Ecological Archives E75-001-S1
Cottingham, K.L. and S.R. Carpenter. Predictive indices of ecosystem resilience in models of north temperate lakes. Ecology, volume 76.
Supplement 1: This file contains the programs and input files used to produce the data used to write our paper.
This document (converted by ESA staff from readme.txt) should contain all of the information needed to use those files.
Please do not hesitate to contact me if you have any questions about these materials.
K.L. Cottingham
UW Center for Limnology
680 North Park St.
Madison, WI 53706
(608) 262-2840
cottingh@limnosun.limnology.wisc.edu
18 April 1994
MODEL DESCRIPTION I: OVERVIEW
Our nonlinear model simulates phosphorus cycling among pelagic biota in a vertically stratified temperate lake during the summer (Cottingham 1993). Thermal stratification separates many temperate lakes into three strata: the epilimnion, metalimnion, and hypolimnion. Nutrient levels are generally low in the epilimnion but increase with depth (Wetzel 1983). Maximum algal biomass occurs in either the epilimnion or metalimnion, depending on water transparency and amount of primary production (Moll and Stoermer 1982). When light levels are sufficient for algal growth, metalimnetic populations exploit the high nutrient levels at the thermocline, creating discontinuous profiles of chlorophyll with depth. However, with increasing nutrient input to the epilimnion, epilimnetic algal populations may become large enough to decrease light levels reaching the metalimnion, shading out metalimnetic algae (Moll and Stoermer 1982). The model allows this possibility.
Size-structured interactions are important to lake predator-prey relationships (Kerfoot and Sih 1987). Our model incorporates size-based distinctions by dividing algae and zooplankton into small and large compartments and by including diel vertical migrations of organisms vulnerable to visual predation. Algal vulnerability to zooplankton grazing depends on cell size and on zooplankton size (Burns 1969, Reynolds 1983). Larger zooplankton, especially Daphnia spp., consume larger algae, although some algae are large enough to escape predation altogether (Porter 1977). Small zooplankton are vulnerable to invertebrate predators such as Chaoborus spp. (Dodson 1972, Neill 1981), while large zooplankton (including Chaoborus) are more vulnerable to visually feeding zooplanktivorous fishes (Brooks and Dodson 1965). In many lakes, large zooplankton and invertebrate planktivores avoid the epilimnion during the day and migrate upwards from the metalimnion and hypolimnion during the night. These migrations are due at least in part to avoidance of fish predation (Lampert 1989; Dini and Carpenter 1991, 1992), and affect predation and recycling rates on daily time scales (Schindler et al. 1993).
Phosphorus enters the epilimnion via external inputs and metalimnetic entrainment; however, when external nutrient loading is low, nutrient recycling by consumers supplies much of the phosphorus available to primary producers (Kitchell et al. 1979, Schindler et al. 1993). Epilimnetic phosphorus is taken up by small and large algae, which sink to lower strata or are eaten by zooplankton. Consumers assimilate, excrete, or egest phosphorus they ingest. Dynamics of all state variables were described by differential equations solved numerically with fourth-order Runge-Kutta integration (Press et al. 1984) using 100 time-steps per day. Surface irradiance changed daily and temperature and thermocline depth changed weekly. All simulations were run for 16 weeks. A full description of model structure, equations, parameters, and sensitivity analysis appears in Cottingham (1993).
We calibrated the model to two contrasting food web configurations: one dominated by planktivorous fish and one dominated by piscivorous fish. These food webs represent extreme contrasts for lakes with fish (Carpenter and Kitchell 1993). Calibration data came from planktivore-dominated Peter and piscivore-dominated Paul lakes at the University of Notre Dame Environmental Research Center near Land O'Lakes, Wisconsin. These lakes are small, steep-sided lakes with minimal littoral zone development (Carpenter and Kitchell 1993) and high metalimnetic algal biomass (St. Amand and Carpenter 1993). Paul Lake has served as a control lake for ecosystem experimentation since 1951 and the fish community has been dominated by piscivorous largemouth bass since 1978 (Carpenter and Kitchell 1993). The food web of Peter Lake has often been manipulated, most recently in 1991 to establish a community dominated by planktivorous golden shiners (Schindler et al. 1993).
