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Recommendations for depletion modelling of granivorous birds

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For simplicity of modelling, we conclude from the above that daily ration approaches are likely to be as useful as functional response approaches. Furthermore, daily ration approaches permit sensitivity analyses to be carried out on a single parameter (dc), rather than two parameters (Th

Fig. 2.4. Accuracy of daily ration (a,b,c) and functional response (d,e,f) models with erroneous parameter estimates. On this resource, the forager has an actual handling time, Th = 1s and an actual search efficiency, a' = 0.01m2s-1. Required daily intake is 7,200 food items, achievable in 2h if food is not limiting. In the functional response models, the forager may spend a maximum of 6h per day feeding. For the daily ration models, the threshold leaving density (or critical density, dc) is calculated on the basis of estimates of Th and a'. In the functional response models, all foraging is conducted according to those estimates. (a,d) Th is underestimated at 0.5s; (b,e) Th is estimated accurately; (c,f) Th is overestimated at 2s. a' is overestimated by an order of magnitude (i), accurate (ii), or underestimated by an order of magnitude (iii).

and a', the effects of which are summarised by their effect on dc). The critical density may be varied through a realistic range of values calculated both by using equation 2.4, above, and also derived from aggregative responses. Critical densities may be derived from aggregative responses by assuming that the critical density is approximately equal to the lowest density at which birds were observed to feed. These are unlikely to be accurate but will give some indication of the correct order of magnitude. Using the functional responses listed in Table 2.1 and aggregative responses shown in Fig. 2.1, critical densities are estimated to lie somewhere between 0.5 and 12.5kJm-2, with a modal estimate in the region of 5kJm-2.
2.3.3 interference models
Interference is a reversible process by which intake rates are affected by the density of foragers within a patch. Increased interference at higher forager densities may be caused by a variety of mechanisms (Goss-Custard, 1980), including disturbance, increased aggression, and increases in the frequency of food theft (kleptoparasitism). Disturbance can result where high forager densities lead to food becoming trampled, or prey species moving to less accessible areas. This is likely to be less serious when feeding on non-motile prey (Goss-Custard, 1970). Aggression is most likely in territorial species and, for many species therefore, is most likely to be an important factor in the breeding season. Kleptoparasitism has been shown to be less frequent when foragers feed on a relatively low value resource that can be quickly swallowed (Sutherland & Koene, 1982). For many species, it has been suggested that interference may be less important than individual foraging efficiency (Caldow et al., 1999; Stillman et al., 2000a). All of these factors suggest that interference may be a relatively unimportant process for modelling foraging in farmland birds, especially during the non-breeding season.
2.3.4 predictions of behavioural models
An important requirement of behavioural models is that the dynamics of the resource populations are well understood. This means that behavioural models will often be most easily applied to highly specialised foragers that feed on only one or a few types of prey, with the most reliable results being produced for seasons in which rates of resource renewal are closest to zero. For the majority of farmland birds that feed on a wide variety of invertebrate and plant species during the breeding season, modelling interactions with their resources during this period is likely to prove highly complex. Behavioural population models that consider the breeding season also, rarely include an explicit treatment of foraging during this season (e.g. (Stillman et al., 2001). Rather, population dynamics in the breeding season are modelled using phenomenological data on fecundities of the type discussed in section 2.2 or, more simply, using data on mean (or density dependent) fecundity in the species of interest. As already observed, relationships between fecundity and resource availability for farmland birds are sparse, and this approach would be restricted to species such as the grey partridge. Furthermore, behavioural foraging models usually predict the point at which foragers will either die, or will leave an area to forage elsewhere. In order to incorporate these into full population models, it is necessary either to assume a closed population (an unrealistic assumption for many species), or to have an understanding of the fate of individuals that do leave to forage elsewhere. Despite the obvious appeal of full population models, therefore, for the majority of species these will be no more informative than models that consider only numbers of foragers that can be supported during the non-breeding season.
