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S. C. Lubetkin, Quantitative Ecology and Resource Management, University of Washington, Seattle, WA 98195–2182, USA (fax +206 543 8798; e-mail firstname.lastname@example.org).
1Biogeochemical tracers such as stable isotopes are often used to determine sources and pathways of organic matter through food webs, which may provide powerful indicators of environmental stress or insight into resource management issues. However, mixing models using such tracers are algebraically constrained and rarely attempted for more than two sources. This makes them of limited value for ecosystems that have numerous contributing sources.
2We developed two complementary mixing models, SOURCE and STEP, which use linear programming techniques with multiple tracers, to estimate the dominant primary producer sources of consumers, and their diets and trophic levels, regardless of the number of sources and trophic steps.
3SOURCE is used to estimate consumers’ direct and indirect uptake of autotrophic sources and their trophic levels. STEP calculates an estimate of a consumer's diet, which may include autotrophs and/or heterotrophs.
4The two models were tested using simulated data sets of producer and consumer tracer values and then used with published δ13C, δ15N and δ34S data from a study tracing organic matter flows in a saltmarsh estuary.
5SOURCE and STEP accurately estimated flows and trophic structures in the simulations, with average errors of 0·07–0·09 for SOURCE and 0·03–0·05 for STEP, depending on the number of tracers used.
6We illustrate two resultant food webs for the saltmarsh estuary showing possible interpretations of the SOURCE and STEP estimates.
7Synthesis and applications. SOURCE and STEP can be used with stable isotope data to estimate accurately consumers’ trophic levels, primary producer dependence and diets, even when the number of potential autotrophic sources or foods is larger than the number of tracers. SOURCE and STEP could be used to assess the roles played by individual species within food webs, to compare food webs across locations or over time, and to examine potential pollutant bioaccumulation in higher order consumers, among other potential applications. The s-plus code for both models is available at http://staff.washington.edu/lubetkin.
Understanding the complex flow of organic matter through food webs is a fundamental pursuit of theoretical, basic and applied ecology. Quantitative knowledge about food web structure has practical applications in measuring the effects of anthropogenic inputs, such as pollutant bioaccumulation in higher order predators (Cabana & Rasmussen 1994), in assessing the impacts of introduced species, and comparison of food webs geographically or temporally, such as may be needed to assess restoration and conservation efforts or to test for the effects of climate change.
Consumer diets, organic matter sources and trophic levels can be estimated using empirically determined tracer levels in mixing models. Most previous mixing models using tracer data have been limited to considering one tracer at a time, even when data from multiple tracers were available (van Dover et al. 1992). Thus, earlier mixing models have often divided food web sources into no more than two groups, such as C3 and C4 plants (Forsberg et al. 1993). In many cases this is an overly simplistic way of partitioning food resources, particularly in estuarine systems, which can have multiple source inputs (Peterson & Howarth 1987; Cifuentes, Sharp & Fogel 1988; Fry 1988).
The purpose of this paper is to describe two companion models, SOURCE and STEP, developed to quantify food web sources and structure with stable isotopes, and to illustrate how they might be used. Our goal was to design linear mixing models using multiple naturally occurring tracers simultaneously, including all potential autotrophic sources or all potential foods, to (i) identify the dominant sources of organic matter supporting the food web, (ii) determine the sequence of trophic steps and (iii) evaluate specific predator–prey linkages. We describe the conceptual framework and algorithm of each model and the testing procedure we used to analyse their mathematical validity. We used SOURCE and STEP with previously published stable isotope data from a saltmarsh estuary on Sapelo Island, Georgia, USA (Peterson & Howarth 1987).
