Abstract Individual-based models (IBMs) predict how animal populations will be affected by changes in their environment by modeling the responses of fitness-maximizing individuals to environmental change and by calculating how their aggregate responses change the average fitness of individuals and thus the demographic rates, and therefore size of the population. This paper describes how the need to develop a new approach to make such predictions was identified in the mid-1970s following work done to predict the effect of building a freshwater reservoir on part of the intertidal feeding areas of the shorebirds Charadrii that overwinter on the Wash, a large embayment on the east coast of England. The paper describes how the approach was developed and tested over 20 years (1976–1995) on a population of European oystercatchers Haematopus ostralegus eating mussels Mytilus edulis on the Exe estuary in Devon, England.
The paper goes on to describe how individual-based modeling has been applied over the last 10 years to a wide range of environmental issues and to many species of shorebirds and wildfowl in a number of European countries. Although it took 20 years to develop the approach for 1 bird species on 1 estuary, ways have been found by which it can now be applied quite rapidly to a wide range of species, at spatial scales ranging from 1 estuary to the whole continent of Europe. This can now be done within the time period typically allotted to environmental impact assessments involving coastal bird populations in Europe.
The models are being used routinely to predict the impact on the fitness of coastal shorebirds and wildfowl of habitat loss from (i) development, such as building a port over intertidal flats; (ii) disturbance from people, raptors, and aircraft; (iii) harvesting shellfish; and (iv) climate change and any associated rise in sea level. The model has also been used to evaluate the probable effectiveness of mitigation measures aimed at ameliorating the impact of such environmental changes on the birds. The first steps are now being taken to extend the approach to diving sea ducks and farmland birds during the nonbreeding season.
The models have been successful in predicting the observed behavior and mortality rates in winter of shorebirds on a number of European estuaries, and some of the most important of these tests are described. These successful tests of model predictions raise confidence that the model can be used to advise policy makers concerned with the management of the coast and its important bird populations.
Coastal areas (estuaries, bays, and extensive flats) provide vital feeding areas for migratory shorebirds (Charadrii and Anatidae), particularly outside the breeding season while they are on passage and/or wintering. These birds are of great conservation value and, around the world, many local, national, and international measures have been taken to protect them. Most of the birds can only feed on their macro-invertebrate and plant food supplies when these are exposed on the intertidal flats over low tide. Many human activities, which are potentially damaging to bird populations, are carried out on the coast (e.g., land-claim, salt production, resource harvesting, recreation), so policy decisions are frequently required on how best to maintain these populations and to reconcile their protection with economic and other activities. However, decisions are frequently not based on quantitative predictions from models that simulate the effect on birds of alternative management and policy options.
The research described here is filling this gap. It does so by developing individual-based models (IBMs) that can be applied rapidly when predictions are required, whether at one or more local sites or to guide a continent-wide policy. The models provide policy makers with quantitative predictions as to the possible negative effects on shorebirds (if any) of one or more of the wide range of human activities on the coast. Simultaneously, they provide quantitative predictions as to the possible positive, or counteracting, effects of, for example, any mitigating measures that might be proposed. Potentially, they therefore provide the means by which informed decisions can be taken on reconciling human activities with the conservation of an important component of coastal biodiversity.
Because the objective of nonbreeding wader and wildfowl conservation policy is to at least maintain present bird numbers, the best measure of the effect of a human activity on birds is the predicted change in population size. Population size in shorebirds is a function of the interaction between (i) the mortality and reproductive rates in the breeding range and (ii) the mortality rate in the nonbreeding range, including the migratory routes (Goss-Custard [1980, 1993], Goss-Custard and Durell , Goss-Custard et al. [1995c, d; 1996b]). We therefore need a method to measure the effect of an activity on the two quantities that are believed to determine individual fitness, and thus demographic rates, in these birds outside the breeding season. The first is the chance of starving, perhaps especially during severe winter weather. The second is the size of the fat reserve that is needed by the surviving birds to fuel migration and, in spring, also to breed successfully after they have reached the breeding grounds. If one can show that the feeding conditions following a proposed change in land use, policy, or management regime would allow the present-day rates overwinter survival and fat storage to continue, there would be no reason to be concerned for the birds (Goss-Custard et al. , Goss-Custard , West et al. ). If, on the other hand, body condition and survival were predicted to decline, population size would be expected to decrease by an amount that depends on (i) the strength of any compensatory density-dependent reproduction on the breeding grounds (Goss-Custard and Durell , Goss-Custard et al. [1995c, d; 1996b]) and (ii) the availability of alternative wintering or passage sites (Pettifor et al. ).
This paper describes how the development of IBMs over the last 2 decades has enabled these predictions to be made with some confidence for shorebirds. The paper starts by describing how the need for a new approach for making such predictions for shorebirds in Europe was identified in the mid-1970s. It then goes on to trace the subsequent development, application, and testing of the models on a number of European estuaries.
2. Identifying the need for a new approach
A threatened water shortage in southeast England during the early 1970s led to the suggestion that a freshwater reservoir could be constructed over the intertidal flats of the Wash, a large embayment in east England (Figure 1). The idea was, in effect, to build a huge container over the intertidal mudflats and sandflats and to store within it freshwater pumped from the upper reaches of the nearby River Ouse. We were contracted to predict the effect that removing a large area of intertidal feeding ground might have on the very large and internationally important populations of shorebirds that depend on the Wash in winter and on passage.
We conceptualized the threat raised by this proposal (and similar proposals) as follows. Removing intertidal flats would reduce the size of the feeding grounds available to the birds. If the birds stayed in the Wash—as would seem likely—the construction of the reservoir would inevitably lead to an increase in the densities of the birds on the feeding grounds that remain. Because the same number of birds would be competing for a reduced food supply, this would increase any competition for food that already occurred in the populations. Alternatively, if competition was not already present in the system, a loss of habitat could introduce it for the first time. The problem, though, was to find out whether or not competition would be increased and, if so, whether bird fitness would be reduced because, in theory, animals can compete for food but at such low intensities that their fitness is unaffected.
As Figure 2A illustrates using mortality over the winter as the measure of fitness, the central issue is whether fitness is already density dependent or is likely to become density dependent if the habitat was removed (Goss-Custard and Durell , Goss-Custard ). At very low bird densities, potential competitors can avoid each other: Density can increase from low levels without competition being intensified and without fitness being affected. But at some point, X, as density rises, competition will begin to reduce fitness. If the system is at present well to the left of the critical threshold density X, habitat could be removed without the birds being affected. But if the loss of habitat raised densities above the critical threshold, the mortality rate would increase. Of course, if the system is already to the right of X, an increase in mortality following habitat loss would be inevitable.
Although the idea was simple, its execution was not. First, it is technically very difficult to measure the winter mortality rate and spring body condition of shorebirds, and very few studies have done so. Second, even if it were technically possible, it would still be very difficult to establish the form of the function across a sufficiently wide range of bird density. The population size of long-lived birds, such as shorebirds, varies by only small amounts from year to year. As illustrated in Figure 2B, 10 years of difficult research might yield only a cluster of points that cannot describe the underlying density-dependent relationship (Goss-Custard et al. [1995c]). Third—and of critical importance—the mortality rates at densities higher than those that currently occur would remain unknown: yet, of course, it is over these densities that fall outside the present-day empirical range for which predictions are required. Finally, the very act of removing habitat may itself change the density-dependent function. For example, if above-average habitat was lost, it is expected that the critical threshold would move to the left and that the subsequent density-dependent slope would increase (Figure 2C). Not only would the present-day density-dependent function shown in Figure 2A be technically very difficult to establish empirically, but all the difficulties and severe uncertainties of having to predict beyond the present-day empirical range would remain.
At the time this work was done on the Wash (1972–1976), the means with which to describe the density-dependent mortality function in present-day and novel conditions were not available. The predictions as to how the birds would be affected by the construction of the reservoir were therefore nonquantitative and based on the findings of field studies but using the same logic as illustrated in Figure 2A (Goss-Custard ). It was, of course, realized that it would have been preferable to provide quantitative predictions but the necessary mathematical models did not exist at that time.
3. Individual-based modeling of the Exe estuary oystercatcher population
So, how could density-dependent functions in shorebirds be determined for both existing environments and for the novel conditions brought about by habitat loss and all the other perceived threats to the feeding grounds of nonbreeding shorebirds, such as harvesting shellfish: disturbance from people, planes, and birds of prey: eutrophication: sea-level rise: and climate change?
As the underlying processes that produce the density-dependent starvation function would be competition among birds for food, it was believed that a study of individual variation in foraging and competitive behavior would eventually provide a means of establishing the parameters of the density-dependent starvation function in existing and novel environments. A study was therefore set up in 1976 on a system that seemed—and proved to be—very suitable for in-depth studies of individual shorebirds. This was the European oystercatcher Haematopus ostralegus that fed primarily on mussels Mytilus edulis on the Exe estuary in southwest England (Figure 3). The small size and easy accessibility of this estuary along with its relative isolation from neighboring estuaries were suitable, as was the oystercatcher whose large size and large prey made it among the easiest shorebird on which to study competitive processes on the feeding grounds. In addition to establishing the density-dependent functions, there was the particular applied project aim of modeling the interaction between the bird and its shellfish prey: The possible conflict between conservationists and the shellfishing industry in Europe was, and remains, a major issue for the managers of the coast (Goss-Custard et al. [1996a, 2004]).
Although a behavior- and individual-based approach was very early seen as the best way in which to derive the density-dependent mortality function (Goss-Custard ), it has taken many years to develop fully operational models. Initially, we called the models “behavior based” to stress the importance of behavioral processes when predicting how birds will respond to a change in their foraging environment. But, more recently, we joined the rest of our scientific community by calling them IBMs following the excellent review of the theory and practice of individual-based ecology by Grimm and Railsback . The first version of the model was published (after much difficulty and long delays imposed by skeptical referees!) in 1995 (Goss-Custard et al. [1995a, b]), the second version was published a year later (Clarke and Goss-Custard ), and the third version was published by Stillman et al. (2000b). Although a fourth version of the model is now available, version 3 is the one mainly used in this paper. The research on the European oystercatcher over 20 years on the Exe estuary and in many other sites throughout Europe, particularly the Netherlands and France, on which the model is based is summarized in Goss-Custard  and Blomert et al. .