We simulated these contrasting food webs at 3 baseline phosphorus loading rates. The lowest level, 0.1 g P L-1 d-1, corresponds to loading rates estimated to occur in Peter and Paul (Carpenter 1992). The highest level, 2.0 g P L-1 d-1, is within the range observed for eutrophic lakes (Vollenweider 1968). The intermediate, 1.0 g P L-1 d-1, may be near a boundary separating trophic state in lakes (Benndorf 1990, Persson et al. 1992).
The model reproduced summer dynamics similar to those observed in lakes that span a range of food web configurations and nutrient loading rates. Increases in daily phosphorus loading rates caused increases in mean chlorophyll concentration within the range predicted from empirical regressions. Simulations at 0.1 g P L-1 d-1 baseline loading generally were within 1 s.d. of the seasonal dynamics in Peter and Paul lakes. Details of the calibration and validation of the model will be addressed in a separate publication.
Before calculating resilience indices and measures, we simplified
model output by combining functionally similar compartments (those that differed
only in lake stratum or size) into trophic levels. For example, the four algal
compartments were aggregated by summing small and large compartments in the
epilimnion and metalimnion. The condensed piscivore-dominated food web had length=6
(phosphorus, algae, zooplankton, Chaoborus, planktivorous fish, and piscivorous
fish), while the planktivore-dominated food web had length=5 (all but piscivorous
fish). These groupings are typical of limnological sampling programs and facilitate
comparison to analyses of real lakes (Carpenter et al. 1992, Persson et al.
1993).
MODEL DESCRIPTION II: DETAILS
Run Conditions
Simulations reported in this paper are driven by input schedules with daily changes in light; weekly changes in epilimnetic temperature and thermocline depth; and seasonally averaged metalimnetic temperature and migration patterns (Cottingham 1993). Operationally, the thermocline (at depth ZE, in m) divides the epilimnion from metalimnion. The epilimnion reaches from the surface to ZE-1 and the metalimnion is fixed at the 2 m layer from ZE-1 to ZE+1.
Phytoplankton
The four phytoplankton compartments take up P via growth and lose P via sinking and predation. The growth function for algal compartments,â , is
â = vmax * f(T) * min { f(L), f(P) }
where Vmax represents the maximum potential growth rate and the functions f(T), f(L), and f(P) describe temperature, light, and P limitation of growth, respectively. I assumed that Liebig's Law of the Minimum would hold for light and nutrients, and that temperature further modifies P uptake rate. A similar formulation was used by Scavia (1980).
The light limitation function, f(L), is
f(L) = (1/sz) * ln { (kL + L) / (kL + L e^(-sz)) }
where kL is the half-saturation constant for light and z is the thickness of the layer (Cottingham 1993). This equation accounts for self-shading of algae in a homogeneous mixed layer. The screening function, s, is from Carpenter (1992):
s = A * (kA * A + kB * B) + Wwhere A is the fraction of light screened by algal chlorophyll (0.017; Reynolds 1983, Brock 1985, Elser 1987), kA and kB are the chlorophyll:P ratios for small and large algae, respectively, and W is the fraction of light screened by other particulates in the water column (0.5; Brock 1985, Elser 1987).
Algal sinking rates in m d-1 were taken from literature values and computed within the program as d-1 rates by dividing by the thickness of each layer.
Temperature Effects
All trophic levels follow the same temperature function with different parameter values. Effects of temperature are based on the maximum water temperature at which activity occurs (Tmax), the optimal water temperature for the activity (Topt), and a parameter which determines the slope of increase to the optimal temperature. This function increases up to the optimal temperature and then decreases to 0 as the temperature reaches Tmax (Kitchell et al. 1977, Scavia 1980, Bartell et al. 1988, Hewett and Johnson 1992):
f(T) = V^x e^{x*(1-v)}
V = (Tmax - T) / (Tmax - Topt)
X = Z^2 * (1 + sqrt(1 + 40/y))^2 / 400
Z = ln( ) (Tmax - Topt)
Y = ln() (Tmax - Topt + 2)
This function is used for algal growth, all predator-prey interactions, and the respiration rate of planktivorous fish.