Two types of prediction may be made using models of dynamics during the non-breeding season. First, as previously discussed, it is possible to estimate the number of forager-days that an area of habitat can support over winter. In a system that can support 100 forager days over winter, the forager-days approach makes no distinction between 100 foragers removing their daily requirement on day 1 of the model, and one forager removing its daily requirement each day for 100 days. Where there is feedback between the foragers and their resource, the level and temporal distribution of depletion may have serious consequences for the resource dynamics. Arable weed communities are such systems and the forager-days approach would be inappropriate in these cases.
The second type of prediction that may be made is an estimate of the likely population size at the end of winter. This is possible using either an energetic flux approach, an individual-based approach with individual variation, spatial constraints or decision error, or an iterative approach. The energetic flux approach combines estimates of rates of uptake, storage and expenditure of energy, in order to model individual energy reserves and to predict the point at which an individual will die. This approach has been pioneered by Stillman et al. (2000b), who noted that their model (constructed for oystercatchers Haematopus ostralagus feeding on mussels Mytilus edulis) was particularly sensitive to energetic parameters, including estimates of energy expenditure and assimilation as reserves. It is unlikely that these parameters could be ascertained as accurately for the majority of farmland birds as for the well-studied oystercatcher.
It is difficult to predict mortality using standard depletion models. In a uniform environment models will predict that all foragers will feed equally until available resources are completely depleted, at which point all the foragers will either leave or die together. Instances of mass mortality have been documented amongst (mostly marine) birds (e.g. Baduini et al., 2001; Camphuysen et al., 2002; Piatt & Van Pelt, 1997) and may occur at a localised scale amongst farmland birds. However, such events are likely to be rare. One way to introduce a more gradual mortality rate, is to introduce individual differences in requirements, error, or spatial constraints into the depletion model. Assigning birds a body mass drawn from a random distribution of plausible masses, means that different individuals will have different daily requirements (according, for example, to equation 2.1). However, variation in body mass would have to be implausibly high in order to stagger deaths over more than a few days. Similarly, only large variations in foraging efficiency (reflected, for example, in different critical thresholds for different individuals), would lead to mortalities being staggered over an entire season. Decision error (owing to individuals having an incomplete knowledge of the best feeding locations), travel costs and sampling restrictions could all be used to introduce staggered mortality into a depletion model. However, decision error would introduce a further unmeasurable assumption, whilst the other approaches are also hard to measure for the majority of farmland birds and would require a spatially-explicit modelling framework. A spatially-explicit model would, in turn, require that resource availability could also be modelled at a spatial resolution appropriate for the heterogeneity of its distribution. The principal food resource of farmland birds in the non-breeding season is weed seeds, which may be extremely heterogeneous in their distribution, even at the scale of individual weed plants. However, models currently available to simulate weed dynamics (e.g. Freckleton & Watkinson, 1998; Lintell Smith et al., 1999) are deterministic, based on mean seed and plant densities over an area. Thus, efforts to incorporate variation in individual fates through these methods may lead to undesirable complications for modelling.
Perhaps the easiest way to determine sustainable populations over winter, is by using an iterative approach (Atkinson, 1998). This approach involves recording the state of the environment at the start of the non-breeding season and then iteratively sending populations through winter. If the population survives, start conditions can be reinstated and a larger population sent through winter. Similarly, if the population fails, a smaller population must be tried. In this way, it is possible to determine (within the limits of the stochastic nature of the model environment) the largest population that can survive the whole winter.