3.1 The model The model is individual based as it tracks the diet, foraging location, and body condition of each individual within the population and whether it starves before the end of winter. The food supply is distributed among a number of discrete patches, each of which may differ in the type (prey species), quantity (numerical density), and quality (size and energy content) of the food items it contains as well as in the duration of the time for which it is exposed through spring and neap tidal cycles.
During each 24-hour day, each bird must consume enough food to meet its energy demands that vary daily according to the ambient temperature. An individual attempts to do this by feeding in the locations, times of the day (day or night), and stages of the tidal cycle where its intake rate is currently the highest. Although all individuals make these choices using the same optimization principle—intake rate maximization—the actual decisions made by each animal differ. This is because their individual choices depend on their individual competitive abilities, which in turn depend on two characteristics. The interference-free intake rate is the rate at which an individual feeds in the absence of interference competition and measures an animal's basic foraging efficiency. The susceptibility to interference measures by how much interference from competitors reduces an individual's intake rate as bird density rises and this, in turn, depends on the animal's social dominance in contests over food items and feeding sites. The model is therefore game theoretic in that each animal responds to the decisions made by competitors in deciding when, where, and on what to feed, as illustrated in Figure 4A.
Survival is determined by the balance between an individual's daily rates of energy expenditure and consumption. Energy expenditure depends on metabolic costs plus any cost of thermoregulation at low temperatures. Energy consumption depends both on the time available for feeding—for example, the duration of the exposure period of the food patches—and on the intake rate while feeding, which in turn depends on the profitability of the prey and on the individual's susceptibility to interference. When daily energy consumption exceeds daily expenditure, individuals accumulate energy reserves or maintain them if a maximum level has already been reached. When daily requirements exceed daily consumption, individuals draw on their reserves. If an individual's reserves fall to zero, it starves, the only source of mortality in the model and the main source of oystercatcher winter mortality in the wild (Goss-Custard et al. [1996c]).
The model also incorporates those aspects of the seasonal change in the food supply that affect shorebirds. The overwinter reduction in the mass of individual prey animals is included, and their numerical density is reduced daily through depletion by the birds themselves and by other mortality agents, such as storms. If shellfishing occurs, the daily harvest is also deducted daily from the shellfish stocks present. Birds disturbed by shellfishers that harvest by hand-picking and by other disturbers, such as walkers, spend time and energy relocating to an undisturbed shellfish bed.
If a feeding patch is removed, either permanently by overfishing or some other form of habitat loss, or temporarily by disturbance, the birds that previously fed on that patch must choose another patch on which to feed (Figure 4B). They do this using the same rate-maximizing optimization decision rules as before. Because wintering shorebirds seem frequently to use this decision rule to decide where, when, and on what to feed (Goss-Custard ), the birds in the model probably react to a change in their feeding environment in the same way as real birds in the wild would respond to a loss of habitat. This is the fundamental reason why a model in which individuals respond to environmental change by making fitness-maximizing decisions are believed to provide a sound basis for predicting to the novel feeding environments brought about, for example, by habitat loss or shellfish harvesting.
3.2 Model development, calibration, and testing Version 3 of the IBM for Exe estuary oystercatchers includes the three main food supplies: (i) 10 mussel beds, which are exposed and available to birds for approximately 6 hours over low water, day and night; (ii) upshore feeding areas, which are additionally exposed for 1 hour on both the advancing and receding tides, again day and night; and (iii) terrestrial fields, which are continuously available, except at night. As in reality, model oystercatchers use the supplementary upshore and field feeding areas when the mussel beds are covered by the tide and the birds had been unable to obtain all their daily energy requirements over low tide from mussels alone.
In constructing the model, a “build-up” approach was adopted. Processes and parameters needed to make the model reproduce the mortality rates observed in the wild over the first five “calibration” winters of the project (1976/1977 to 1980/1981) were included, step-by-step, in the sequence that our understanding of the system suggested would be the most likely to succeed. The first version was rather simple and did not include some potentially important processes, such as feeding on supplementary food supplies (Goss-Custard et al. [1995a]). Only when such processes had been incorporated could the model predict the winter (September to March) mortality rates that had been observed over the calibration period. This is not surprising because natural history characteristics, such as feeding in fields over high water, form part of the animals' survival strategy whose exclusion from the model would inevitably reduce the chances that the model would predict mortality rates accurately.
Once the model was able accurately to replicate the observed mean mortality rate recorded over the five calibration years, it was used to generate a density-dependent function. This was done by running a series of simulations with differing numbers of oystercatchers, using for the food supply the average abundance and quality of mussels recorded over the 8-year study of the mussel population of the Exe estuary (McGrorty et al. [1990, 1993], McGrorty and Goss-Custard [1991, 1993, 1995], Stillman et al. [2000c]). The model predicted that mortality would be density dependent (Figure 5). This prediction was supported when a further six field estimates of the winter mortality rate of oystercatchers from the 1980s and 1990s became available. These data showed that bird density on the mussel beds and the winter weather both significantly affected the proportion of birds that survived the winter (Durell et al. [2000, 2001, 2003]).
3.3 Application of the model to the Exe estuary oystercatchers and shellfishery The observed mortality rates were more variable than the rates predicted by the model because annual variations in the ambient temperature and wind strength and in the flesh-content of the mussels—which has an important influence on intake rate (Goss-Custard et al. )—could not be included in the model simulations. However, it was concluded that the model predicted moderately well that mortality would be density dependent and that it could therefore be used to explore the effect of shellfishing on bird fitness.
Both model predictions and field data suggest that mortality became density dependent above bird densities of approximately 20–25 birds/ha of mussel bed; that is the critical threshold density X in this system lies approximately in that range (Figure 5). As bird density over the 25 years of the study had varied above and below this range, it seemed that some shellfishing could sometimes take place without reducing bird fitness, depending on the standing crop of the mussels and as long as not too many mussels were harvested.
But how many could be harvested before the birds would be affected? To explore this, the model was used to vary the quantity of mussels that remained in autumn after shellfish harvesting and at the time when the birds return to the estuary. Food abundance was expressed as the mass of mussel flesh (measured as ash-free dry weight [AFDM]) available on September 1. The effect of varying this quantity was simulated for both measures of fitness: that is winter mortality rate and body reserves in spring. In addition, simulations were run with and without the disturbance from shellfishers associated with harvesting by hand over low water, rather than by dredging from a boat over high water (Stillman et al. ).
The results show that, as the ratio between the biomass densities of the mussels and birds decreases, both measures of fitness are at first unaffected (Figure 6). But both measures of fitness decreased sharply below a critical ratio of about 200. This is equivalent to a “critical threshold” of 61 kg AFDM/bird in September, a concept that is perhaps easier to comprehend than the critical ratio and is therefore used later in this article. The reduction was greater if the shellfishers disturbed the birds while harvesting mussels. Over the study period, the stock of mussels in autumn was sometimes above and sometimes below the critical ratio and threshold. The model therefore quantified the abundance below which the mussel stock should be prevented from falling if bird fitness was to be maintained at its previous level. On the Exe estuary, the policy required to maintain the oystercatchers at their previous level of fitness is to prevent shellfishing reducing mussel stocks in September below the critical ratio of 200 or the critical threshold of 61 kg AFDM/bird.
An important point is that the shellfishery regime should aim to maintain stocks above the critical threshold level in the long term and not just in the current year. Some shellfishing techniques, such as low intensity hand picking, have no demonstrable effect on long-term mussel abundance, so stocks can probably be reduced each winter to the threshold value. But trawling mussels can reduce stocks in the long term by reducing recruitment (McGrorty et al. , Stillman et al. ). The shellfishing regime should therefore also prevent the cumulative impact of harvesting over successive years from reducing autumn shellfish stocks to below the threshold values (Stillman et al. ).
Inspection of Figure 6 shows that the reduction in fitness below the critical ratio and threshold seems to be rather slight, and some might argue that harvesting could be conducted down to lower levels of shellfish abundance without much affect on the size of the bird populations. However, this is not so because even small reductions in fitness can substantially reduce population size in long-lived animals such as shorebirds. This was demonstrated using a separate, demographic model of the population dynamics of shorebirds (Goss-Custard [1980, 1981], Goss-Custard and Durell , Goss-Custard et al. [1995e, d, 1996b]). Simulations with this model show, for example, that over the range of annual mortality rates found in oystercatchers the long-term equilibrium population size is sensitive to the mortality rate, whether there is strong or weak density dependence on the breeding grounds (Figure 7). The reason for this is that, whereas an increase in mortality from 3–6% represents only a small absolute increase, it actually represents a doubling of the annual mortality rate, with a consequently large impact on equilibrium population size (Goss-Custard [1980, 1981]), especially if the increased mortality falls especially heavily on young birds, which are the “seed corn” of future generations (Goss-Custard and Durell ). If the management priority is to maintain oystercatcher populations at their existing size, even quite small decreases in fitness due to shellfishing need to be avoided.
4. Applying the Exe oystercatcher model to other species and systems
As might be imagined, the first reaction to the appearance of this IBM was that it would be very difficult to apply it to other, less amenable species because, even in a species selected for its ease of study, the model had taken so long to develop and test. Although it is true that developing and testing the Exe estuary oystercatcher–mussel IBM had taken about 20–25 years of often time-consuming effort, it is now possible to parameterize and calibrate the model for other species and systems comparatively quickly and usually well within the time usually available for environmental impact assessment. Indeed, as will be shown later, successful models can now be produced and tested in just a few months!
