1LPBN is a parameterized simulation model of flows of nitrogen (N) in an ecosystem of cyanobacteria, grass and grazers, based on the N dynamics of a grazed Puccinellia lawn in an intertidal marsh on Hudson Bay. This system shows two alternate stable states: (a) lawns that either support a foraging population of lesser snow geese, or are not grazed by geese; and (b) exposed saline sediments that support little or no vegetation. The model represents the flow of N from cyanobacterial fixation, the major N input into the system, to the geese that migrate in autumn; those that do not return represent the major N output from the system. We have modelled N fixation, the transformations of N in the soil, plant growth, lawn regeneration, and goose grazing and grubbing.
2The model simulates steady-state flows of N similar to those observed in the field at zero and at moderate goose density, and it also simulates the transition to the state of zero plant biomass, a consequence of increased grubbing at high goose density. The simulated steady-state flows are found to be more sensitive to changes in the parameters that describe N fixation and goose biology, than to similar changes in the parameters describing plant biology.
3Because the model shows the alternate stable states and the transition between them, with values for the state variables that are consistent with field data, we conclude that N dynamics are crucial in determining the stability of the real salt marsh-goose system. The determining factor is the loss of the input of N from fixation when lawn area is reduced because the rate of goose grubbing exceeds that of plant re-establishment.
In recent decades, the North American Mid-Continent population of lesser snow geese has increased steadily at about 7% per annum (Abraham & Jefferies 1997). This is probably the result of the birds feeding on agricultural crops on their wintering grounds and along the flyways (Abraham & Jefferies 1997; Jefferies et al. 2003), which has increased winter survival of the birds (Francis 1999). Increased numbers of snow geese have returned to the Arctic breeding grounds each year and the intense foraging has led to loss of vegetation. At one site in the vicinity of La Pérouse Bay, Manitoba, on the Hudson Bay coast, where there were once extensive intertidal grazing lawns dominated by the grass, Puccinellia phryganodes (Trin) Scribn. & Merr., the vegetation has been lost or severely damaged, so that surface sediment is exposed in some 2500 ha of coastal marsh (Jano et al. 1998). The loss of vegetation has led to hypersalinity of the soil (Iacobelli & Jefferies 1991; Srivastava & Jefferies 1996), and depletion of soil organic matter (Wilson 1993; Wilson & Jefferies 1996). These changes in soil properties make re-establishment of vegetation extremely slow (Handa & Jefferies 2000; Handa et al. 2002).
Because N availability strongly affects overall productivity in this marsh, the effects of increased numbers of geese on N dynamics may alter drastically the productivity and stability of the system. We have used a modelling approach in order to examine the sensitivity of the system to changes in the goose density. General ecosystem models such as CENTURY (Parton et al. 1967) and GEM (Rastetter et al. 1991) simulate N flow and other models simulate the effects on N flow of such disturbances as grazing and fire (Risser & Parton 1982; Seagle et al. 1992; Pastor et al. 1998), or they model grazed lawns (e.g. Thornley & Cannel 2000) or the uptake of different forms of nitrogen into plants (Leadley et al. 1997). None of these is entirely appropriate to simulate the effects of different densities of geese on the flow of N within grazed Arctic intertidal marshes, where there appear to be alternate stable states, and where there are different modes of consumption of forage by grazers.
Previous models of systems with alternative stable states have emphasized saturation of herbivore feeding (Noy-Meir 1975) or enhancement of N loss (Rietkerk & Van de Koppel 1997) as explanations for the multiple stable states and catastrophic behaviour in the system. Recently, Mulder & Ruess (2001) have developed a spatially explicit model of the effects of changes in grazing intensity by black brant geese on small-scale (within patch) sea arrowgrass dynamics. However, because the above models have not been parameterized, they can only describe the behaviour of the systems in qualitative terms. Here we present a simulation model (LPBN) of N flow in a patch of grazed lawn in the intertidal marsh discussed above. The two alternative stable states upon which the model is based are intertidal lawns, grazed or ungrazed, and exposed sediment devoid, or largely devoid, of vegetation (Hik et al. 1992).