For SOURCE and STEP to work, the tracers and sampling procedure must conform to some basic assumptions, which require the following. (i) All of the equations must be linearly independent. (ii) Consumer uptake of the tracer must be linear with respect to tracer concentration in the food source, and the assimilation rate of the tracer must be known. This implies that a threshold level of the tracer is not required before uptake occurs, nor is there a saturation level after which no more of the tracer is incorporated into its tissue. Even if certain tracers do not strictly meet this assumption, if uptake is approximately linear over the range of values of interest, the tracer may still be used in these models. (iii) Each source must have a distinct tracer signature. SOURCE and STEP are based solely on the tracer signatures of the organisms in the system; therefore, the models predict that any organisms having similar tracer signatures will function similarly, regardless of how different their biological roles in the ecosystem may be. To be considered distinct, the source tracer signatures must exceed a certain difference measure (see Nearest neighbour distance). (iv) Tracer signatures of all major organic matter sources contributing to the food web must be known. That is, if a food source is important enough to be a significant contributor to many consumers, directly or indirectly, a sample of the source has been obtained and its tracer signature analysed. (v) At the level of the individual organisms, variation in tissue level fractionation has been accounted for in the sampling protocol, and the tracer levels are representative of the whole organism or that part eaten by consumers (Gannes, O’Brien & del Rio 1997; Olive et al. 2003). (vi) In SOURCE we also assume that all organisms have the same fractionation rates for each tracer. In reality, there will be some variability about the magnitude of trophic level shifts and in the average tracer values for each organism (Post 2002; Olive et al. 2003). We have incorporated variability into a set of simulations to compare the performance of the models with ‘perfect’ and ‘variable’ data (see Model validation).
In these models, the working definition of an organism's trophic level is the number of times an average unit of primary production was metabolically processed to become the tissue of the organism of interest; autotrophs are defined as trophic level zero. Trophic levels are not restricted to integers. Thus a strict herbivore would have trophic level 1, but a consumer that eats both plants and other animals would have a trophic level greater than 1, proportionally reflecting that mixture (equation 9).
source: primary producer inputs and trophic level
SOURCE estimates the mixture of autotrophic sources that have been assimilated, directly or indirectly, into a consumer's tissues and the consumer's trophic level. By mass balance, consumer tracer signatures will reflect the mixture of food or prey in the diets, and through them the mixture of autotrophic sources at the base of each trophic pathway. In addition, the tracer signatures reflect the biofractionation that occurred at each trophic transfer.
Mathematically, n autotrophic sources contribute fractions s1 to sn to a consumer's diet. Each source i contributes its own characteristic tracer values, ti, ui and vi, to the consumer. If the average trophic level shifts for tracers t, u and v are α, β and γ, respectively, the consumer's overall tracer levels, Ct, Cu and Cv, are a reflection of its diet and trophic level, L, such that:
s1 + s2 + s3 + ··· + sn = 1(eqn 1)
s1t1 + s2t2 + s3t3 + ··· + sntn + αL = Ct(eqn 2)
s1u1 + s2u2 + s3u3 + ··· + snun + βL = Cu(eqn 3)
s1v1 + s2v2 + s3v3 + ··· + snvn + γL = Cv(eqn 4)
where 0 ≤ si ≤ 1.
As there are more sources than tracers, it is mathematically impossible to find an exact solution to this mixing model. We cannot solve for all of the variables simultaneously, so we iteratively solve sets of linear equations for subsets of sources. For example, with three tracers there are four equations, so we could consider any combination of four variables simultaneously and find a unique solution for that combination. In practice, the subset of unknowns solved for in SOURCE always includes the organism's trophic level because excluding the trophic level would disregard the effects of biofractionation. SOURCE solves for L and all possible mixtures of as many sources as there are tracers to find valid corner point solutions (i.e. s1, s2, s3 and L, then s1, s2, s4 and L, and so on through all combinations of three sources; Table 1 and Fig. 1). For each combination of three sources and L there is exactly one mixture that will have Ct, Cu and Cv, which is found using Gaussian elimination (Hillier & Lieberman 1990). Each resultant mixture is a corner point of the solution space (Hillier & Lieberman 1990).
Table 1. Solutions for each set of sources and the centre of mass estimates
Indicates which source was not included in the subset considered; that source therefore contributes a fraction of 0 to the final mixture.
These source combinations, while having the mathematically valid solutions shown here, violate the physical constraint that all fractions (sA–sD) contributing to the mixture must fall between 0 and 1. These sets of fractions did not enter the estimation of fractional ranges and were not used in calculating the centre of mass estimate.