The reason for this increased speed of application is that so many of the parameter values can be extracted from the literature, often in the form of allometric functions. These are the six groups of parameters that are needed to model a particular system:
(i) Bird energetics,
(ii) Prey energy content,
(iii) Functional responses,
(iv) Interference functions,
(v) Food supply, exposure time, weather, and
(vi) Human activities, for example, fishery.
The parameters that describe the local environment (e.g., tidal exposure periods for food patches, prey abundance) and the levels of disturbance (e.g., how frequently birds are disturbed and for how long, the shellfish harvest) must, of course, be obtained locally, although often the parameters are already known from previous local field surveys. The parameters for the bird energetics (e.g., daily energy requirement, energetic cost of flight after a disturbance) and the energy content of the prey (e.g., energy density, assimilation efficiency) are readily available from the literature. The parameters of the functional response (the intake rate of the birds as a function of prey abundance) and of the interference function (the reduction in intake rate due to competition as bird density increases) are the two parameter groups (i) on which most research time was spent in developing the oystercatcher model for the Exe estuary and (ii) for which it was, until recently, most difficult to obtain the parameter values for other species. It was the discovery of ways to obtain these two groups of parameter groups very rapidly that has allowed the models to be applied speedily to new species and systems. The way this has been done is described here.
4.1 Functional response Intake rate in shorebirds is usually measured as the AFDM of prey consumed per second of active foraging; that is excluding time spent on digestive pauses and other activities, such as preening. Although the widely applied Holling type II (“disc equation”) captures the general shape of these functional response, field studies on shorebirds have shown that the asymptote is not set by handling time, as assumed by the disc equation, because only about half the foraging time is spent in successfully or unsuccessfully attacking and handling prey, the rest being devoted to searching (Goss-Custard et al.[2006a]). Therefore it was not possible simply to measure handling time as a function of prey size and then use the disc equation to derive the parameters of the functional responses for the IBM.
A review of 30 functional responses in shorebirds showed that intake rate in free-living birds varied independently of prey density over a wide range, with the asymptote in most studies being reached at very low prey densities (<150/m2), as the example of oystercatchers eating mussels on the Exe estuary illustrates (Figure 8). This means that when researchers have measured intake rate in shorebirds for a wide range of purposes without going to the lengths of describing the functional response, most of their data would have been collected at or near the asymptote of the functional response. This, in turn, means that if we were able to predict these “spot” estimates of intake rate from parameters that were themselves easy to measure (e.g., prey size), it would be possible to predict the asymptote of shorebird functional responses easily and rapidly.
Accordingly, a multivariate analysis was carried out on 468 spot estimates of intake rates obtained from 26 shorebirds; none of these estimates came from studies of the functional response. This analysis identified 10 variables, representing characteristics both of the prey and of the shorebird, that accounted for 81% of the variance in logarithm-transformed intake rate. But four variables accounted for almost as much (77.3%), these being bird size, prey size, whether the bird was an oystercatcher eating mussels, or breeding. The four variable equations predicted all but 2 of the observed 30 estimates of the asymptote of the functional response with an average discrepancy of only 0.2% (Figure 9). An equation of just four variables therefore predicted the observed asymptote very successfully in 93% of cases. Indeed, if the birds were not breeding and were not oystercatchers eating mussels, equally reliable predictions were obtained using just two variables, bird and prey masses, both of which are very easy to measure.
The shorebird IBM now calculates the asymptote of the functional response of a bird of average foraging efficiency instantaneously from this allometric equation using just bird size (a state variable in the model) and the current mass of the prey in the patch in question, which is updated daily in model simulations. Rather than spend seven winters in the field measuring the asymptote of the functional response of oystercatchers eating mussels or earthworms Lumbricidae, as had been necessary for Exe estuary oystercatchers, the asymptote can now be measured instantaneously.
The gradient of the functional response was measured as the prey numerical density at which intake rate had risen half way to the asymptote. A multivariate analysis of the 23 estimates of the gradient of the functional response of shorebirds suggested the gradient was steeper when prey were small but less steep when the birds were large, and especially in oystercatchers (Goss-Custard et al.[2006a]). However, because all the available estimates were used to derive this equation, its predictive power has yet to be tested. But because in all but two cases the gradient was very steep, the IBM model for shorebirds uses an average value obtained across 21 field estimates. In shorebirds, intake rate seems generally to rise to half way to the asymptote at the very low prey densities of <150 prey/m2 that, as Figure 8 illustrates, is a very low density for most shorebird prey. Any imprecision in the estimate of the gradient arising from this simplifying assumption is therefore likely to have only a trivial influence on model predictions when interference competition is strong but could be important when depletion competition is important.
4.2 Interference function Interference is the reversible decrease in intake rate as the density of the birds, or competitors, increases. In shorebirds, it occurs for two main reasons (Goss-Custard ). Many mobile macro-invertebrate prey, such as some polychaete worms, escape a foraging shorebird by, for example, withdrawing rapidly into a burrow beyond the reach of the bird's bill. As bird density increases, therefore, an increasing proportion of the prey become unavailable to the birds whose intake rate therefore declines: This is termed interference through prey depression. In other cases, individual prey items are large and take some time to pick up and swallow (i.e., to “handle” them), as in oystercatchers eating shellfish. This can make it both possible and profitable for a dominant bird to attack and steal the food from a subdominant bird that has either found a spot where a prey is present or is in the process of handling a prey it has found. As bird density increases, it becomes increasingly difficult for subdominant birds to avoid dominant individuals so the frequency of theft increases and the intake rates of the subdominants declines: This is called interference through prey stealing.
It required several years of intensive fieldwork to measure the interference function in Exe estuary oystercatchers eating mussels in which interference arises through prey stealing (Figure 10). The two parameters that describe the function are (i) the threshold bird density at which intake begins to decrease and (ii) the slope of the decrease thereafter. A long period of fieldwork was required because, although the threshold was rather constant, the slope in oystercatchers eating mussels differs among individuals of different feeding methods and social dominance (Ens and Goss-Custard , Goss-Custard and Durell , Stillman et al. ), with the stage of the winter, and among birds of different age (Goss-Custard and Durell ). Given the crucial importance of interference in some shorebird systems (Goss-Custard et al. , West et al. ), it was imperative to find more rapid ways of estimating the parameters of the interference function if the shorebird IBM was to be applied successfully to different species and systems.
Recent research has shown that the strength of interference competition can be predicted in three ways (Stillman ): (i) from previous empirical studies of a range of species; (ii) from the predictions of a separate IBM of interference, and (iii) from the foraging behavior of the bird species in question and the mobility and antipredator escape responses of the prey.
Interference can be assumed not to occur in shorebird species consuming very small and immobile prey (e.g., the small snail Hydrobia spp.) that are (i) rapidly swallowed, so the birds cannot steal them, and (ii) have no antipredator response. Interference in a species consuming mobile prey with antipredator responses where the prey are too small to be stolen by competitors can be assumed to occur through prey depression as the prey can escape when they detect a shorebird predator. The threshold and slope of interference through this prey depression has been measured in some cases, notably redshank Tringa totanus eating the crustacean Corophium volutator (Yates et al. ), or can be predicted from a model of depression interference (Stillman et al. [2000a]). In shorebird species taking large prey that take a relatively long time (>5–10 seconds) to consume, interference arises through prey stealing and through the need for subdominants to avoid other birds that might steal their prey. The threshold and slope of the interference function in shorebirds eating large prey have now been measured in some systems (Stillman et al. [1996, 2002], Triplet et al. , Yates et al. ). But they can also be predicted for other systems from a small number of very easily measured variables (e.g., time required to deal with one prey, the running speed of the shorebird) by using another, quite separate IBM of interference through prey stealing (Stillman et al. [1997, 2002]). The predictions of this model have been tested with generally encouraging results (Figure 10). The development of the models for interference through either prey depression or prey stealing means that the parameters of the interference function can be obtained instantaneously for a system for which no field estimates are available. It is no longer necessary to carry out many years of time-consuming fieldwork, as was required for Exe estuary oystercatchers.
5. Application of the shorebird IBM to the management of shorebird populations
The development of ways by which the parameters of the functional response and interference function in shorebirds could be rapidly obtained opened the way for applying the model to solve coastal management problems other than the mussel fishery of the Exe estuary. Figure 11 summarizes the issues that have been addressed or are being addressed currently. Some are described here in more detail to illustrate how the shorebird IBM can be used to solve practical management problems.
5.1 Shellfishing and oystercatchers The largest numbers of birds that depend on commercially exploited shellfish occur in Europe during autumn and winter, over which period the birds must survive in good condition in order to migrate to the breeding grounds in spring. Although the birds can eat other prey species when shellfish stocks are low, these alternatives often do not allow birds to survive as well, and in such good body condition, as when shellfish are abundant (Camphuysen et al. [1996, 2002], Smit et al. , Atkinson et al. ). As a result, the size of the oystercatcher population wintering in the Wash, UK, for example, decreased sharply over the 1990s when shellfish stocks were frequently very low (Atkinson et al. ). Even more precipitous declines in the numbers of mollusc-eating birds (oystercatcher, knot Calidris canutus, eider duck Somateria mollissima) occurred in the Dutch Wadden Sea where winter shellfish stocks have frequently been at very low levels over the last 2 decades (Beukema and Cadée ).
If bird populations are to be maintained, sufficient shellfish to meet their demands must remain for them after harvesting. But the question then is what constitutes sufficient shellfish? A common approach has been to base this decision on the biomass of shellfish required by the whole bird population to satisfy its energy demands (Lambeck et al. ). One oystercatcher requires approximately 10 kg AFDM of shellfish flesh to satisfy its energy requirements from autumn to spring (Stillman et al. ). Ten thousand birds would therefore consume 100,000 kg (AFDM) by spring, equivalent to approximately 3,000 tons of shellfish wet weight (including shells).