We present here for the first time annual N budgets for grazed and ungrazed lawns prepared from empirical data (Wilson 1993) and used in constructing the model. LPBN simulates changes in these budgets in time. It both interpolates and extrapolates, to provide day-by-day and year-by-year summaries of the N contents of compartments in the system and N flows between them. It represents goose foraging as a combination of grazing and grubbing. The objectives were: (a) to show that a parameterized model of N flows alone can simulate alternate steady states (grazed or ungrazed lawns or exposed sediment); (b) to study the processes that lead to the apparent stabilities of lawns, and those that lead to the transition between the alternate states; and lastly (c) to determine what additional empirical studies would improve the model's accuracy and hence our ability to predict the system's behaviour.
LPBN was developed and its parameter values were chosen to describe N transport in the intertidal salt marsh at La Pérouse Bay, Manitoba, Canada (58°45′ N, 93°30′ W). This site is representative of the extensive intertidal salt marshes that occur along the coast of the Hudson Bay Lowlands, a consequence of isostatic uplift and availability of sediment (Andrews 1973; Kershaw 1976; Martini 1982).
Data on N pools and fluxes in this marsh, collected between 1978 and 2001 (cf. Jefferies 1988; Wilson 1993; Bazely & Jefferies 1996), are used to establish yearly N budgets for the system. The first budget is based on steady-state grazing conditions where there is neither a net gain nor a net loss of N from the system over the year. Amounts of N in the initial standing crop, in the net primary production of above-ground biomass (NAPP), in plant tissue removed by geese, in faeces, and in the biomass of geese, were measured or estimated collectively by Cargill & Jefferies (1984a,b), Bazely & Jefferies (1985), Bazely & Jefferies (1989a,b), Hik & Jefferies (1990) and Wilson (1993). Data of the volatilization of N from faeces are given in Ruess et al. (1989). Some N is lost from the system via denitrification (G. Blicher-Mathiesen, unpublished data). Plant litter was calculated as NAPP less that removed by grazing. Amounts of N in rainfall and imported or exported by tides, or lost at spring run-off, were measured directly (R.L. Jefferies, unpublished data), or estimated from the elemental composition of Arctic rainfall (Nadelhoffer et al. 1992).
Although the total soil N in the intertidal marsh in the budget was 165 g m−2 (Fig. 1a), the total soil N pool in the model was taken to be 85 g m−2, the amount of N present in the top 2.7 cm of soil, where the graminoid roots are located. The total plant uptake of N from the soil was set at 2.47 g m−2 year−1. This was based on the N content of above- and below-ground biomass and data of rates of net N mineralization (Cargill & Jefferies 1984a,b; Hik & Jefferies 1990; Wilson & Jefferies 1996) and amounts of soluble organic N in soils (Henry & Jefferies 2002).
A budget for an ungrazed intertidal marsh is shown in Fig. 1(b). The initial standing crop in June (and hence above-ground plant N) is higher than that in the grazed marsh. The rate of N fixation by cyanobacteria is lower by a factor of 5, which is attributed to plant litter accumulation on the soil surface and possibly to competition for resources such as phosphorus or water from the increased standing crop of vascular plants. Overall NAPP is reduced by about 20% compared with that of grazed sites, because of the absence of faecal recycling and the increased standing crop.
Most models of plant growth explicitly include carbon dynamics. However, growth of both plants and geese at the chosen site is N-limited (Cargill & Jefferies 1984a). Plant growth is unlikely to be carbon limited in the modelled lawn, which is only about 2.5 cm high and has a leaf-area index that is about 0.2 (Bazely 1984). Consequently, PAR is little attenuated at the ground surface and plant leaves in the lawn receive essentially direct sunlight. Even in ungrazed exclosures, attenuation of PAR is minimal, especially in spring and early summer before the growth of above-ground vegetation, when most of the annual N fixation occurs (Bazely & Jefferies 1989a).