There will be mathematical solutions for all the combinations of sources, but some are not physically or biologically possible because not all the fractions fall between 0 and 1. In the example we discard the second and fourth solutions, which are physically impossible (Table 1). The combination of the two acceptable corner points outlines a solution space that is much smaller than the four-dimensional space where sA+ sB+ sC+ sD= 1 and all the fractions fall between 0 and 1. The corner points represent the extreme ranges of the fractions that contributed to the mixture, similar to the ranges found by Phillips & Gregg (2003).
With larger numbers of sources, there are more corner point solutions to consider, and more than two may fall within the allowed solution space. Corner points often cover a wide range of values but can occur in clusters where a specific set of sources is repeatedly estimated to contribute a large fraction to the consumer. We wanted to condense this list of possible solutions and broad estimate ranges to one estimate that incorporated the trends within the set of corner points, and evaluate its accuracy and precision. We used the arithmetic mean of the coordinates of the biologically possible solutions, which is the centre of mass of the outlined solution space (Fig. 2). We also calculated the standard deviation, directly incorporating the range of fractions found during the iteration process. Means and standard deviations are calculated source by source (Table 1). When there are more than two corner points in the acceptable solution space, the means are not necessarily the mid-points of the fractional ranges.
step: direct calculation of consumer diet pathways
We can use the tracer data to determine the direct links between consumers and their diets with STEP. Here we model each consumer's signature as a composite of the foods it ate modified by one metabolic fractionation. Mathematically, STEP is represented as:
f1 + f2 + f3 + ··· + fm = 1(eqn 5)
f1t1 + f2t2 + f3t3 + ··· + fmtm = Ct − α(eqn 6)
f1u1 + f2u2 + f3u3 + ··· + fmum = Cu − β(eqn 7)
f1v1 + f2v2 + f3v3 + ··· + fmvm = Cv − γ(eqn 8)
where f1 to fm are the fractions that each of the m potential foods contribute to the consumer's diet. Whereas the sis in SOURCE are measures of the autotrophic sources the consumer assimilates directly and indirectly, the fis in STEP quantify direct incorporation of foods. Again, each fraction must fall between 0 and 1. The pool of potential foods may, in theory, include all organisms in the system for which there are tracer measurements. Thus, cannibalism is allowed, where cannibalism refers to consumption of food organisms that have the same general trophic level and feeding practices, and hence similar tracer signatures. (In cases where biological knowledge of the system would give sufficient reason to exclude one or more organisms as potential foods, the number of unknowns to examine for that organism can be reduced.)
STEP simultaneously solves for combinations of as many foods as there are equations (i.e. with three tracers there are four equations, and we can examine sets of four foods at a time) and calculates the centre of mass of the acceptable corner points where all the fractions fell between 0 and 1, generating estimates of each consumer's diet. STEP is applied to each consumer in the system (and each could potentially have a unique list of possible foods to consider). Once all the direct diets have been estimated, trophic levels are calculated for each consumer. Each consumer's trophic level is a weighted average of the trophic levels of its foods plus one:
f1L1 + f2L2 + f3L3 + ··· + fmLm + 1 = L(eqn 9)
where the fis are the fractions each food contributes to the consumer's diet, the Lis are the trophic levels of those foods, and L is the trophic level of the organism of interest. Once the diet of one consumer is known, that information can be used with the information about its diet items’ diets, and so on, iteratively through the food web, to calculate an estimate of the consumer tissue's primary producer origins. These results can then be compared with SOURCE model estimates.
We used two methods to evaluate the models. The first assumed perfect data (e.g. there was no error in measured tracer signatures or variability between samples of the same species) for the tracer signatures of all the organisms in the system and precisely known trophic level shifts. These simulations made it possible to analyse how the models perform under ideal conditions and focused on the mathematical validity of the centre of mass estimates. The second simulation introduced variability into the tracer signatures and trophic level shifts, such as would be expected in real data from multiple samples (Post 2002; Olive et al. 2003), to examine how natural variability in the data would affect the models’ performances.
The strategy behind the simulations was to create known flows through four different sized ‘food webs’, use those flows to calculate the resultant tracer signatures in primary and secondary consumers, and then evaluate how well SOURCE and STEP performed in recreating the flows using only the tracer signatures. Each of the simulated data sets consisted of randomly assigned trophic links from producers to primary consumers and secondary consumers. We used those links to calculate signature amounts of three tracers in each consumer based on the tracer signatures of the producers. The tracer signatures also had a trophic level component, with trophic level shifts analogous to δ15N, δ13C and δ34S (strong, weak and none). We created flows based on four to seven sources in each set of simulations. All simulations and analyses were performed using s-plus.