Predation
Predator-prey interactions are mediated through a functional response that allows for interference competition among predators (DeAngelis et al. 1975) and selectivity among multiple prey categories (O'Neill 1969, 1976; Bartell et al. 1988, DeAngelis et al. 1989a). The equations for predation on prey compartment i by predator j (i,j) have the form
i,j = Cmax * f(T) * wi,j * Bi * Bj * { 1 / (Bj + wk,j Bk) }
where Cmax is the maximum potential consumption rate for predator j, f(T) is the temperature function for predator j, wi,j is the selectivity of predator j for prey i, B is the amount of phosphorus in a compartment, k is the index over prey types, and n is the number of prey types consumed by predator j. Predation rates are governed by predator biomass when prey are abundant, by prey biomass when prey biomass is low, and by biomass of both predator and prey at intermediate prey densities (DeAngelis et al. 1975, O'Neill 1976, Bartell et al. 1988). This functional response can stabilize nonlinear systems (DeAngelis et al. 1975).
Selectivity coefficients indicate the probability that a prey type will be eaten if encountered (Scavia 1980). Preferred prey items have higher selectivity coefficients than less preferred prey items. The selectivity of the most- preferred prey item is 1.0, with other selectivities are scaled relative to that value (Vanderploeg and Scavia 1979).
Zooplankton and Chaoborus
Small zooplankton prey upon small algae and large zooplankton prey upon both small and large algae. The selectivity of small zooplankton for small algae was 1.0, because there is only the one prey item. The selectivity of large zooplankton for large algae was adjusted from 0.7 (Scavia 1980) at low P loading rates to 0.2 (Scavia 1980) at high P loading rates in order to produce blooms of large algae at the higher nutrient load. This adjustment anticipates an enrichment-induced shift in composition of larger algae from partially edible chrysophytes and dinoflagellates (St. Amand and Carpenter 1993) to relatively inedible cyanobacteria (Reynolds 1983, Kitchell 1992).
Zooplankton consumption is divided among assimilation, excretion, and egestion. Assimilation depends on prey type: zooplankton assimilate 50% of small algal P (Peters 1983, Scavia et al. 1988, Carpenter 1992) but only 20% of large algal P (Scavia 1980, Carpenter 1992). A constant fraction of consumption (40%; Peters and Rigler 1973, Carpenter 1992) is recycled via excretion back to the available P compartment. Consumed food not assimilated or excreted is assumed to be egested and lost from the system.
Epilimnetic zooplankton are lost to predation by Chaoborus or small fish, depending on zooplankton size. In the metalimnion, both zooplankton compartments have small, linearly density-dependent mortalities that represent losses from the model system.
Chaoborus eat small zooplankton, then excrete, egest, grow or get eaten by fish. Assimilation of prey P is assumed to be 67% (Cressa and Lewis 1986) and the selectivity is 1.0 because small zooplankton are the sole prey item.
Excretion rates for Chaoborus and fish are determined through mass balance bioenergetics models (e.g. Kitchell et al. 1977, Elser et al. 1987). The P budget for all consumers can be summarized as (Kraft 1992):
Cp = Gp - Up - FpConsumed P (CP) is partitioned among growth (GP), excretion (UP), and egestion (FP). This differs from the carbon budget in which there are losses due to respiration (RC):
Cc = Gc - Rc - Uc - Fc
In the model, Chaoborus P excretion rates are estimated by the same procedure as used by Schindler et al. (1993) for field data. Calculations are made from regression models of C respiration rates since C excretion is negligible (Swift 1976). Excretion calculations are made in each time step based on food web type and current temperature, using the mean weight of individual chaoborids in the calibration lakes: 0.35 mg in Tuesday Lake and 1.0 mg in Paul Lakes (Soranno et al. 1993).
Fish
Parameters for fish dynamics are taken from established bioenergetics models (e.g. Kitchell et al. 1977). The planktivores (bluegill, Lepomis macrochirus) begin each simulation as a population of average sized fish (4 g; D.E. Schindler, personal comm.). The piscivores (largemouth bass, Micropterus salmoides) begin as the average sized fish in Paul Lake (160 g; Xi He, personal comm.).
Fixed daily growth (G) and mortality (D) rates (d-1) were derived from field data. I assumed exponential growth during the summer (Wt = Wo e^(Gt)) and used average fish weights at the beginning and end of the season to calculate growth rate. G was 0.01 d-1 for planktivores and 0.003 d-1 for piscivores (Schindler et al. 1993). Daily mortality of planktivores and piscivores were 0.01729 and 0.005328, respectively. The realized growth rate was the sum of mortality and calculated growth rates.