We set out to model the entire system as far as possible on the basis of its N contents and flows alone. We did not provide for spatial heterogeneity or grazer choice: LPBN represents a uniform area of indefinite size. Grazer density and the grazing process in the steady state are also represented as uniform, both in space and in time, within the limits of a structured calendar describing the model year. Under natural conditions, geese are constantly moving across the landscape of the Cape Churchill Peninsula that includes La Pérouse Bay. The calendar provides a year of 120 days, representing the ‘season’, the period from 20 May (goose arrival, Cargill & Jefferies 1984b), to 15 September (the onset of frequent frosts). LPBN is intended to describe three steady states:
• the steady state in the absence of grazers (e.g. in exclosures); and
• the steady state of exposed soil caused by reduction in lawn area (the effects of foraging by increased numbers of geese).
The model is intended to simulate both intra-annual and long-term trends in N levels in the system over periods of the order of 1–50 years. It is not meant to simulate changes over longer periods, and it does not represent the longer-term processes brought about by isostatic uplift and vegetation change.
LPBN runs under MATLAB™ 5.1, and was constructed using the SIMULINK™ 2 toolkit (both from The Math Works Inc., Natick, MA, USA). Integration is done with MatLab 5.1's built-in solver ode15s: other built-in solvers also worked well. A set of files for defining the model and for setting its parameter values, supervizing the runs and presenting the results, is available from the first author (email@example.com).
LPBN is an eight-state model, representing a system of seven N compartments (Fig. 2a). The system is open to the outside world. The compartments represent the density of N in: unavailable organic materials in soil (H), available organic solutes in the soil solution (O), available inorganic solutes in the soil solution (I), plant shoots (S), plant roots, rhizomes and stolons (R), bodies of geese (G) and goose faeces (F). A further state (Fig. 2b) represents lawn area (A), the fraction of the area of the modelled site occupied by lawn, as opposed to exposed (grubbed) soil. Note that there is no explicit representation of plant or animal numbers or biomass, the compartments represent N density in mass per unit area. For each state there is a differential equation of rates of change in terms of individual transport processes (Table 1a) and rates of flow for the model parameters (Table 1b). The state equations are integrated over 120 days to provide time-courses over the growing season; further equations (Table 1c) relate the initial states at the beginning of the season and to the final states at the end of the previous season.
Table 1. Equations for the model. (a) State equations, giving the rates of change of the state variables in terms of the rates of individual transport processes. (b) Equations for these rates in terms of model parameters. (c) Equations for linking successive integrations of state equations (model years) giving the initial values of the state variables – in the first day (0), of a given year (t) – in terms of the preceding final values for the last day (120) of the previous year (t − 1). Quantities (e.g. of N per unit area) are represented by Roman capitals, rates (quantities per unit area and time) by italic capitals and model parameters by lower-case letters. Greek letters are used for those coefficients determining the carry-over of states from year to year
Ms + Mr + Ts–J – H
Soil organic N:
F + H – Uo– Rm
Soil inorganic N:
Rm + Ls–Ui
Uo + Ui–P – B − Mr
Ga + Z +B – C–Gd
Area grubbing rate:
N grubbing rate:
qg Dr b
G tg + (B + Z) (1 – r)
N fixation rate:
A fmax– S fs
Goose N arrival rate:
Goose N departure (total):
Leaching rate to shoots:
Leaching rate to soil:
Root mortality rate:
Shoot mortality rate:
Root-to-shoot transport rate:
A I R p
Shoot N turnover rate:
Inorganic N uptake rate:
R I ui
Organic N uptake rate:
R O uo
qg gmax/[1 + (A kg/s)3]
Shoot N density:
Root N density:
the rates and functional relationships
The rates of flow are functions of the contents of the relevant compartments (Fig. 1a,b, Table 1b). Few of these functional relationships are known, although the available annual budgets give average annual rates for many of the flows (Fig. 1). In the absence of contrary evidence, we have generally assumed for simplicity that the flow is proportional to the content of N in the source compartment.