We used SOURCE and STEP to estimate the diet fractions, primary producer sources and trophic levels of the consumers using data from two or three tracers simultaneously. Realistically, individual diet fractions can range between 0% and 100%, but we accepted corner points with individual diet fraction estimates ranging from −5% to 105%. The purpose of finding the corner points is to characterize the range of fractional values each source or food can contribute to the mixture, and the models can only consider mixtures of limited complexity. Because the actual mixture may contain fractions from many more sources than any individual corner point solution can include, the expanded corner point acceptance region is a compensatory measure to allow borderline cases to contribute to the overall estimate of the mixture. However, not all corner point solutions are used in calculating the centre of mass estimate because that can lead to nonsensical estimates.
We evaluated the models’ performances by calculating the differences between the model-estimated diet fractions, primary producer sources and trophic levels and the true values of those quantities. For true values of the contributing organic matter sources s1 to sn, the error of each estimate is ɛi= si′ − si for i in 1 to n, where si′ is the centre of mass estimate of the contribution from the ith organic matter source. For example, if a consumer was estimated to have 60% of its diet derived from primary producer A and the true value was 55%, the resulting ɛi was 0·60 − 0·55 = 0·05.
nearest neighbour distance
Both models use only the tracer signatures to distinguish between source inputs, so those values must be distinct if the organic matter sources are to be considered separately (Sackett 1989; Macko & Ostrom 1994; Michener & Schell 1994). In order to determine how much the difference between any two signatures affects the accuracy of the calculated estimates, we developed a ‘nearest neighbour’ criterion that related source separation to the discrimination power of the models (Lubetkin 1997). We defined the squared normalized distance between the source signatures in r-space, where r is the number of tracers measured. Each of the r tracers may have a distinct range of values that it can take, and the ranges may differ from tracer to tracer. The relationship of the squared nearest neighbour distance (NND2) between points in r-space to the | ɛi | can be used to determine the minimum value of separation for two sources to be considered distinguishable using their tracer levels. Note that two sources may have statistically significantly different tracer values, using a two-sample t-test or some other measure (Rosing, Ben-David & Perry 1998), without being far enough apart in tracer space to be distinguishable by the models.
We applied SOURCE and STEP to δ13C, δ15N and δ34S data on organic matter flow through a saltmarsh estuary on Sapelo Island, Georgia, USA (Peterson & Howarth 1987; data used with permission). There were six primary producer sources: Spartina alterniflora, Juncus roemerianus, Pinus taeda, Quercus virginica, creekbank algae (spring data) and phytoplankton. No value for δ34S was listed for creekbank algae (Peterson & Howarth 1987) so we used an estimated value of 15‰ for δ34S in the models, which would represent a mixture of surface water sulphate (21‰) and sulphides (10‰; B. Peterson, personal communication). We used the signatures of 14 estuarine consumer organisms collected in the autumn, which fell into a variety of overlapping feeding categories, such as suspension feeder, deposit feeder, suspension-deposit feeder, omnivore and predator (table 4 in Peterson & Howarth 1987). As δ34S is less commonly measured than δ13C and δ15N, we examined the results of SOURCE and STEP using just δ13C and δ15N and then with all three isotopes, to illustrate how the inclusion of more information affected the estimates.
The SOURCE model yielded fairly accurate estimates of primary producer sources. The errors were distributed roughly normally around zero (Fig. 3). Because the centre of mass estimate is an arithmetic mean, the estimates of s1 to sn are pulled away from the most extreme ends of the ranges, and thus SOURCE has a tendency to overestimate those sources that contribute relatively small amounts to the consumer's tracer signature and underestimate the importance of those that are more dominant. When all three tracers were used, 75–95% of the estimates had | ɛi | < 0·10. Of those estimates, more than half were within 0·05 of the true values (Table 2). As expected, simultaneously using a greater number of tracers improved the estimate accuracy. When estimating the trophic level using SOURCE, the tracers with the larger trophic level shifts gave more accurate estimates of the dominant primary producer sources (Table 2). SOURCE accurately estimated the consumers’ trophic levels when used with a tracer with a strong trophic level shift (Fig. 4).