But as the model of Exe estuary oystercatchers showed, it may take considerably more than 10 kg AFDM per bird to ensure that oystercatchers survive the winter in good condition (Figure 6). This finding was confirmed when the shorebird IBM was applied to a further four European oystercatcher–shellfish systems (Figure 12). As on the Exe estuary, the probability of oystercatchers starving in these four systems was extremely low and independent of shellfish stock at the high shellfish abundance of 70–120 kg AFDM/bird (Goss-Custard et al. ). It increased sharply below this range with the increase beginning over the range 20–61 kg AFDM/bird, depending on the system. The increase seemed to begin earlier where mussels, rather than cockles Cerastoderma edule, were the predominant shellfish (Figure 12).
The data show that, to ensure that most oystercatchers survive until spring, there must be present at the time of their arrival in autumn from two to six times the amount of shellfish that oystercatchers actually eat, even though they can and frequently do take alternative prey. In other words, the “ecological food requirement” greatly exceeds the “physiological requirement.” Reductions in shellfish biomass available per oystercatcher increases mortality rate (and reduces the birds' body condition) by simultaneously increasing the intensity of interference competition and the rate of food depletion.
To maintain oystercatcher populations in areas harvested for shellfish, stocks must not be allowed to fall below two to six times the shellfish biomass that the birds consume. Providing enough food to meet their physiological requirement is simply not enough. Until recently, the policy in The Netherlands for regulating shellfishing on their important tidal flats was to ensure that 60% of the food requirements of mollusc-eating birds, such as the oystercatcher, remained after harvesting, on the grounds that the remaining consumption would be provided by alternative prey, such as polychaete worms and other bivalves. We now know that this provision was inadequate by a factor of three- to tenfold and accounted for the decline in the numbers of these birds in The Netherlands that has occurred over the last 2 decades. The policy has now changed so that a much more of the shellfish stocks are reserved for the birds.
Recent work with the shorebird IBM has shown how shellfishing policy can be formulated in such a way as not only to improve the feeding conditions of the birds but also simultaneously to increase the yield to the shellfishery (Caldow et al. ).
At the Menai Strait in north Wales, the overwinter consumption of 242 tons of commercially harvestable mussels (>40 mm) by oystercatchers in 1999–2000 was worth £133,000 ($226,000), which represented 19% of the value of the landings. Traditionally, the shellfishery operates by dredging young mussels in their first year of life (the “seed mussels”) from the sublittoral and laying them onto commercial lays in the intertidal zone where they can be tended and subsequently harvested but where they are very vulnerable to frequent attack by oystercatchers. An IBM of this system showed that the losses to oystercatchers of the largest and commercially harvestable mussels could be considerably reduced by repeatedly changing the shore level at which the mussels were situated at different stages of their development from seed mussels to harvestable adults (Caldow et al. ).
Caldow et al.  proposed that the seed mussels should first be laid upshore where losses to oystercatchers would only be slight because most oystercatchers do most of their feeding at the middle and low shore levels where larger mussels with a high flesh-content occur. Even though the mussels are small, very few could be eaten at these higher shore levels by sublittoral predators, such as crabs Carcinus maenas, which can inflict enormous damage on young mussels in the sublittoral zone. As the seed-mussels grow over the next 2 to 3 years, they should be moved progressively further downshore where the growing conditions are better: downshore, the mussels are covered by water, and so able to feed for a greater proportion of the time. This would be the stage at which the mussels would be most at risk from oystercatchers because of their relatively high flesh content, and thus high food value to the birds, and frequent exposure to the birds over the low tide period. The mussels would spend their last season before being harvested very far down the shore where, by virtue of being covered by the tide for so much of the time, their growth rates and flesh content would be further increased. By that stage in their development, the mussels would be large enough to avoid most of the predation from crabs that, at earlier stages, would have taken so many of them. Although these large mussels would be very attractive to oystercatchers, the birds would consume very few of them because, at these very low shore levels, the mussels would only seldom be exposed over dead low water to predation by birds.
The results suggested that adopting this management practice would increase the return to the commercial fishery from the seed mussels by up to fourfold even though it would mean that more mussels would be lost to oystercatchers during the early stages of the cultivation process. By accepting greater losses of mussels earlier in the cultivation cycle, rather than later as at present, not only would the yield to the fishery be increased but also the feeding conditions for oystercatchers would be improved. Caldow et al.  concluded with the encouraging message that, with appropriate management, the interest of shellfish growers and competing shorebirds do not necessarily have to conflict but can both benefit simultaneously from a change in shellfishery management practice.
5.2 Disturbance from people and raptors Disturbance causes birds to spend energy flying away and to lose feeding time while relocating to different feeding areas, where the increased bird densities may intensify competition from interference and, if of sufficient duration, from increased rates of prey depletion. The shorebird IBM has been used to investigate whether disturbance from people walking on the mussel beds of the Exe estuary could or already does reduce the fitness of oystercatchers and to explore ways of regulating disturbance so that it has no effect on the birds (West et al. ). Here we show how the shorebird IBM can also be used to establish how frequently shorebirds can be put to flight before their fitness is affected and to establish simple policies for managing disturbance.
Goss-Custard et al.  showed how IBMs can be used to establish “critical thresholds” for the frequency with which shorebirds can be disturbed before their chances of dying of starvation are significantly increased. By way of an example, they used oystercatchers wintering in the Reserve Naturelle of the baie de Somme, France where oystercatchers start the winter eating cockles but switch to other prey, such as the polychaete worm Nereis diversicolor, if the cockles disappear later in the winter, mainly through storms.
Field studies in the baie de Somme showed that oystercatchers were disturbed into flight up to 0.513–1.73 times per daylight hour by people and over-flying raptors, depending on the winter. As Figure 13 indicates, simulations with the IBM showed that the highest disturbance frequency would only have increased mortality by a small amount in the winter of 1995–1996, a mild winter in which cockles remained abundant throughout. Most birds could continue eating cockles throughout that winter and did not have to switch to the less profitable ragworms. In contrast, in the similarly mild winter of 1997–1998 during which most cockles disappeared, mortality started to increase at 0.5–0.6 disturbances/daylight hour, frequencies of disturbance that were common on the baie de Somme (Figure 13). The critical disturbance threshold was only 0.2–0.3 disturbances/daylight-hour in 1996–1997, the third winter modeled, when, in addition to cockles becoming depleted, a prolonged period of cold weather occurred in January and many birds arrived from The Netherlands where many of the intertidal flats were ice covered (Hulscher et al. ).
The critical disturbance thresholds were similar whether the disturbances were caused by people staying for 20 minutes or by rapidly over-flying raptors (Goss-Custard et al. ). Therefore, a quite simple policy rule can be devised for the management of disturbance in the Reserve. A frequency of flying caused by people and/or raptors in autumn and early winter of <1.5/daylight hour can be allowed. However, if the cockles become considerably depleted by the end of December, especially if they are small, the frequency of disturbance—from both people and raptors—should be kept below 0.5/hour. This could mean that in winters with many raptors, people should not be allowed to disturb the birds at all. Similarly, no disturbance from people should be allowed during prolonged cold spells.
This policy guideline is practicable for the baie de Somme. It is very easy to detect from a very simple sampling program when cockles become scarce. It is also straightforward to estimate the frequency with which birds are put to flight by people or by raptors. Although it is of course impossible to control the number of disturbances due to raptors, it is possible to see when the disturbance from people causes the total number of disturbances from raptors and people to exceed the critical threshold, whereupon access to the feeding areas by people can then be stopped. Similarly, it is easy to prevent access by people during cold spells, just as hunting and catching birds is also prevented during periods of severe winter weather.
This example of the baie de Somme has shown that IBMs can be used to establish practicable critical disturbance thresholds, as well as to predict the effect of existing frequencies of disturbance on a particular estuary as shown by West et al. . The level of the critical threshold will depend, of course, on estuary-specific features, and the same values cannot be regarded as being applicable everywhere. The precise value of the critical thresholds will depend on the bird species and the many factors known to affect the ability of shorebirds to survive the winter in good condition. But by using the oystercatchers of the baie de Somme as the test system, Goss-Custard et al.  have demonstrated that IBMs provide a means by which critical disturbance thresholds can, for the first time, be obtained.
5.3 Spread of the grass Spartina onto upshore mudflats In many parts of the world, the spread of cord grass Spartina spp. onto mudflats is believed to threaten the shorebirds that feed on the high-level mudflats that expose first on the ebbing tide and are covered last as the tide flows back. By covering the mudflat with vegetation among which many birds will not or cannot forage, spreading Spartina not only removes an area of food supply but may also reduce the amount of foraging time available to shorebirds by preventing them from feeding at the start and end of the exposure period. There is some evidence to suggest that the numbers of wintering dunlin Calidris alpina decreased in those British estuaries where Spartina had spread extensively but remained stable in those where the grass had not spread (Goss-Custard and Moser ). The causality of this correlation has, of course, been questioned because it is uncertain whether it was the spread of the Spartina itself or the changes in the estuary environment that encouraged the spread of the grass that was responsible for the decrease in dunlin numbers (Goss-Custard and Moser ). Further tests are needed to investigate the impact of Spartina, if any, on shorebirds that feed on high level mudflats.
A recent IBM of dunlin feeding in the baie de Somme, France, has provided modeling evidence that Spartina can indeed reduce shorebird fitness (S.E.A. le V. dit Durell, unpublished information). Dunlin feed on the upper mudflats over which Spartina has been spreading in recent years at an annual rate of 20–40 m. The shorebird IBM showed that, at such a rate of spread, the fitness of dunlin begins to be reduced after only a few years and that their mortality rate could double after only 10 years of Spartina spread (Figure 14). This happens even though the food supply and duration of the exposure period elsewhere remains the same, suggesting that Spartina should be removed if a reduction in dunlin fitness is to be avoided.