2.4 Conclusions
2.4.1 predictive models for granivorous birds
In sections 2.2 and 2.3 we discussed the techniques available for predicting the response of farmland bird populations to changing food supplies. In terms of flexibility to a variety of species and applicability to novel circumstances, depletion models appear to have the greatest utility of the approaches discussed. For predicting population trends into the future, however, it is necessary to have population models that run from year to year, including dynamics in both the breeding and non-breeding seasons. The limitations of depletion models in this regard are discussed in section 2.3. It seems likely that given current constraints on available data, long-term population forecasting will be most feasible using a model that predicts overwinter populations using an iterative depletion model but a phenomenological model of breeding season dynamics based on empirical data. The latter should link breeding dynamics to resource availability wherever possible.
To apply such models at the level of a farmland bird community also poses many problems. Dietary divisions between the majority of species are indistinct. A major review of farmland bird diets (see Objectives 3 and 5) shows that assessments of a single species’ diet vary markedly according to both the methodology used (crop-content, gut-content or faecal analysis), and the year and location of studies. The former confounds attempts to classify the diet of a species exactly, whilst the latter indicates that most species have some flexibility of diet to reflect what is available. Furthermore, for farmland birds, niche separation is often at a very fine scale (e.g. different species may specialise in foraging in different parts of a single field) and modelling competition and coexistence between species with broadly similar diets requires more information than is currently available. Thus, it may be possible only to model certain types of birds with broadly different diets (e.g. grain specialists or weed seed specialists), to determine the likely consequences of a given change in the dynamics of resources for these broad groups.
2.4.2 limitations of available data
The preceding discussion indicates several areas in which available data may limit the range of approaches available for predicting the response of farmland bird populations to changing food supply. These limitations apply both to phenomenological models and to behavioural models. In particular, further research in four main areas is desirable to enhance the power of predictive models. Aggregative responses
As discussed in Section 2.2.1, aggregative responses are surprisingly time-consuming and difficult to collect. Nevertheless, they represent one of the most straightforward tools available to modellers for predicting the consequences of changing food availability. At present, very few examples of aggregative responses for farmland birds are available from published literature. Collecting aggregative responses for more farmland birds should be a priority, not only to aid the resolution of problems discussed in Section 2.2.1, but also to provide more information on the distribution and abundance of resources for farmland birds. Food supply and demography
A crucial limitation of both phenomenological population models and the breeding season component of behavioural models, is the lack of data linking demography to food supply for granivorous birds. Many autecological studies of farmland passerines produce excellent data on the reproduction and survival of these species. However, measuring food supply should also be standard practice within such studies, in order that the critical link between food availability and both summer and winter density dependence may be better understood. Density dependence is unlikely to be easily inferred from studies of only a few years’ duration and, consequently, long-term studies, such as that of the grey partridge, remain of very high value. Critical food densities
The estimation of food densities below which foragers cannot obtain their required daily intake is important for depletion models, regardless of the mechanism used to model foraging. Approximate values for critical densities can be determined from aggregative responses and this provides a further reason for increasing the number of aggregative responses available for farmland birds. Critical densities may also be derived from functional responses. More studies, both in the field and in aviaries, are needed to determine functional responses more accurately, and for a greater number of granivorous species. Given the tight allometric relationships between body mass and energetic requirements, it might also be expected that functional responses would show similar scaling rules for species with similar diets. For example, small seeds may be hard to find (high a') but very quick to eat (low Th), whereas large seeds are likely to be easy to find (low a') but take longer to deal with (high Th). Considerably more functional responses are required if such generalities are to be determined. Breeding season diet and relation of demography to food dynamics
The year-round dynamics of many arable weeds and their seed production are reasonably well understood (see Objective 4). However, during the breeding season, many farmland bird species adopt a much broader and more flexible diet, including a wide variety of seeds and invertebrates (see Objectives 3 and 5). We have already alluded to the dearth of studies linking demography to the availability of these foods but for behavioural models of foraging in the summer, it is also necessary to understand spatio-temporal patterns of their abundance and depletion. This, in turn, demands detailed studies of diet, foraging ecology and resource availability for farmland birds during the breeding season.

In conclusion, simulation modelling techniques available for predicting the consequences of changing food supply for farmland birds, have largely outstripped the availability of data required to parameterise them. Detailed predictions are possible only for a limited number of species in a restricted set of circumstances. Current simulations should rely most heavily on depletion modelling, as this approach has the greatest generality and requires the least specific inputs. Concurrent efforts should be focussed on gathering specific data on a greater range of species, such that more general rules can be discerned from species-specific approaches.

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