Table 2. Percentage of SOURCE and STEP estimates of diet and primary producer input fractions within certain tolerances using the perfect simulated data. S, the tracer was strongly fractionated; W, a weakly fractionated tracer; N, a tracer not significantly fractionated
S, W, N
The overall accuracy of the STEP model was higher than that of SOURCE (Tables 2 and 3). Again, the error was approximately normally distributed. For this model, which set of two tracers was used did not seem to be important, reflecting that the trophic level effects were accounted for at the outset. The estimates of proportional contributions of primary producers to consumer diets using STEP (average | ɛi | ranging from 0·03 to 0·05) were closer to the true values than the SOURCE estimates (average | ɛi | ranging from 0·07 to 0·09) (Table 3), as were the indirectly calculated trophic levels.
Table 3. Average absolute differences between si and si′ and fi and fi′ over all four sets of consumers. Tracers’ designations as in Table 2
Simulation data type
S, W, N
Variable tracer signatures and trophic level shifts
In the simulations with variability in the data, SOURCE estimates had an average standard deviation of 0·20 when all three tracers were used, and 0·28 when the two strongly fractionating tracers were used, while STEP had lower average standard deviations of 0·14 and 0·16, respectively. Generally, the standard deviations from the centre of mass estimates from both models were greater than the true error of the estimates (Fig. 5). Therefore, the standard deviation could be used as a conservative estimate of the models’ precision.
Incorporating variability into the tracer signatures and fractionation rates did not greatly affect SOURCE's and STEP's estimation accuracy of the primary producer dependence or diet fractions (Table 3), but slightly reduced their ability to estimate the consumers’ trophic levels. However, even with the decreases in estimate accuracy, more than 86% of the trophic level estimates were within 0·3 trophic levels of the true values (| Ltrue–Lestimate |) and more than a third were within 0·1. (Note that these are not fractions, so a difference of 0·3 does not represent 30% error.) The errors in estimated trophic levels were highly correlated with the differences between the assumed and actual trophic level shifts of the most strongly fractionating tracer (Fig. 6).
nearest neighbour distance
The magnitude of the error (| ɛi | = | si′ − si |) increased rapidly when NND2 was less than 0·10, and fractional inputs from sources with distinct tracer signatures (NND2 > 0·10) were estimated more accurately than fractional inputs from sources with similar tracer signatures (Fig. 7). We suggest that sources with a NND2 less than 0·10 be pooled together, and those with NND2 greater than 0·10 be considered distinct.
After calculating their isotopic nearest neighbour distances, Pinus taeda and Quercus virginica were grouped together as ‘upland plants’ but all other sources were distinct (Table 4).
Table 4. The isotopic tracer or tracers (δ13C, δ15N and/or δ34S) that sets each source or source group apart from the others with a nearest neighbour distance greater than 0·10. For example, Spartina is isotopically distinct from creekbank algae based on their δ15N and δ34S values
δ13C, δ15N, δ34S
δ13C, δ15N, δ34S
With SOURCE, the average standard deviation was 0·14 when all three isotopes were used, and 0·25 when only δ13C and δ15N were used with the Sapelo Island data. The average standard deviation for STEP was 0·08 when all three isotopes were used, and 0·10 with just δ13C and δ15N. All four values were lower than the corresponding values from the simulations.
Although Peterson & Howarth (1987) considered only three sources (Spartina, upland plants and phytoplankton), in part due to the difficulty involved in getting ‘clean’ samples of creekbank algae (B. Peterson, personal communication), our results from SOURCE agree with their analyses on the whole. Based on δ13C and δ34S, Peterson & Howarth (1987) concluded that most consumers in the marsh and estuary assimilate a mixture of Spartina and phytoplankton, with suspension feeders being more dependent on plankton and deposit-suspension feeders more reliant on Spartina. SOURCE estimates of phytoplankton contribution using all three isotopes ranged from 20% to 50% for suspension feeders, but accounted for less than 20% of the organic matter input to the suspension-deposit feeders (Fig. 8). Our results were heavily influenced by the inclusion of δ34S, but with or without that isotope our estimates included creekbank algae as an important contributor to the food web, along with Spartina and phytoplankton. This may just reflect that creekbank algae have δ13C and δ34S intermediate to Spartina and plankton. For three of the four suspension feeders (small Mugil cephalus, Palaeomonetes pugio and Penaeus setiferus), the inclusion of δ34S shifted the balance from being based more on Spartina with δ34S to more on creekbank algae without it, but the total of the two fractions (Spartina+ creekbank algae) remained roughly the same in both sets of isotopic analyses (Fig. 8).