5.4 Habitat loss, sea-level rise, and mitigation The intertidal feeding areas of shorebirds are reduced when they are covered by various developments, raising concern that this reduces the fitness of shorebirds by reducing the extent of their food supplies. When this happens in some parts of the world, the law requires that effective mitigation be provided to remove the potential threat from the development to the birds. Until recently, however, there was no means by which the effectiveness of any proposed mitigating measure could be tested. Indeed, there was no means to establish whether the birds would be affected by the habitat loss in the first place and so whether the mitigation—which may be very expensive—was even required.
An IBM of dunlin wintering on the Seine estuary illustrates how the shorebird IBM can both explore whether habitat loss will reduce shorebird fitness and, if so, whether a proposed mitigation would be effective at returning fitness to its previous level (Durell et al. [2004; 2005]). The proposal was to extend the port at Le Havre at the mouth of the Seine onto 105 ha of intertidal mudflats and to provide by way of mitigation an area of mudflat further upstream by removing vegetation from an area of mudflat, which would allow the birds to feed there (Figure 15). This would provide an area of upshore mudflat of comparable quality to that which was lost under the port development. The IBM of this system showed that the initial loss of the mudflat under the port development would indeed be likely to increase the winter mortality rate of the dunlin. It also shows that an equivalent area of mudflat would have to be freed from vegetation for it to be returned to its previous level (Figure 16). This can easily be understood. Because the loss of habitat was predicted to increase the mortality rate of the birds, mortality must be density dependent. Therefore, to return the mortality rate to its preport level, an equivalent area of habitat of a comparable quality to that which is lost must be provided. The policy advice emerging from this study, therefore, was that it was necessary to clear about 100 ha of vegetation if the proposed mitigation was to be fully effective.
Stillman et al.  used a multispecies version of the shorebird IBM to predict how a loss of intertidal habitat resulting from sea-level rise in the Humber estuary, UK, would affect the overwinter survival of nine shorebird species; dunlin, common ringed plover Charadrius hiaticula, red knot Calidris canutus, common redshank Tringa totanus, gray plover Pluvialis squatarola, black-tailed godwit Limosa limosa, bar-tailed godwit L. lapponica, Eurasian oystercatcher, and Eurasian curlew Numenius arquata. This model extended the previous shorebird IBM by predicting survival rates in a community of interacting species, and not just in a single species. Stillman et al.  found that shorebird survival was most strongly influenced by the biomass densities of annelid worms and of the bivalve molluscs the cockle and clam Macoma balthica. A 2–8% reduction in intertidal area (the magnitude expected through sea-level rise and industrial developments) decreased predicted survival rates of all species except dunlin, common ringed plover, red knot, and Eurasian oystercatcher.
5.5 Measuring and monitoring habitat quality for shorebirds Conservation managers responsible for estuaries are often required to monitor their site to ensure that the conservation status of any bird species for which the site is considered important is not affected by deterioration of their habitat or by disturbance of the birds themselves. For example, in Europe, the EU Habitats Directive 92/43/EEC, Article 6 states: “Member States shall take appropriate steps to avoid, in the special areas of conservation, the deterioration of natural habitats and the habitats of species as well as disturbance of the species for which the areas have been designated, in so far as such disturbance could be significant eg … .”
The conservation importance of an estuary is usually measured in terms of the numbers of birds using it, but monitoring bird numbers on an estuary is not in itself a reliable way of assessing whether the quality of a site is beginning to fall. There may be a lag between a decline in habitat quality and a detectable fall in bird numbers, particularly in long-lived species because established birds may be reluctant to change site. More importantly, the numbers of a bird species on a particular site depend not only on the conditions in that site but also on conditions elsewhere in the species' nonbreeding range and on the breeding grounds (Goss-Custard ). A decline in bird numbers at one site might be due to a decline in the quality of that site but it might also be caused by an increase in the quality of sites elsewhere in the nonbreeding range. A decline in the reproductive rate or an increase in mortality on the breeding grounds would reduce the size of the greater population to which the birds belong and hence could, in turn, reduce the number of birds using the site, even though its quality had not changed.
As bird numbers are not a reliable way to assess the quality of a site, another method is required. In migratory shorebirds population size is a function of the interaction between (i) the mortality and reproductive rates in the breeding ranges and (ii) the mortality rate in the nonbreeding range, including migratory routes (Goss-Custard [1980, 1981, 1993], Goss-Custard and Durell , Goss-Custard et al. [1995d, 1996b]). Because the objective of shorebird conservation in the nonbreeding range is to maintain the conservation status of birds, the best measure of habitat quality for an estuary is one that, directly or indirectly, determines these demographic rates. For migratory shorebirds during the nonbreeding season, this means that habitat quality should be measured in terms of its effect on the number of birds that die of starvation during the nonbreeding season, perhaps especially during severe winter weather. If it can be shown that feeding conditions in the site are sufficient to maintain present-day rates of survival at the current bird population size and in the current climatic conditions, then the quality of the estuary is being maintained. If the population using the estuary declines despite this, then the cause of that decline needs to be sought elsewhere.
West et al. demonstrated how a multispecies shorebird IBM could be used to assess the quality of the Wash estuary, UK, for eight overwintering shorebird species. The results identified the prey species that were most important for maintaining the present-day survival rates of the birds. Birds began to starve when the estuary-wide food biomass density in autumn fell below about 5 g AFDM m−2. Survival rates fell below 90% by the time prey biomass declined to 4 g AFDM m−2 (Figure 17). West et al. therefore suggested that the quality of the Wash estuary for shorebirds could be monitored by regular surveys of the food supply designed to establish whether the food supply had fallen below the 99% confidence limit of the biomass density required to maintain the present-day rates of survival (Figure 17). They also concluded that, as the survival of all species in the model remained high until there were as many as 20 disturbances per hour, the present low frequency of disturbance on the Wash does not threaten their survival. Importantly for estuary management, West et al. concluded that, for the Wash, prey density was a more important factor in shorebird survival than habitat area, and therefore habitat loss, and identified those shorebird species that were most vulnerable to changes in site quality.
6. Should we have confidence in the predictions of the shorebird IBM?
Managers of the coast need to know whether the predictions of the model provide a reliable source of advice as to the policies that they should adopt. Much time and effort has therefore been spent on testing the predictions of the shorebird IBM partly for scientific reasons, of course, but also to be able to advise policy makers on the confidence with which they can treat its predictions.
Two questions can be asked about whether the model predicts real events reasonably well. One asks whether the model captures with precision the behavior of real birds in the system being modeled. Because the predictions for fitness are derived from the behavior of the birds in the model, and because decision making by fitness-maximizing individuals is the fundamental feature of the model, it is vital that the model adequately represents the behavior of real birds. The other question is whether the model accurately predicts the fitness measures that, for policy makers and scientists alike, is really the “bottom-line” for evaluating the model's success or failure.
On the Burry Inlet, oystercatchers over the low tide period feed either on the cockle bed or the main long-established mussel beds with mature mussels or on what is locally called “mussel crumble” that consists of young mussels attached to cockles squeezed out of the sand by their own high density. West et al.  showed that the shorebird IBM predicted reasonably well the distribution of the birds across these three feeding areas (Figure 18). In the absence of interference, all the birds would have fed on the mature mussel beds because these provide the highest interference-free intake rates. As the mature mussel beds occupied a rather small part of the whole Inlet, the model must have represented reasonably well the interference among birds that prevented all but a small proportion of them from feeding on the mussel beds. The model could very easily have made a very poor prediction, but in fact it did not do so.
On the Exe estuary, surveys of the mussel stocks in autumn as the oystercatchers arrived from the breeding grounds and in spring at the time of their return showed that, on average over 8 years, 12.1% of the mussels in the size range consumed by oystercatchers disappeared over the winter, and exclosure experiments confirmed that the vast majority of the mussels that disappeared had been taken by oystercatchers (Goss-Custard et al. ). The model predicted that just under 12% would be taken by oystercatchers, which is reasonably close to the observed value (Figure 19). Again, there is plenty of opportunity for the model not to have predicted with such accuracy. Apart from the possibility that the daily energy requirements of oystercatchers may not be accurately represented in the model, both the model birds and real birds eat many prey other than mussels, including intertidal prey when the mussel beds are under water, such as the clam Scrobicularia plana, and earthworms Lumbricus spp. in the fields at both high and low water. That the model predicted with reasonable accuracy the mortality inflicted on the mussels by oystercatchers suggests that it represents the important processes of mussel consumption with reasonable precision.
It is widely recognized that when shorebirds are hard-pressed to obtain their food requirement they extend the amount of time they spend feeding over the period during which the intertidal flats are exposed and therefore accessible to them (Goss-Custard et al. ); indeed, often birds continue feeding over high tide on terrestrial habitats, though this is not considered here. The ability of the model to predict the amount of time spent feeding on the intertidal flats over a tidal cycle by the average bird is therefore a particularly important test of the model because it represents a measure of its overall success at representing the ease or difficulty with which the birds are able to meet their energy demands. A number of comparisons have now been made between the predicted and observed amounts of time spent feeding over a single tidal cycle in daylight by a range of shorebird species of very different body size, an important correlate of feeding time (Goss-Custard et al. ). Generally, the observed values match the predicted values quite closely (Figure 20). This again suggests that the model performs reasonably well at predicting the observed behavior of the birds.
6.2 Mortality and body condition The most critical tests, of course, are of the model's ability to predict the observed mortality rates and body condition of shorebirds, these being the two “proxy” measures of fitness that should be used to assess the impact of a coastal management policy on the well-being of shorebirds (Goss-Custard et al. ). So far, no test has been possible for body condition, but several are now available for mortality predictions.
Four of the tests concern oystercatchers eating mussels (Exe estuary) or a mixture of cockles and mussels (Wash; Burry Inlet). In the case of the Burry Inlet, the winter mortality rate of oystercatchers was not measured but the behavior of the birds implies that it was very low. The birds spent a large amount of time roosting when shellfish were available on the Inlet and earthworms were available in nearby fields, implying that the feeding conditions were very good and therefore that few birds would have died of starvation. In the remaining cases, the winter mortality of the birds had been measured in the field.