Dynamics of fish compartments also depend on prey categories that were not dynamic but were input to the model as a daily schedule. Fish eat littoral benthos in addition to the modeled compartments (zooplankton, Chaoborus, and smaller fish). Selectivities for different prey categories were set from field observations and literature values. Planktivores were assumed to choose indiscriminately among large zooplankton, Chaoborus, and benthos (Xi He, pers. comm.); therefore all selectivities were set to 1.0. Piscivore selectivities were derived from Swenson (1977). Planktivorous fish were the preferred prey (selectivity=1.0), while Chaoborus and benthos were less preferred (both selectivities=0.4).
In Paul Lake, benthic food items comprise uup to 75% of piscivorous fish diets (Hodgson et al. 1993). However, I did not model benthos as an explicit compartment because I lacked due to insufficient empirical data for model fitting and calibration. I compromised by including littoral prey such that modeled fish growth reflected observed field growth rates. Total consumption is modeled as the sum of known growth and loss processes. In terms of the C budget, I fix total consumption at a level that allows fish to grow at observed rates after losses to respiration, excretion, and egestion. Total consumption (d-1) is equal to a constant times the sum of respiration and growth rates (derived in Cottingham 1993). Respiration rates are calculated in each time step using standard bioenergetics equations and empirically derived parameters for largemouth bass and bluegills (Hewett and Johnson 1992). These rates are converted to g fish (g fish)-1 d-1 using the oxycalorific conversion (3.24, Elliot and Davidson 1975) and assuming that individual fish have a caloric density of 1000 cal g-1 (Hewett and Johnson 1992). Fish mass is updated weekly based on the growth rate and the equation Nt = No ert.
Because total consumption is calculated as a per-day rate, there is no need to convert from the C budget to the P budget. The fraction of consumption that is assimilated (GP + UP) is 0.72 for most fishes (Nakashima and Leggett 1986). By mass balance, then, the fraction of consumption that is egested is 0.28 and the fraction excreted is 0.72-GP. GP has been fixed for the season, so the amount of P excreted by fish per unit time is the product (0.72-GP)*CP (Kraft 1992).
PASCAL PROGRAMS
All simulations were done using programs created in Borland Turbo Pascal Version 6.0. There are code (.pas) and compiled (.exe) versions of each Pascal program on the disk. One caution: some of each program is coded specifically for Turbo Pascal, and will require changes for other compilers.
PROGRAMS USED:
These programs require three input files (see next section) in order to run. Given those files, they run a simulation as described in the methods and create a series of output files used by the post-simulation processing programs. The planktivore and piscivore versions of the programs differ only in the algorithms for linearizing the model output into the matrix form.
INPUT FILES
(1) SYSINPUT.DAT contains the initial conditions and most of the parameters for interactions among state variables. It's not an easily understandable file; however, a key to the parameters listed in the file appears on the next page.
There are twelve versions of the SYSINPUT.DAT file, corresponding to planktivore- and piscivore-dominated food webs; P loading rates of 0.1, 1.0 and 2.0 g L-1 d-1; and reference (no pulse perturbation) and perturbed (pulse perturbation) simulations.
Food Web |
P Load |
Ref. File |
Perturb. File |
Planktivore |
0.1 |
||
1.0 |
|||
2.0 |
|||
piscivore |
0.1 |
||
1.0 |
|||
2.0 |
Before running the model, be sure to copy one of those twelve files to SYSINPUT.DAT, or you will get a run-time error.
(2) WEEKLYIN.DAT contains the weekly input schedules. There should be 16 lines in the file, one for each week in the four-month field season from mid-May to mid-September. This file contains the epilimnion and metalimnion depth and temperature as well as the vertical migration patterns for large zooplankton, Chaoborus, planktivorous fish, and piscivorous fish.
The data contained in this file are from Paul Lake, 1991 (Carpenter et al. unpublished data).
(3) DAILYIN.DAT contains the daily input schedule for light at the lake surface, taken from 1991 data (Carpenter et al. unpublished data).