Peterson & Howarth (1987) found little evidence for upland plants as an important source of organic matter in the food web, a result we quantified using SOURCE. There was only one consumer (small Mugil cephalus) where the estimated contribution of upland plants exceeded 14%. Peterson & Howarth (1987) noted that although the borrow pits in which Mugil cephalus lived were not surrounded by forest, Mugil cephalus had a δ34S quite similar to upland vegetation. Thus, inclusion of δ34S greatly influenced the resultant SOURCE estimates of primary producer contributions to small Mugil cephalus. Based on δ34S, Peterson & Howarth (1987) also speculate that small Mugil cephalus use more Spartina, directly and through trophic transfer, and the large Mugil cephalus use more plankton, findings consistent with our SOURCE model results (Fig. 8).
We also estimated direct interactions between organisms using STEP. We did not limit the possible foods of any of the consumers and ran STEP without assuming anything about the biology of the system. Using STEP, suspension feeders, suspension-deposit feeders and deposit feeders had estimated diets heavily based on Spartina, plankton and creekbank algae (Fig. 9). Diet estimates from STEP characterized higher order consumers as directly consuming a smaller percentage of primary producers than lower level consumers (Fig. 9). Oddly, STEP often estimated that three consumers (small Mugil cephalus, Uca pugnax and Littorina irrorata) were large components of suspension, suspension-deposit and deposit feeders’ diets. It is ecologically unlikely that suspension and deposit feeders prey on themselves, but small Mugil cephalus had δ13C and δ34S isotopic signatures very similar to a mixture of Spartina and upland plants, Uca pugnax was isotopically similar to creekbank algae, and Littorina irrorata had an intermediate isotopic signature falling between Spartina and creekbank algae. All three also had low δ15N values. Therefore, estimates of predation on these three consumers by suspension and deposit feeders were probably artefacts of the similarities of their signatures with the primary producer signatures and the limited number of potential foods that can be considered at one time. We suggest that, just as the NND2 criteria are checked on primary producers before running SOURCE, the NND2 of all the organisms in the food webs be calculated before using STEP (S. C. Lubetkin, L. L. Conquest & J. E. Zeh, unpublished data).
Because of the increased number of potential food sources, STEP examined many more combinations of foods than SOURCE did. The smaller fractions estimated using STEP (fi ≤ 10%) probably reflect a few mathematically possible but biologically unlikely combinations of food sources that yielded the appropriate consumer isotopic signatures. These low percentage STEP diet estimates probably are not significant contributors to the diet (Lubetkin 1997). (In the simulations, small diet components were overestimated and larger ones underestimated. That may be the case for the diet estimates of these consumers.)
The trophic levels calculated using SOURCE and the results of STEP with equation 9 (Table 5) covered a range of values without distinct modes, reflecting the overall omnivory and diet overlap of many of the organisms. Peterson & Howarth (1987) used a trophic level shift of 2‰ for δ15N and we used 3‰. Several SOURCE trophic level estimates fell below 1, indicating that at least for those organisms we used too large a trophic level shift (Macko et al. 1982; Post 2002). STEP had slightly higher trophic level estimates, reflecting the method by which they were calculated. The only STEP trophic level estimates below 1 were those of Uca pugnax and Littorina irrorata when δ34S was not included. Both consumers had slightly negative fractional diet estimates that have been rounded to 0, and it is those small negative values that resulted in STEP trophic level estimates less than 1.