arrival, departure, feeding, excretion
N in geese (G) is increased from zero and decreased to zero within a season by the timed processes of arrival (Ga) and departure (Gd) (Tables 1a, b and 2). Between these events, the processes of feeding and excretion affect G. Feeding includes grubbing (B) of root N (R) on days 1–24 (Table 2) and thereafter grazing (Z) on shoot N (S). The grubbing rate, in the absence of a detailed study, is assumed to be proportional to G. There are no studies that define the rate equation for grazing by snow geese, so we have assumed the rate to be given by a generalized Michaelis-Menten function (Type III of Holling 1959; see also Johnson & Parsons 1985). We take the change in grazing rate on day 36 (hatch) to be proportional to the change in number of grazers, rather than to goose N, on the basis of a study that shows gosling growth rate from hatch to be nearly linear in time (Gadallah & Jefferies 1995). Thus, after day 36 (Table 2) grazing by goslings and adult females is assumed to increase the grazing population, and hence the rate, by a factor of 5, the effective clutch size being taken as 3. This is based on an actual average clutch of 4 and a gosling summer mortality of 0.5 over the summer (Cooch et al. 2001). We have assumed for simplicity one death at hatch, i.e. an effective clutch size of 3, to calculate the increase in number of individuals grazing after hatch, and for overall N budgeting we have assumed the second gosling death to happen at departure, where it is subsumed into the loss of goose N over the winter.
Table 2. Timings of processes during the model year. Times in days at which individual processes are changed in rate during the model year of 120 days. Processes not listed are on at constant rate the whole year. Day 1, goose arrival; day 24, thaw; day 36, goslings hatch; day 84, goose departure
The parameter for over-winter goose N survival (σ) is multiplied by departing goose N on day 120 (G120) in order to calculate the initial G (G0) on arrival in the following model year (Table 1c). Our over-winter N loss (1 – σ) thus results from part of the gosling mortality over summer, and goose mortality during migration and on the wintering grounds. For these reasons our over-winter survival (of N) is not directly comparable with reported values of individual survival. Yearly variation in first-year survival is in any case large, with values ranging from 0.07 to 0.70 for the different colonies, while mean adult survival between 1970 and 1988 at La Pérouse Bay was about 0.83 (Francis 1999). In the simulations we have taken σ to be between 0.4 and 0.7 in different runs. The excretion rate (C) is the sum of rates of ingestion of N (B + Z) not retained by digestion and the goose protein turnover rate (tp). The rate of retention of ingested N is proportional to the rate of ingestion and that of N turned over is proportional to G. Retention (r) is set at 0.67, and tp was estimated from data for Barnacle geese (Prop & Black 1997), from the intercept of their linear regression of excretion against ingestion rates.
Constant daily fractions of faecal N (F) are either volatilized (V), or leached into the soil (Ls) and absorbed by plant compartments (Lp). When relative lawn area is < 1 as a result of grubbing, all faecal N is assumed to land on the area of the intact lawn.
root and shoot death, denitrification, dehumification, mineralization
Humus N (H) increases as a result of the daily turnover of shoot N (S) and year-end mortality of S and of root N (R), and decreases as a result of dehumification (H), the rate of which is assumed to be proportional to H. Dehumification is defined in the model as a flow from H to the available organic N (O). N is lost from O to inorganic N (I) as a result of mineralization (Rm). Mineralization, which is assumed to be proportional to organic N, describes the net effect of the turnover of the soil microbial N pool, a small rapid-cycling pool not explicitly represented in LPBN. Input of N to inorganic N (I) is the sum of the mineralization rate and leaching of faeces (F). I decreases as a result of denitrification (J), which leads to a loss of N from the system.