{Line 1} Initial Values of Each State Variable: epilimnetic P, epilimnetic small algae, epilimnetic large algae, epilimnetic small zooplankton, metalimnetic P, metalimnetic small algae, metalimnetic large algae, metalimnetic small zooplankton, large zooplankton, Chaoborus, planktivorous fish, piscivorous fish
{Line 2} Epilimnetic flushing rate, entrainment from metalimnion to epilimnion
{Line 3} Metalimnetic flushing rate, entrainment from hypolimnion to epilimnion
{Line 4} Small algae parameters: maximum growth rate, half-saturation for light, half-saturation for nutrients, sinking rate, P to chlorophyll ratio
{Line 5} Large algae parameters: maximum growth rate, half-saturation for light, half-saturation for nutrients, sinking rate, P to chlorophyll ratio
{Line 6} Small zooplankton parameters: metalimnetic death rate, maximumconsumption rate
{Line 7} Large zooplankton parameters: metalimnetic death rate, maximum consumption rate
{Line 8} Chaoborus maximum consumption rate
{Line 9} Planktivore epilimnetic death rate, maximum consumption rate, unused value, gross rate of increase (growth+death rates), initial weight
{Line 10} Piscivore epilimnetic death rate, maximum consumption rate, unused value, gross rate of increase (growth+death rates), initial weight
{Line 11} Small zooplankton eating small algae: assimilation, excretion, selectivity
{Line 12} Large zooplankton eating small algae: assimilation, excretion, selectivity
{Line 13} Large zooplankton eating large algae: assimilation, excretion, selectivity
{Line 14} Chaoborus eating small zooplankton: assimilation, excretion, selectivity
{Line 15} Selectivity of planktivores for large zooplankton
{Line 16} Selectivity of planktivores for Chaoborus
{Line 17} Selectivity of planktivores for benthos
{Line 18} Selectivity of piscivores for Chaoborus
{Line 19} Selectivity of piscivores for planktivores
{Line 20} Selectivity of piscivores for benthos
{Line 21} Number of weeks to run, number of days in a week, interval between saving data, unused parameter, day of perturbation
{Line 22} Small algae Tmax, Topt, Q10; large algae Tmax, Topt, Q10; small zooplankton Tmax, Topt, Q10; large zooplankton Tmax, Topt, Q10
{Line 23} Chaoborus Tmax, Topt, Q10; small fish Tmax, Topt, Q10; small fish respiration Tmax, Topt, Q10; large fish Tmax, Topt, Q10
{Line 24} unused parameter; baseline P loading rate
{Line 25} Chaoborus respiration rates at 5, 10, 15, 20, 25 degrees
{Line 26} perturbation magnitudes (up to 10)
OUTPUT FILES
NONEQNTR.DAT non-equilibrial nutrient turnover rates (NTR) One file per call to a .PAS file. Columns are baseline P loading rate, size of pulse perturbation, compartment # (0=whole system; 1=phosphorus; 2=algae; 3=zooplankton; 4=Chaoborus; 5=planktivores; 6=piscivores), NTR based on inputs, NTR based on outputs, amount of P in the compartment (standing stock)
MATRIX_?.MAT flux matrix for the linearized model. One file per simulation (1 in reference simulations; 10 in perturbation simulations). The ? runs from 0-9, in order of simulation.
INPUTS_?.MAT vector of inputs to the linearized model. One file per simulation (1 in reference simulations; 10 in perturbation simulations). The ? runs from 0-9, in order of simulation
SYSTAT_?.SEP daily output for each compartment of the model, one file per simulation (1 in reference simulations; 10 in perturbation simulations). The ? runs from 0-9, in order of simulation.
Columns are day of simulation, baseline P loading rate, size of pulse, dummy, epilimnetic P, epilimnetic small algae, epilimnetic large algae, epilimnetic small zooplankton, metalimnetic P, metalimnetic small algae, metalimnetic large algae, metalimnetic small zooplankton, large zooplankton, Chaoborus, planktivores, piscivores.
SYSTAT_?.AGG daily output for each of the aggregated compartments of the model. One file per simulation (1 in reference simulations; 10 in perturbation simulations). The ? runs from 0-9, in order of simulation.Columns are day of simulation, baseline P loading rate, size of pulse, dummy, phosphorus, algae, zooplankton, Chaoborus, planktivores, piscivores.
FLUXES_?.TOC not needed for this paper
** VERY IMPORTANT! **
Because each program creates output files with the SAME filenames,
these output files must be renamed with unique names between subsequent program
runs.
RESILIENCE INDICES
We evaluated two indices of resilience in reference simulations of the model: the dominant eigenvalue and nutrient turnover rate. Resilience is positively related to both of these indices (DeAngelis 1992).