Table 5. Estimated trophic levels for Sapelo Island consumers calculated using SOURCE and STEP with δ13C, δ15N and δ34S or δ13C and δ15N. BC, from Bighole Creek; LC, from Lab Creek
Model Isotopes used
C, N, S
C, N, S
Crassotrea virginica (BC)
Crassotrea virginica (LC)
Mugil cephalus (small)
Mugil cephalus (large)
Omnivores and predator
Bairdiella chrysura (predator)
There were no STEP estimates for the diets of small Mugil cephalus or Littorina irrorata when δ34S was included (Fig. 9). These two consumers had low trophic level estimates (Table 5), suggesting that the standard fractionation values used in SOURCE and STEP were not accurate reflections of the trophic level shifts of these organisms (Macko et al. 1982).
Using the estimated trophic levels as the basis for the overall web structure together with the STEP and SOURCE estimates, we constructed food web diagrams for Sapelo Island with and without δ34S stable isotope data (Fig. 10a,b). It is beyond the scope of this paper to address the ecological implications of the webs or estimated relationships, which were discussed by Peterson & Howarth (1987). We present these webs to illustrate that the models’ mean estimates can be used to suggest ecologically realistic food web structures. Different webs could be constructed using other values from within the range of possible diet estimates. The web including δ34S (Fig. 10a) had fewer links than the web using δ13C and δ15N (Fig. 10b) because the additional information from the δ34S allowed for more precise estimates, and more spurious links were excluded.
We have shown that SOURCE and STEP are flexible models that can be used with site-specific numbers of primary producers, consumers and tracers, yielding fractional ranges for contributing sources and foods, and accurate estimates for systems meeting their basic assumptions. When the biology does not meet the mathematical criteria, the results from SOURCE and STEP will have to be interpreted with caution. For example, it may be difficult to get a good estimate of the tracer signatures of all the motile organisms in the system. If sources or foods that contribute significantly to the food web are not sampled, consumer signatures may have little relation to the tracer signatures that are known. Unfortunately, SOURCE and STEP cannot be used to predict tracer signatures of ‘missing’ sources. Such predictions would require that all the other fractional estimates of the consumer's diet or primary producer inputs were measured precisely and that the only unknown to solve for was the signature of the missing contributor. Additionally, there are cases where all of the corner points fall outside the acceptable solution space, most probably representing ‘diffuse solutions’ (Phillips & Gregg 2003) or instances where the model assumptions were not met.
More accurate results should be obtainable when more tracers are used simultaneously, but the number of suitable tracers may be limited. As the number of tracers increases, it is more likely that not all tracers will meet the models’ assumptions. Isotopic data could be pooled with or replaced by other tracers in either model, such as fatty acids or amino acids (Kirsch et al. 1998). Bioaccumulating pollutants are potential tracers but have serious constraints (Jackson & Schindler 1996; Jarman et al. 1996). The spatial distribution of the pollutant could be patchy or poorly quantified. Rates of bioaccumulation might not be known or might vary considerably among organisms, both as individuals and as species. Furthermore, unlike stable isotopes, pollutants may have deleterious effects on the organisms that ingest them, a consequence that is unfortunate both for the organism and the modeller, who may not have sufficient biological knowledge to include this factor into the existing mixing models.
Realistic levels of tracer signature variability did not significantly hamper the models’ performances. Unless the variability within samples of a specific organic matter source or food is great compared with its NND2, the within-sample variation will be overridden by the difference between its average value and the average value of its nearest neighbour (see also Peterson & Howarth 1987; Table 2). To put this in perspective, a range of 2‰ or 3‰ in one isotopic ratio will have little effect on the model outcome if its closest neighbour is separated from it by 7–10‰. Unfortunately, this lack of sensitivity to small isotopic ratio differences translates into low detection power of small or subtle differences in behaviour over time and space and between individual organisms.
We used average trophic level shifts in the simulations and case study, but that may not be appropriate in all cases. Incorrectly assumed fractionation rates may be averaged out in the SOURCE model, but they can lead to skewed direct diet estimates in STEP that propagate down the food web (Post 2002). STEP can easily be used with species-specific trophic level shifts if necessary, but as long as the variability in the fractionation rates is small compared with the overall range of signatures considered, the diet composition estimate accuracy should remain high.