uptake of nitrogen, senescence of puccinellia shoots
R increases as a result of uptake from I and O (Ui and Uo, respectively) at rates proportional to R × I and R × O, respectively. Taking R to be proportional to root size, i.e. to explored volume, and assuming essentially all soluble inorganic and organic N in this volume is taken up, these two rate constants should have the same values. Although the uptake kinetics of organic and inorganic N by Puccinellia roots saturate at high concentrations, uptake over the range of concentrations present in the soil solution is approximately linear (Henry & Jefferies 2003). S increases as a result of transport (P) and the leaching of F. The sum of these gains in S is used to calculate net above-ground primary production. Uptake and transport processes start on model day 24, slow to 24% of their initial rates at day 84 (Cargill & Jefferies 1984b) and stop completely on day 120 (Table 2). Over winter, 20% of S and 80% of R are carried over to the next year, and the remainders are transferred to H, representing senescence (Table 1c).
grubbing and re-establishment of vegetation
Complete cover by a lawn corresponds to a relative lawn area (A) of 1.0. Grubbing represents an immediate proportional loss of root N (R) and A. The soil compartments lose N through grubbing only from a grubbed area where vegetation does not re-establish in the same season. The fraction of soil N uncovered by grubbing and not reclaimed by lawn re-establishment in the same growing season is permanently removed during the winter.
n fixation by cyanobacteria
Higher rates of N fixation are known to occur at lower shoot N (in grazed lawns), but little is known about the mechanism or the form of the dependence. For simplicity, we have assumed a fixation rate that diminishes linearly with an increase of shoot N. The N fixation rate per unit area of lawn is multiplied by area to give the N fixation rate. It is assumed that any N fixed on exposed soil is lost from the system (cyanobacterial mats dry out and are blown away by the wind in summer).
parameter values, initial conditions and calibration
The choice of parameter values and initial conditions (Table 3) was based, where possible, on the results of previous studies conducted in the intertidal zone at La Pérouse Bay. Specifically, rates of N mineralization and denitrification, and pool sizes of H and S, were obtained from Wilson (1993) and Wilson & Jefferies (1996), and N fixation rates and measures of N leaching from faeces were provided by Bazely & Jefferies (1989a) and by Ruess et al. (1989). N uptake kinetics and pool sizes for I and O (primarily soluble amino acids) were obtained from Henry & Jefferies (2002). An active soil depth of 2.7 cm was used to convert data in volume-specific units to area-specific units. Grazing rates were taken from Hik & Jefferies (1990), Hik et al. (1991) and Gadallah & Jefferies (1995). Unpublished studies of the re-establishment of lawns by J. McLaren give a rate of about 1 cm year−1 for re-growth of Puccinellia perpendicular to a cut edge. We assumed that grubbing exposes 1 m of edge in each square metre of lawn. Volatilization data (Ruess et al. 1989) were obtained from a study conducted in marshes at La Pérouse Bay.
Table 3. List of (I) symbols and values for parameters and (II) symbols and initial values for state variables
Winter regeneration of area
(the quantity grubbed in recent year)
Winter survival of shoot N
Winter survival of root N
Winter survival of goose N
1.2 × 10−3
g−1 m2 d−1
Regeneration of grubbed area
3.0 × 10−4
Maximum N fixation rate
3.1 × 10−2
g−1 m2 d−1
Slope of N fixation vs. shoot N
3.0 × 10−2
Maximum grazing rate
7.8 × 10−2
2.0 × 10−4
1.6 × 10−5
Leaching to shoots
3.0 × 10−2
Leaching to soil
7.0 × 10−2
2.0 × 10−2
Transport from roots to shoots
1.2 × 10−2
Fraction of N retained by geese
Goose N turnover
2.5 × 10−3
Death rate of shoots
8.0 × 10−3
Uptake of inorganic N
3.0 × 10−2
g−1 m2 d−1
Uptake of organic N
3.0 × 10−2
g−1 m2 d−1
1.0 × 10−3
II. Initial values of state variables
Other parameter values were plausible estimates. For example, shoot and root over-winter survival (ζ and ρ) were based on the comparison of values of late-season biomass with values for the early season. The grubbing rate parameter (b), for which consistent data were not available, was adjusted to allow the re-establishment of the lawn to match the grubbing losses under steady-state conditions in the presence of geese. The grazing affinity (kg) was set to give a grazing rate near saturation at documented values of S, which seems intuitively reasonable. The power in the equation for Z was set at 3 (Johnson & Parsons 1985), but trials at 1, 2, 3 and 4 showed no major differences.