The dominant eigenvalue, max, is the eigenvalue closest to zero in the matrix of flux rates of a linearized model. When real parts of all eigenvalues are less than zero, the system equilibrium is stable (DeAngelis 1992). We estimated max from a donor-controlled linear matrix model parameterized from time courses for each trophic level. Translation of the daily nonlinear model output to the linear system followed the framework and assumptions of Carpenter et al. (1992). max was calculated by loading the flux matrix (FLUXES_?.MAT from the Pascal program) into the Matlab software package and using the built-in eigenvalue calculation algorithms.
Nutrient turnover rate (NTR, the ratio of external nutrient flow to nutrient standing stock) is much simpler to estimate than max, yet appears to yield comparable information about ecosystem recovery when nutrient input is the factor which limits recovery (Harwell et al. 1977, 1981; DeAngelis et al. 1989a; Carpenter et al. 1992; DeAngelis 1992). NTR integrates aspects of nutrient inputs and recycling and can be estimated directly from field data on nutrient input rate, nutrient output rate, and the total amount of nutrient in the system (Carpenter et al. 1992). We estimated NTR by calculating phosphorus input and output rates for the whole system, then dividing the average of these by the average system standing stock of phosphorus (Watson and Loucks 1979). This is done within the Pascal program, and saved in the file NONEQNTR.DAT
We ran 10 simulations of each of the six combinations of food web structure and baseline phosphorus loading rate under perturbed conditions. Each perturbation was a single pulse of nutrients into the epilimnetic available phosphorus compartment on day 50 of the simulation. The 10 different pulse perturbations ranged in magnitude from 5-50 g/L. Preliminary investigations indicated that all food web-baseline phosphorus loading combinations responded little to pulse perturbations of available phosphorus <5 g/L and failed to recover from pulses >50 g P/L.
There are many ways to measure resilience after a perturbation has occurred (e.g. Bloom 1980; DeAngelis 1980, 1992; Steinman et al. 1991, 1992). Resilience is most commonly defined in terms of the recovery of each system component to its predisturbance state (Pimm 1984). As a result, most measures require enormous amounts of data on the perturbation and pre-perturbation state of each component -- information that may be difficult or impossible to obtain in the field. In a model, this is less difficult. We began by calculating the displacement between reference and perturbed simulations, summed over all trophic levels:
DN(t) = sqrt ( { (Npi(t) - Nri(t))^2 / (Nri(t)^2) ) }Npi is the amount of nutrient stored in trophic level i in the perturbed simulation, Nri is the amount of nutrient storred in trophic level i in the reference simulation, n is the number of trophic levels, and t is the time since perturbation. DN(t) was determined for each day from the day of perturbation until the end of the simulation. Percent displacement was obtained by multiplying DN(t) by 100. Return times were then calculated from formulas developed in other modeling studies. The first proposed formula for return time integrated the displacement (DN(t)) over the period from the time of perturbation (t=0) through the end of a simulation (TF; Harte and Morowitz 1975, O'Neill 1976):
Unscaled return time = DN(t) dtThe integration is from the day of perturbation until the end of the simulation, and trapezoidal integration at one-day intervals was used to approximate the integral. In more recent literature, return times have been calculated from the integral of displacements, scaled by the displacement on the day of the perturbation (DN(0); DeAngelis 1980, 1992; DeAngelis et al. 1989a):
Return time scaled by Initial Disp = {DN(t) / DN(0)} dtHowever, when there were delays between the perturbation and the time of the maximum displacement, equation (3) could be misleading (DeAngelis et al. 1989a). Consequently we also tried a third measure, the integral of the displacements scaled by the maximum displacement observed after the perturbation (DN(m)):
Return time scaled by Maximum Disp = {DN(t) / DN(m)} dt
Return times calculated from equations 3 and 4 estimate the mean time ittakes for the displacement of each trophic level to decay to e-1 (37%) of its initial or maximum value (DeAngelis 1992). As a result, return times could be quite long even for a tiny perturbation. We were interested in recovery rates following perturbations large enough to cause a change in the system, and so did not calculate scaled return times when initial or maximum displacements were <1.0 (100%).
Each of these measures was calculated for each perturbation-baseline nutrient loading-food web combination using automated batch files.
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