥ f*:

N=N0.(eqn 13)

The number of bird-days supported (S) is found by integrating equation 13 with respect to t within the range t= 0 to T:

image(eqn 14)

Integration then gives:

image(eqn 15)

Substituting T and 0 for t gives:

S=N0T.(eqn 16)

Substitution of T with equation 12 and simplification gives:

image(eqn 17)

Appendix 3

derivation of the daily ration models incorporating additional food loss

Absolute amount lost per day

This model makes the same assumptions as the simple ration model, with the exception that food density is reduced both by depletion by the foragers and other sources. The additional sources of loss remove food at a constant absolute daily rate. The rate at which food abundance decreases is therefore:

image(eqn 18)

where Lconst= constant rate of reduction in food abundance due to sources other than the animals themselves. This equation extends equation 5 of the simple ration model by including a term for constant alternative food losses. As in the simple ration model, all animals are identical and so the number surviving at any time is given by equation 6. Substituting equation 6 into equation 18 when f ≥ f* (i.e. when all animals are alive) gives:

image(eqn 19)

Integration then gives:

F=arN0tLconstt,(eqn 20)

where a= integration constant. We assume that F=F0 when t= 0 and so a=F0. Substituting F0 for a and rearranging gives:

F=F0− (rN0+Lconst)t,(eqn 21)

where F0= initial food abundance. Food abundance is converted to food density by substituting equation 4 for F and F0. Rearranging and simplifying as in the simple model (equations 10 and 11) then gives:

image(eqn 22)

The time for which the animal population is supported (T) (i.e. the time taken for food density to be reduced from f0 to f*) is found by setting f to f* and t to T and rearranging:

image(eqn 23)

As in the simple ration model, the number of animals alive at any time t between 0 and T is found from equation 13. The number of bird-days supported (S) is found by integrating equation 13 with respect to t within the range t= 0 to T (equations 14–16). Substituting T with equation 23 gives:

image(eqn 24)
Proportional daily loss

This model makes the same assumptions as the constant absolute food loss model except that the additional sources of loss remove food at a proportional rate. The rate at which food abundance decreases is therefore:

image(eqn 25)

where Lprop= proportional rate of alternative food loss and 0 ≤ Lprop ≤ 1 (see below for how Lprop can be measured in the field). This equation extends equation 5 of the simple ration model by including a term for proportional food loss due to factors other than depletion.

As in the simple ration model, all animals are identical and so the number surviving at any time is given by equation 6. Substituting equation 6 into equation 25 when f ≥ f*(i.e. when all animals are alive) gives:

image(eqn 26)

Taking the reciprocal of both sides gives:

image(eqn 27)

Integration then gives:

image(eqn 28)

where a= integration constant. We assume that t= 0 when F=F0 and so:

image(eqn 29)

Substituting equation 29 into equation 28 gives:

image(eqn 30)

which can be simplified to give:

image(eqn 31)

Rearranging to express F as a function of time and simplification gives:

image(eqn 32)

Food abundance is converted to food density by substituting equation 4 for F and F0. Rearranging and simplifying as in the simple model then gives:

image(eqn 33)

The time for which the animal population is supported (T) (i.e. the time taken for food density to be reduced from f0 to f*) is found by setting f to f* and t to T and rearranging:

image(eqn 34)

As in the simple ration model, the number of animals alive at any time t between 0 and T is found from equation 13. The number of bird-days supported (S) is found by integrating equation 13 with respect to t within the range t= 0 to T (equations 14–16)). Substituting T with equation 34 gives:

image(eqn 35)
Measuring Lprop in the field

Lprop can be measured when the rate of loss due to the animals themselves is zero (i.e. r= 0), as in an exclosure. Then equation 33 can be simplified to:

image(eqn 36)

Which can be rearranged to give:

image(eqn 37)

Ancillary