The model's initial conditions and parameter values were adjusted to reproduce the compartment contents and the annual fluxes given in the empirically derived N budgets (Wilson 1993), under two sets of conditions:
• In a steady state, in the presence of geese at densities that do not lead to soil exposure. As indicated above (σ), the over-winter survival of goose N, was set at 0.4 for this to occur (this is the fraction of exported N returned at the beginning of the season).
• In a quasi-steady state, in the absence of geese, with humus N rising over time (Table 4).
Table 4. Comparison of budgets with model output for stable states in the presence and absence of geese (Fig. 1a,b)
Annual N flows (g m−2 y−1)
Net primary production
Peak shoot N (S)
Peak root N (R)
Annual net export of goose N (G)
Total grazed and grubbed
Sensitivity analysis was performed to quantify the relative influence of changes in input parameters and initial conditions on the steady-state output values of the annual net export of N (goose N at departure less that of the previous arrival), net plant primary production and lawn area. Parameter values and initial conditions were modified independently to correspond to plus or minus 10% of their values under steady-state goose grazing, and the percentage deviations in output relative to the steady-state were plotted over a 50-year time course. A further sensitivity analysis was performed to explore the effect of goose survival on reduction of lawn area. For this analysis, the lawn area at the end of the season was plotted over a 50-year time course for 0.02 unit increments of over-winter survival of geese starting at a value of 0.4. The model was also re-parameterized to run under steady-state conditions in the absence of organic N uptake by plants, in order to assess the ability of the model to match empirically derived N budgets under these conditions.
The model, like the budgets it was based on, describes a system with a dominant N source (N fixation) and a dominant N sink (net export of goose N). In a true steady state, these must be equal, i.e. in the steady state, the model describes throughput of N from fixation to goose over-winter mortality. There is also a dominant N buffer in the system, the humus, which can absorb differences in these rates over many decades. On a decadal time-scale, the contents of other soil and plant compartments simply adjust to give the required throughput, at least within the range of stability of the model system. On a time-scale of 1–2 years, however, the contents of these intermediate compartments are critical in determining the behaviour of the model. Setting it to represent an initial steady state requires a number of cycles of adjustment of the N pools in the following compartments: shoots (S0), roots (R0), soluble soil inorganic N (I0) and soluble soil organic N (O0). Thus the model, initially based on a steady-state budget, can, because it represents the underlying processes, simulate both steady and non-steady states of the system, including the more realistic state in which goose N increases until the system crashes.
The steady-state annual fluxes between major N pools in the model, with values of 0 or 1.0 g m−2 for goose N at the start of the season (qg) were brought into agreement with those in the empirically derived static budgets for grazed or ungrazed conditions (Table 4). G (goose N) and peak S reached the same values for the model and the budget, although the model over-estimated fixation and NAPP by 10% and 38%, respectively. With qg at 0, the model was qualitatively correct in showing lower fixation and primary production and higher S than their respective values with qg at 1.0, but it still overestimated fixation.