For comparing observed and predicted mortality rates for the Exe estuary, the data were grouped into years of low bird numbers and years of high bird numbers because the winter mortality rate is density dependent. For the Wash, the data were grouped into years of abundant and scarce shellfish because this has a huge effect on the winter survival of oystercatchers (Atkinson et al. ). As Figure 21 shows, the model predicts the average mortality rates observed over these four conditions with good precision. It should be noted, however, that the model predicts the mortality rates observed in individual years less well, probably because in any one year a parameter that varies annually (such as the body condition and therefore energy content of the mussels) was not known.
Recently, the predicted and observed mortality rates have been compared in a species other than the oystercatcher (Goss-Custard et al. [2006b]). This is a particularly important test because the model was used retrospectively to predict an increase in the mortality rate of redshank following a loss of habitat. This loss occurred after a large area of habitat had been removed by the construction of a barrage across the mouth of the estuary that turned an area of intertidal mudflats into a freshwater lake, unusable by shorebirds. The data on the food supply were rather sparse, and no new fieldwork was carried out so it was necessary to calibrate the model by varying one parameter (feeding efficiency at night) until the model reproduced the observed, prebarrage mortality rate. Despite this, the model predicted the observed postbarrage mortality rate after habitat loss with encouragingly good precision (Figure 21).
A number of lessons could be drawn from this study: (i) The IBM can predict observed effects of habitat loss on wintering shorebirds with good precision; (ii) the model can make very good predictions for mortality in wholly new environmental circumstances for which it is very difficult or currently impossible to measure mortality rates directly; (iii) the model can give good predictions without a huge and expensive fieldwork program being carried out first—often a high proportion of the necessary data are already available; (iv) the calibration procedure adopted does not prevent the model from producing reasonable predictions so that the (usual) absence of some important parameter values is not fatal to the approach; and (v) so long as the necessary data are available, the model may not take many months to set up.
7. Recent additions to the model
The latest versions of the model include a number of features that make it now possible to apply to a wider range of species, systems and scenarios. These are:
(i) Decision rules: The birds in the model can now be allowed to make their decisions on the basis of satisficing rather than on the basis of maximizing their rate of intake. That is, instead of always choosing the option that maximizes their instantaneous intake rate, the birds can choose an option that will simply enable them to meet their demands, even though they could be met more rapidly by taking another option.
(ii) Predators: The birds in the model can now decide to avoid an option that carries a risk of being eaten by a predator that is greater than the current risk of starving. This allows model birds to choose, for example, a feeding area that is free from predators but provides a lower intake rate than they could achieve in another, less safe area.
(iv) Multisite: Previously, a separate model was constructed to enable birds to move among different sites (such as estuaries) within their wintering range (Pettifor et al. ). This facility has now been incorporated in the latest version of the shorebird IBM so that simulations can be either run for a single site or for a group of sites, depending on the choices made by the modeler.
A formal description of this latest version of the model, which is called “MORPH,” is given in Appendix 1.
8.1 Underlying principles There were two reasons for developing the shorebird IBM. First, many responses by birds to environmental change are behavioral; for example, they change feeding location and the species and/or size of prey taken (Goss-Custard and Durell ). Second, the models must often predict how bird fitness would be affected by, as yet, untried management scenarios and policy options, which therefore lie outside the present-day empirical range; that is, they are without precedent. To have confidence in its predictions, the models had to operate on basic principles, underpinned by theory, that will still apply in the new scenarios rather than on present-day empirical relationships, which may no longer hold in the scenarios for which predictions are required (see below).
The principles on which the approach is based can therefore be summarized as follows:
(i) Well-established behavioral decision rules—so that animals in the model are likely to respond to environmental change as real ones would.
(ii) Individuals in the model differ in their competitive abilities otherwise it would not be possible to predict the mortality rate (Goss-Custard et al. ).
(iii) Natural history details are included—because these represent the adaptations that are critical to survival.
(iv) Calibration may be necessary—because we do not know every parameter value precisely.
(v) Validation of predictions—otherwise the predictions should not be believed.
8.2 Present and future applications Taken together, the tests of the model's ability to predict the behavior of the birds and the choices they make and the consequences for their mortality rate are beginning to raise confidence in its ability to be used as a guide to policy.
The shorebird IBM was, of course, developed specifically to evaluate policy options in terms of their effect on survival and body condition. They are providing predictions such as (i) if an intertidal area were to be reclaimed (e.g., to extend a dock or to remove salt pans), the extra numbers of birds dying of starvation over the winter, or failing to reach their target premigratory body mass in spring, would be n1; (ii) if shellfishing by hand were to be introduced at such-and-such an intensity, the combination of disturbance caused by the fishermen and the reduction in the abundance of the birds' shellfish food would increase the numbers of birds starving over the winter, or failing to fatten up, by n2; (iii) if aquaculture (e.g., oyster racks) were to be introduced, the loss of bird feeding areas under the racks would increase the numbers dying or failing to fatten by n3; (iv) if the numbers of people recreating on the mudflats were to double, the extra numbers of birds dying or failing to fatten would be n4; (v) if water abstraction was predicted to reduce by 25% the average abundance of the invertebrates or intertidal plants on which the birds feed, n5 more birds would die, or fail to fatten up, than at present; (vi) if estuary morphology were to change in such-and-such a way and so change vegetation and invertebrate abundance, n6 more or less birds would starve or fail to fatten up; and (vii) if a mitigating measure were to be built (e.g., by breaching the seawall and allowing fields to revert to mudflats), the number of birds starving or failing to fatten would be reduced by n7.
The quantity n is a measure of the “ornithological cost” of pursuing a particular activity—if any, for n could often be zero. So if, for example, 500 of a population of 10,000 birds starve at present, the model might predict that, under scenario (i), the number dying would increase to 750. It thus gives the “bird cost” of doing something—or even of not doing something and leaving the management regime or policy as it is. In the same way, the models can predict the “bird-gain” from putting in place a mitigating measure, so long as the food biomass density provided by the proposed measure can be predicted with reasonable confidence. The model might predict, for example, that the mitigating measure would save 250 birds that would otherwise have starved. And because they can predict density-dependent functions, behavior-based models can also predict the effect of a particular scenario on the long-term equilibrium size of the population (Goss-Custard et al. [1995b], Stillman et al. ).
This paper has illustrated the wide range of environmental issues on the coast to which the model has been applied. In fact, the model is being used in an increasing number of systems to this purpose, as reference to the CEH Web site will testify: http://www.ceh.ac.uk/birds/
Appendix: Morph—an individual-based model to predict the effect of environmental change on animal populations
A1. Introduction This appendix describes the latest version of our IBM, MORPH. The new model is based on the same fundamental principle as our previous models—that individuals behave in ways that maximize their own chances of survival and reproduction. It also tracks the location, behavior and ultimate fate of each individual in the population and incorporates variation in the foraging abilities of different individuals. However, the new model is much more flexible than the original models (hence the name MORPH) and can be used to address a much wider range of environmental issues. For example, our previous models contained wader- or goose-specific assumptions, and assumptions applicable to certain coastal areas but not others (e.g., the pattern of the tidal cycle). The new model removes these limitations and contains virtually no species-or system-specific assumptions. However, once parameterized, it can still be applied to specific species or systems.
We have described the model using a standard protocol, designed by Grimm and Railsback  (Individual-based modeling and ecology, Princeton University Press, Princeton). The general protocol ensures that IBMs are fully described in a way that is clearly understandable and would enable the models to be recreated by others. Clear communication of IBMs has often been a problem in the past, which has limited the widespread use of some models (Grimm and Railsback ). The protocol has been used to avoid this problem. Model description is divided into the following sections: Purpose, Structure, Processes, Concepts, Initialization, Input, and Submodels. A general overview of the main elements of the model is initially presented (Purpose, Structure, Processes, and Concepts), followed by a more detailed mathematical description (Initialization, Input, and Submodels). A number of technical modeling terms are used in this description, as these are required to describe the model unambiguously. However, each of these terms is defined when first used.
A2. Purpose The overall purpose of the model is to predict how environmental change (e.g., habitat loss, changes in human disturbance, climate change, mitigation measures in compensation for developments, and changes in population size itself) affects the survival rate and body condition in animal populations. The model does this by predicting how individual animals respond to environmental change by altering their feeding location, consuming different food or adjusting the amount of time spent feeding. The central assumption of the model is that animals behave in ways that maximize their chances of survival. The model itself does not predict reproductive rate but its survival and body condition predictions can be input into other models that do make this prediction.
The model has been designed to be very flexible so that it can produce both general predictions (when parameterized in a very simple way) and predictions for specific systems (when parameterized using detailed system-specific data). The model can read in equations as parameters and so can potentially represent very wide ranges of species or systems.
A3. Structure The model itself contains only very general aspects of behavior and ecology, applicable to a wide range of systems. The basic assumptions of the model are as follows:
(i) Time progresses in discrete, fixed duration, time steps.
(ii) Space is divided into a number of uniform habitat patches, with fixed location and area. The location of patches is either represented by coordinates on a two dimensional grid or by latitude and longitude on the earth's surface.
(iii) Habitat patches contain a number of resources that can be consumed by foragers.
(iv) Resources contain components that are assimilated into foragers when resources are consumed.
(v) Foragers remain at the same location during a time step, either on a patch or traveling between patches, but move between time steps.
(vi) Foragers alter their location and the food they consume in order to maximize their chances of survival.
(vii) The model defines the following five entities (objects within the model).
(viii) Global environment—State variables (values used to describe the global environment) that apply throughout the modeled system.
(ix) Patches—Locations with local, patch variables (values used to describe patches) containing resources and foragers. Foragers may have travel costs when moving between patches.
(x) Resources—The food consumed by foragers. Foragers can simultaneously consume one or more resources from a patch. Such collections of resources are termed diets.
(xi) Components—Elements within resources that foragers assimilate into their bodies.