Intra-annual variation in N pools (Fig. 3) was quantitatively consistent with trends typically observed in the field (Cargill & Jefferies 1984b; Bazely & Jefferies 1989a,b). S reached a maximum at day 36, after which it fell to a stable value. After departure, S increased again until the end of the model year. Soil N increased rapidly at the beginning but fell with time. H (humus N) also declined during the model year, but recovered with the influx of N in plant litter at the end of the year. Intra-annual trends in compartment contents were similar with qg at 0 or 1.0 g m−2, except that S increased in the absence of grazing until the end of the year (Fig. 3). S in the absence of grazing also differed from that when geese were present. With either value of qg, all compartments were in a steady state from year to year except H, which showed a steady gain in the absence of grazing.
The annual net export of N was highly sensitive to the initial conditions and to changes in most goose and soil parameters, but relatively insensitive to long-term changes (> 5 years) in plant parameters, which elicited only short-term, low amplitude responses (Fig. 4). Specifically, the net export of N was sensitive to increases and decreases in retention (r), turnover (tg), grazing (gmax and kg), fixation (fmax and fs), dehumification (h) and initial humus (H0), and to decreases in area regeneration (e) and initial area (A0). Increased over-winter survival of geese (σ) resulted in a strong initial increase in the annual net export of N, but over the long-term a decrease occurred as a result of a decrease in A (area of lawn available) and S.
Net above-ground primary production (NAPP) had a different sensitivity to changes in goose parameters than the annual net export of N (Fig. 5). In particular, increases in retention (r) resulted in decreases in NAPP. Increases in NAPP with increased gmax and s were proportionally lower than those exhibited by the annual net export of N. The sensitivity of NAPP to changes in soil and plant parameters and initial conditions (not shown) was similar to that of the annual net export of N.
Relative area of lawn (A) was sensitive to increases in σ and moderately so to changes in fixation (fmax), grazing (gmax and kg) and grubbing (b and e) (Fig. 6a). Decreases in A with increased σ were approximately linear up to a value of 0.48. Above that value of σ, the output from the system declined sharply after 10 years or less (Fig. 6b). A more rapid decline in A occurred when grubbing rate (b) was increased (Fig. 6c).
The model could be re-parameterized to represent a steady-state budget without any uptake of O (soluble organic N) by the plants. Neither the annual net export of N nor NAPP were affected if mineralization (Rm) was raised to 2.25 g m−2 y−1, but this is outside the range of the empirical estimate (0.43–1.72 g m−2 y−1, Wilson & Jefferies 1996). However, if organic N uptake was included, Rm was 0.68 g m−2 y−1, which was within the empirical range.
This work has shown, for the first time, that a parameterized model can exhibit transitions between alternate stable states: thus it sets a course for future modelling of such ecosystems. It also confirms our view that a N-limited system such as that modelled here can reasonably be modelled on the basis of its N alone. Thus the model is in these senses a landmark, but we are also conscious of its potential for continued improvement. Even at this stage of development, however, it can act to guide decisions about the likely relevance of data before they are gathered.
While we have obtained, and here present, day-by-day output from the model (Fig. 3), it should be clear that this is not primarily a model intended to offer daily data; its prime purpose is to model year-by-year trends. When there are many more data available about such things as process rates and daily goose numbers, it will be possible to adapt the model to provide more realistic daily outputs. This would need among other things daily weather information. We see the model's ability to grow, in this and other directions, as a major feature.
sensitivity to parameters
When parameter values were increased by 10% from their steady-state values, the individual effects both on the annual net export of N and on primary production are different (Figs 4 and 5).
The annual net export of N was sensitive to a wide range of variables. These included not only over-winter survival of geese, those controlling the grubbing rate, the retention and the turnover of N by geese, but also the rate of N fixation. Even after a model run of 50 years, new steady state conditions were not yet established after such shifts. In contrast, when most plant and soil variables were increased by 10%, a new steady state was established after 15 years, although in the first 5 years changes in the annual net export of N were often abrupt (Fig. 4b,d).
We found that NAPP is particularly sensitive to goose survival and retention of N. An increase in the retention of N leads to a fall in NAPP and a rise in goose numbers following the increase in utilization of primary production by geese.
The steady-state budget (Fig. 1a) was prepared to represent an idealized steady state in the ecosystem, from data for the real system that was close to, but not in, a steady state (goose numbers were rising steadily). The steady-state budget was used to set initial conditions and rates in the model, which could then, by small adjustments, simulate either a steady state or a non-steady state.
stability and transition to the state of exposed sediments
At low goose N density, LPBN represents a remarkably stable system, which changes little over a range of goose N densities. However, one of its most interesting features is that it can be in a state where a small increase in goose N survival can elicit a progressive decrease in lawn area, leading to the collapse of the system (Fig. 5b,c). An increase in (over-winter) N survival from 0.4 to 0.5 shifts the system from a steady state to one where it collapses in less than 12 years. If the survival is increased to 0.7 then the collapse occurs in 7 years. Survival of adult geese on the wintering grounds and on migration increased significantly from 0.78 to 0.88 between 1970 and 1988 (Francis 1999) and since that time there have been further increases. During this period substantial loss of vegetation occurred at La Pérouse Bay. Although values of survival based on N cannot be easily converted into population estimates of survival, it is significant that values of 0.7 or less lead to collapses of the system similar to actual events on the inter- and supratidal marshes at La Pérouse Bay between 1985 and 1993. When the N survival values based on nitrogen are high, the system does not reach a steady state, but collapses, in contrast to the simulation based on an N survival value of 0.4. These rapid effects reflect the historical changes that have occurred in the intertidal marsh in the late 1980s and the early 1990s (Jefferies & Rockwell 2002). Intensive grubbing resulted in much of the intertidal marsh changing to mud flats, recognized as an alternative stable state of the marsh (Hik et al. 1992; Handa et al. 2002). The outcome is similar to those predicted by the mathematical models of catastrophic shifts in systems and alternative stable states by Van de Koppel et al. (1997) and Van de Koppel et al. (2001) for semiarid grasslands and tidal flats. However, in the present model, the collapse of the system is precipitated by a disruption of N fixation, rather than by the enhancement of N losses through erosion.
organic n uptake
LPBN models the N flux through the plants as driven by the availability of both organic and inorganic soil N, as it has been found that Puccinellia phryganodes can readily take up soluble organic N (Henry & Jefferies 2002). The model shows that if plant uptake was based solely on inorganic N then the rate of mineralization would have to be much higher than observed. Thus, the model is consistent with the finding that plants rely on both organic and inorganic N.
tests of the model
Formal tests of the model, such as its reproducing data on the present ecosystem not used in its parametrization, or its ability to model a different N-limited grazed ecosystem, are highly desirable, but are not available to us at this time. When such tests are possible, they will be carried out. Meanwhile the model's utility is perhaps best shown by its actual need for data to complete its ability to predict, and then to test its predictions.
Our experience with LPBN emphasizes the requirement for further experimental studies, especially of the properties of the major sources and sinks of N, which are critical in determining the stability of the modelled system. Most importantly, quantitative studies of the rate of N fixation as a function of environmental variables connected to plant standing crop or grazing rate, over the range from ungrazed lawns to bare mud, are obviously needed. So also are quantitative data on the grubbing and grazing behaviour of geese, to replace the present plausible guesses. As we have seen, these are critical to the stability of the modelled ecosystem.
At present, LPBN is not a spatially explicit model, but it could be extended to include spatial dimensions along which there is variation in lawn properties and grazer choice. The effects of intense grazing, as opposed to the effects of grubbing, might be included, but this will increase the demand for quantitative data on the functional response of geese to the availability of forage species.
We thank the Natural Sciences and Engineering Research Council of Canada for a grant to RLJ and an Ontario Graduate Scholarship in Science and Technology to HALH. We also thank Lindsay Haddon, Johan van de Koppel and an anonymous referee for helpful criticisms of the manuscript.