(xii) Foragers—Animals that move within the system attempting to maximize their survival and body condition. One or more forager types/species may be present within the modeled system.
Table A1 lists the state variables (values used to describe a model entity) of each entity. The global state variables are the major driving variables in the model system. Patch variables may depend on these global variables. Patches contain one or more resources, which in turn contain one or more components. Foragers have a range of possible diets, which are simply a collection of resources. Foragers consume diets, from which they assimilate components. Components can either have positive, neutral, or negative effects on foragers. Foragers are not forced to consume diet but instead may occupy a patch and not feed.
Table A1. State variables used to describe model entities.
State variable description
Zero or more environmental variables that apply throughout the modeled system
Surface area/volume of patch
Zero of more patch-specific environmental variables
Density on patch
Density of each resource on each patch
Density in resource
Density of each component within each resource on each patch
Forager type/species to which forager belongs
Zero or more forager-specific constants that remain constant throughout a simulations
Zero or more forager-specific variables that can change throughout a simulation
Coordinates of foragers location
One if forager is moving between patches, zero if forager is occupying a patch
Patch number being occupied by forager during current time step (zero if forager is moving between patches)
Diet number being consumed by forager during current time step (zero if no diet is being consumed)
Proportion of time feeding
Proportion of time feeding during current time step (zero if forager is moving between patches)
Diet consumption rate
Rate at which diet is being consumed during current time step and averaged over previous and predicted for future time steps
Component consumption rate
Rate at which a component is being consumed during current time step and averaged over previous and predicted for future time steps
Component assimilation rate
Rate at which a component is assimilated into the body during current time step and averaged over previous and predicted for future time steps
Component metabolic rate
Rate at which a component is metabolized/excreted from the body during current time step and averaged over previous and predicted for future time steps
Component reserve size
Amount of a component within the body's reserves during current time step
A4. Processes The model defines the following processes (the transfer of information or model entities between [other] model entities):
(i) Change in resource density. Changes in the density of a resource on a patch caused by consumption by the foragers and/or other factors.
(ii) Change in component density. Changes in the density of a component in a resource.
(iii) Forager immigration. The movement of foragers into the system.
(iv) Forager decision making. The optimal patch and diet selection of foragers and decisions to emigrate from the system.
(v) Forager emigration. The movement of foragers away from the system.
(vi) Forager movement between patches. Movement of foragers between patches. Movement may have associated costs and may take more than one time step.
(vii) Forager diet consumption. The transfer of resource components into foragers when diets are consumed.
(viii) Forager physiology. Change in the size of a forager's component reserve due to the balance of consumption and metabolism.
(ix) Forager mortality. Death of foragers.
A5 Concepts The following concepts (basic characteristics common to all IBMs) are represented in the model.
A5.1 Emergence The following phenomena emerge from the interaction between individual forager traits and global and patch variables, resource and component densities, and forager constants and variables:
(i) Resource depletion. The amount of each resource consumed by foragers from each patch during each time step.
(ii) Forager distribution and diet selection. The location of each forager and its diet during each time step.
(iii) Proportion of time foragers spend feeding. Proportion of each time step each forager spends feeding.
(iv) Forager component reserve size. The amount of each component within each forager's reserves during each time step.
(v) Forager mortality and emigration. The number of foragers remaining in the system after a given number of time steps.
A5.2 Adaptation Foragers adaptive traits (behavior through which foragers maximize their fitness[i.e., survival and reproduction]) are their location and diet selection. During each time step, foragers select the patch/diet combination that maximizes their perceived fitness, or emigrate from the entire system if this has a higher perceived fitness than any patch/diet combination.
A5.3 Fitness A number of fitness components are assumed to affect the survival of animals and hence their overall fitness. Fitness components may have negative or positive affects on survival. Each fitness component has associated submodels (see below) to calculate the true probability of surviving the fitness component during a time step. The combined true survival probability for all fitness components is the product of the survival probabilities associated with each fitness component (see below). Each fitness component has a fitness measure, calculated using a submodel (see below), which animals use to assess the fitness consequences of different decisions. The combined fitness measure is the product of the fitness measures associated with each fitness component (see below). The forager selects the patch and diet combination (including no diet) that maximizes its combined fitness measure, or emigrates from the system if this has a greater fitness measure than any of the possible patch and diet combinations. Once the forager has selected a patch and diet, the consequences of this decision are determined by true probability of survival. Both true survival probability and fitness measure submodels can depend on any combination of global, patch, resource, component, or forager state variables.
A5.4 Prediction Foragers remember their foraging success during a given number of previous time steps. This memory is used to calculate average state variables over previous time steps (see Table A1 for a list of these variables). Foragers can also predict their future foraging success, over a given number of time steps, taking into account the time taken to move from their current location to a target patch. In making these predictions, the model assumes that foragers do not know the future values of any state variables, resource or component densities, or the location of other animals. Instead, state variables, resource and component densities, and the location of other foragers are all assumed to remain the same as in the current time step.
A5.5 Interaction Foragers interact within patches through the consumption of a shared resource (depletion competition). The number and/or density of other foragers within a patch can also affect any of a forager's state variables, and fitness measures and true survival probabilities. These effects can be either positive or negative, depending on the submodels used. Increased competitor numbers or density can either increase consumption rate (facilitation) or decrease consumption rate (interference competition), again depending on the submodels used. Foragers can only interact within patches. The actual mechanisms of interactions within patches are not incorporated explicitly.
A5.6 Sensing The amount of knowledge foragers have can be varied. This can range from perfect knowledge of the complete system during the current time step, through complete knowledge of local patches, to no knowledge at all. Similarly, the amount of knowledge a forager has of its own state, both during the current time step and previous and future time steps, can be varied. Foragers base their decisions on the fitness measures associated with different patches and diets (or no diet). The fitness measure may or may not be related to the true probability. Foragers will tend to avoid patches and diets with low fitness measures. Depending on the relationship between the true survival probabilities and fitness measures, this can mean that foragers avoid safe patches and diets (i.e., high true survival probability) because these are perceived as dangerous (i.e., low fitness measure) or select dangerous patches or diets because these are perceived as safe. The model does not explicitly represent any sensing mechanisms.
A5.7 Stochasticity The amount of stochasticity (random variation in model predictions) can be varied. Any state variables, except for patch size and location, and forager type/species can be stochastic. The probability of a forager (or the individuals within the forager (see below)) dying during a time step is a stochastic event unless the probability is zero or one.
A5.8 Collectives Collectives (groups or aggregations of foragers) are included in the model. These are represented by the number and/or density of foragers on each patch and arise from the patch and diet selection of foragers. Collectives are not represented as social groups, instead each individual behaves independently albeit with its behavior influenced the number and/or density of competitors on different patches. Super-individuals can be incorporated, with each forager (super-individual) representing more than one individual. The number of individuals within a forager is set at the start of a simulation but can decrease through time as some individuals within the forager die. In contrast, all individuals within a forager simultaneous immigrate to or emigrate from the system.
A5.9 Scheduling Time is represented using discrete time steps that are of constant duration. Figure A1 shows the sequence of events during each time step. Global events are processed first, followed by patch events and then forager events. Finally, results are displayed and saved. The order in which foragers are processed can either be random or based on the value of a specified forager constant. Once the order of foragers has been determined, foragers are updated one at a time during each time step (asynchronous scheduling). This means that all forager events (immigration, patch and diet selection, movement and emigration, diet consumption, resource depletion, and forager mortality) are applied to one forager before the next forager is processed.
A5.10 Observation The results used to test the model depend on the particular system for which it is parameterized. All state variables can be displayed and saved during each time step.
A6 Initialization The initial values of state variables are either read from a parameter file, created using random numbers, or calculated from state variables defined earlier in the parameter file. The sequence of random numbers is itself randomized at the start of each simulation so that replicate simulations using the same set of parameters will produce slightly different predictions. All global and patch variables are initialized at the start of the simulation. Forager state variables are initialized once the forager has immigrated into the system, and so foragers immigrating at different times may have different initial state variables.
A7 Input The particular data used to parameterize the model will depend on the particular system to which they are applied. However, Table A2 lists the basic set of parameters, which would be required for any system. Parameters can either be single values, values for each time step read in from a file, or an equation (submodel) to calculate values during each time step.
Table A2. Basic set of parameter values/submodels required by the model.
Number and names of global variables
Value/submodel for each global variable
Number and names of patches
Size of each patch
Location of each patch
Value/submodel for each patch variable on each patch
Number and names of resources
Initial density of each resource on each patch
Submodel for change in density (excluding consumption by foragers) of each resource on each patch
Number and names of components
Value/submodel for density of each component in each resource on each patch
Number and names of diets
Number and names of resources in each diet
Number and names of forager types/species, and type/species of each forager
Number and names of forager constants
Value of each forager constant for each forager
Number and names of forager variables
Value/submodel for each forager variable
Value/submodel for time to move between patches
Number and names of diets consumed by forager type/species
Rule to determine whether patches can be located
Rule to determine whether fitness measure can be assessed on a patch
Value/submodel for diet consumption rate
Value/submodel for maximum diet consumption rate
Value/submodel for assimilation efficiency of each component in each diet
Value/submodel for rate of metabolizing each component while feeding, resting and moving between patches
Value/submodel for target reserve size for each component
Number of fitness components
Value/submodel for fitness measure for each fitness component
Value/submodel for true survival probability for each fitness component
Value/submodel for expected fitness measure on patches on which fitness measure cannot be assessed
Value/submodel for expected fitness measure of emigrating
A8 Submodels Many of the model's submodels will be read in as equations from the parameter files. In these cases the particular submodels will depend on the specific system to which the model is being applied. These are termed parameter submodels in the following sections. However, a number of submodels are incorporated into the model itself. The following sections describe the submodels used to represent each of the models processes.
A8.1 Change in resource density Resource densities change on patches due to (i) nondepletion change and (ii) depletion when foragers consume diets on a patch.
Nondepletion change is calculated at the start of each time step, except the first time step, using a parameter submodel that determines how resource density is updated at the start of each time step:
where R= new resource density at start of current time step, Rprevious= old resource density at end of previous time step, f(p1, p2, … , pn)= a submodel containing n parameters. The submodel's parameters may be any number of global or patch state variables.
After the resource density has been calculated at the start of each time step, the density of each diet is updated. Diets are simply a collection of resources, and so the density of a diet is simply the sum of all of the resources it contains:
where Rdiet= diet resource density, r= resource number, N= number of resources in diet, and Rr= density of resource r.
Depletion is incorporated by reducing the amount of a resource in a patch by the amount consumed by foragers. Foragers consume diet rather than separate resources, and so the model needs to calculate the amount of each resource consumed in the diet. The model assumes that resources are consumed in proportion to their relative density within a diet. The amount of a resource consumed within a diet is therefore given by:
where E= amount of resource consumed (eaten), Ediet= amount of diet consumed, R= density of resource and Rdiet= density of diet. The density of the resource on the patch is updated by assuming that depletion occurs uniformly throughout the patch:
where R= new density of resource, Rprevious= previous density of resource before depletion, A= size (area/volume) of patch, and E= amount of resource eaten.
Depletion occurs continually during a time step (as foragers are processed asynchronously). Diet densities are updated every time depletion occurs.
A8.2 Change in component density Component density within each resource on each patch is either read in as a single value that applies throughout the simulation or read in as a parameter submodel to calculate values during each time step. Component density submodels can depend on global or patch state variables.
The density of a component within a diet is a weighted mean of the component density within each of the resources contained in the diet:
where Cdiet= density of component in diet, r= resource number, N= number of resources in diet, Cr= density of component in resource r, and Rr= density of resource r.
A8.3 Forager immigration The probability of immigrating to the system for each forager type/species is read in as a single value that applies throughout the simulation or read in as a parameter submodel to calculate values during each time step.
A8.4 Forager decision making Foragers in the model make three types of decisions:
(i) Patch choice,
(ii) Diet choice, and
(iii) Emigration from the system.
The model uses the same submodel to determine how foragers make these decisions. The model's basic assumption is that foragers behave in order to maximize their fitness, which in turn is assumed to be measured as the probability of survival. Reproductive components of fitness are not considered directly as these are outside of the scope of the model. Model foragers test the fitness consequences of moving to different patches, consuming different diets, consuming no diet, or emigrating from the system. The list of possible diets depends on the diets consumed by the forager type/species to which the forager belongs. Foragers select the combination that maximizes their combined fitness measure.
Foragers do not necessarily have perfect knowledge of their survival probability when moving to different patches or consuming different diets. This uncertainty operates at two levels:
(i) Ability to assess fitness measures and
(ii) Accuracy of fitness measures (i.e., their relation to the true survival probability).
A8.4.1 Ability to assess fitness measures. Figure A2 shows how the ability to assess the fitness measures associated with different decisions is incorporated. Foragers are assumed to be able to assess fitness measures associated with consuming different diets on their current patch. Other patches fall into one of three different categories. (1) Foragers may know the location of a different patch and be able to assess fitness measures on the patch. They can assess the survival consequences of moving to this patch consuming any diet, and know the values of all of the patch's state variables during the current time step. (2) Foragers may know the location of a patch but not be able to assess the fitness measures associated with different diets. They cannot assess the survival consequences of consuming different diets and are unaware of any of the patches state variables. However, they do have an expected fitness measure on this patch (Fexpected), which is used to compare this patch with others. (3) Patches may be of unknown location and so cannot be considered as potential locations to move Emigration from the system also has an expected fitness measure, which is used to determine whether emigration is the decision that maximizes survival (Femigrate).
A8.4.2 Accuracy of fitness measures. For patches on which survival consequences can be assessed, the following process is used to calculate fitness measures, which may be unrelated to the true survival probability. Survival is assumed to be affected by a number of fitness components. Each fitness component has an associated submodel to predict its fitness measure given the foragers state (including average state over previous time steps) and any combination of global or patch state variables. The combined fitness measure is found from the product of the fitness measures associated with each fitness component:
where Fassessed= combined fitness measure for all fitness component, c= fitness component number, N= number of fitness components, fc= fitness measure for fitness component c. Fassessed is calculated for all possible combinations of patches and diets, including the option to occupy a patch but not feed. The forager selects the patch and diet combination that maximizes either Fassessed or Fexpected (depending on whether fitness can be assessed on the patch), or emigrates if Femigrate exceeds any of these values. In the event that more than one decision maximizes survival, the forager takes a random option, but weighed by patch area, or remains in the system if Femigrate equals the probability associated with remaining in the system.
Once the forager has selected a patch and diet, it is allowed to move and consume its selected diet (see below). The consequences of this decision are then determined by true probability of survival:
where Strue= true probability of surviving all fitness measures, c= fitness component number, N= number of fitness components, sc= true probability of surviving fitness component c. The assessed fitness measure may or may not be related to the true probability. Foragers will tend to avoid patches and diets with low fitness measures. Depending on the relationship between these and the true survival probabilities, this can mean that foragers avoid safe patches and diets (i.e., high true survival probability) because these are assessed as dangerous (i.e., low fitness measures) or select dangerous patches or diets because these are perceived as safe.
A8.5 Forager emigration and movement between patches Foragers move when they emigrate from the system or change patches. Emigration is assumed to be instantaneous, with foragers leaving the system during the same time step in which they decide to emigrate. Movement between patches may or may not be instantaneous. When movement is instantaneous, foragers move to a target patch as soon as they decide to move and then spend the whole of a time step on the target patch. Otherwise, movement may take one or more time steps. If movement takes one or more time steps, foragers are assumed to reach a patch at the start of a time step. This means that they are able to respond to the local conditions on the patch (i.e., decide which diet to select or move to another patch), as these may not have been fully known when the forager initially decided to move to the patch. The time to travel between patches is calculated from a parameter submodel:
where Tmove= time to move between patches and f(p1, p2, … , pn)= a submodel containing n parameters. The submodel can depend on any global variables, the relative location of patches, and any forager constants or variables.
For simplicity, the movement submodel assumes that movement always takes a whole number of time steps. The submodel checks that the relative values of movement time and time step length always result in movement time exactly equaling a whole number of time steps:
where Ntimestep= movement time in time steps, Tmove= time to move between patches and ttimestep= duration of one time step.
While moving, foragers' metabolize their component stores at a rate determined by the moving metabolic rate. The change in component reserve size while moving is:
where Cfinal= final component reserve size after moving, Cinitial= initial component reserve size, and Mmoving= rate of metabolizing/excreting component while moving.
For simplicity, it is assumed that foragers cannot make any decisions while moving between patches, but can die while moving.
A8.6 Forager diet consumption and physiologyFigure A3 shows how diet consumption and physiology are incorporated into the model. Foragers consume resources from which they assimilate components. The rate of consuming resources may be limited if the maximum rate of consumption is exceeded. The rate of assimilating components depends on the density of components within resources and the efficiency with which components are assimilated. Metabolism removes components from the forager's system. The balance between the rates of assimilation and metabolism determine the amount of a component within the forager's body.
A submodel parameter is read in to calculate the diet consumption rate of foragers of each forager type/species. The efficiency of assimilating each component from the resources in the diet to the body (i.e., the proportion of the component in the diet that is transferred to the body) is also read in as a submodel parameter. Both submodels can depend on any forager constant or variable, patch state variable, or global variable. The rate of assimilating a component is calculated from:
where Iassim= rate of assimilating component, a= efficiency of assimilating the component, Cdiet= density of component in the diet and Idiet= rate of consuming the diet.
The amount of the component assimilated during a time step also depends on the proportion of time spent feeding during the time step. The proportion of time spent feeding can be limited in two ways:
(i) Regulation of diet consumption rate and
(ii) Regulation of component reserve size.
A8.6.1 Regulation of diet consumption rate. A submodel is used to calculate the maximum diet consumption rate (Imax) during a time step. The maximum proportion of time that can be spent feeding (Pmax) is calculated from:
A8.6.2 Regulation of component reserve size. If the forager were to feed for Pmax of the time step, its component reserve size at the end of the time step would be:
where Cfinal= final component reserve size at end of time step, Cinitial= initial component reserve size at start of time step, Mfeeding= rate of metabolizing/excreting component while feeding, and Mresting= rate of metabolizing/excreting component while resting. The model uses a parameter submodel to calculate the target component reserve size (Ctarget) during any time step. The required proportion of time needed to exactly match this target, or approach it as closely as possible, is found by setting Cfinal to Ctarget, Pmax to Ptarget, and rearranging the previous equation:
where Ptarget= proportion of time that forager needs to feed for to match its target or approach it as closely as possible.
The actual proportion of time spent feeding depends on the value of Pmax and the values of Ptarget for each component in the diet. The model attempts to exceed or match the target reserve size for each component, with the constraint that the proportion of time feeding cannot exceed Pmax. It does this by comparing the maximum value of Ptarget with Pmax:
where Pfeed= proportion of time feeding during time step. The component store size at the end of the time step is then found from:
A8.7 Forager mortality A submodel parameter is read to calculate the probability of surviving each fitness component based on a forager's state and any combination of global or patch variables. A uniform random number generator is used to determine whether each of a forager's individuals is killed by any of the fitness components. In the event of two or more fitness components killing an individual, one is selected at random. When all the individuals in a forager are killed, the forager is removed from the simulation.
JG-C is very grateful to Rollie Lamberson for the invitation to attend the RMA conference where this paper was presented. Both of us would like to thank our colleagues in the Centre for Ecology and Hydrology who have worked so closely with us over so many years to develop the IBM models described in this article. We would also like to thank the many other colleagues from Belgium, Denmark, France, Germany, Portugal, Spain, The Netherlands, and the United Kingdom who in many ways also contributed significantly to the development of the shorebird IBMs.