The premise of SOURCE and STEP is similar to that of Phillips & Gregg (2003). Both methods find the fractional ranges of these underconstrained systems. Phillips & Gregg (2003) examine every biologically possible solution and evaluate which mixtures have the right tracer signature, a method that becomes computationally intense when the number of potential sources or foods increases. In contrast, SOURCE and STEP find the outer bounds of the possible mixtures that have the required tracer signature, examining far fewer points for any given mixture. The food web example with seven sources and two isotopes from Phillips & Gregg (2003) would require examining more than 1·7 billion partitions at a 1% increment or over 32 million at an increment of 2%. SOURCE would take 21 corner points to sketch the outer boundaries of the solution range. ‘To avoid misrepresenting the uniqueness of the results’, Phillips & Gregg (2003) also discourage focusing on a ‘single value such as the mean’. The arithmetic means we present here should not be viewed as anything more than a characterization of the true solution. Many other possibilities contained within the area bounded by the corner points would also be valid. However, even with the comparatively tiny sample of points used to compute them, the centre of mass estimates summarize trends in the solution space, including where those estimates are not very precise. By maximizing the utility of data from underconstrained systems, the estimates from SOURCE and STEP provide an objective, quantitative basis for deciding where to concentrate further study resources.
We agree with the assessment of Gannes, O’Brien & del Rio (1997) that there need to be more experiments to deal with the assumptions made in stable isotope ratio analyses. We would expand upon that to suggest further experimental work on other tracers and theoretical work on dynamic models involving multiple tracers. At the level of individual consumers, there needs to be further development of models that integrate tracer signatures of changing diets (Fry & Arnold 1982; Owens 1987; Rau et al. 1992; Michener & Schell 1994; Yoshioka, Wada & Hayashi 1994; Olive et al. 2003), particularly utilizing multiple samples from organisms and tissue-specific turnover rates (Peterson & Fry 1987; Michener & Schell 1994). At the food web level, there are some difficulties in interpreting the trophic relationships using tracer data. Spatial and temporal variability of feeding relationships could be addressed by sampling along physical or biological gradients, as did Peterson, Howarth & Garritt (1986), or by repeated sampling of the same area through time to evaluate short-term changes in organic matter inputs or ontogenetic changes in consumers (Yoshioka, Wada & Hayashi 1994). Future efforts should take a more dynamic approach to incorporate the mechanisms regulating the stable isotope ratios and other biomarkers in producers and consumers, explicitly involving the biochemistry and bioenergetics of the system.
SOURCE and STEP are complementary approaches, each with its own strengths. SOURCE is the more appropriate model to use when the system is undercharacterized and when tracer signature data are incomplete. SOURCE is also useful for generating a baseline for comparison; when both models are used, it is easy to evaluate the consistency of estimates produced by both models, especially on the trophic level shift and its assumed uniformity across organisms in the system. STEP estimates may be easier to ‘groundtruth’ than SOURCE estimates, and can be compared with feeding observations or stomach content analyses. STEP would be easier to modify than SOURCE to include estimates of assimilation efficiency for individual consumers if there is a significant difference between what a predator consumes and what it assimilates (Gannes, O’Brien & del Rio 1997).
We see potential for SOURCE and STEP in at least five broad practical categories: restoration, conservation, pollution assessment, resource management and climate change studies. With SOURCE, we might evaluate the influence of an introduced species, such as Spartina alterniflora on estuarine food webs in the Pacific Northwest (C. A. Simenstad, S. C. Lubetkin, V. T. Luiting, J. R. Cordell, D. J. Stouder & K. L. Fresh, unpublished data). By understanding the feeding habits and predation risks to an endangered species, conservation efforts could be directed to optimize its survival chances. When used with stable isotope data, SOURCE and STEP yield a measure of trophic level that could be used with pollutant concentration data to measure possible bioaccumulation in higher order consumers, as well as monitoring contaminant movement through food webs. The differences in feeding behaviour of two similar species living in the same habitat, such as wild and hatchery salmon (Oncorhynchus spp.), could be explored with STEP to assess if they are competing for food, impacting the food web differently, or being differentially preyed upon. Long-term data collection might reveal seasonal and interannual changes in feeding or migration patterns, as seen in baleen whales (e.g. Eubaelena australis; Best & Schell 1996).
SOURCE and STEP are first-order models that work very well when the assumptions required by the simple mathematical framework are met. Some model assumptions may have tenuous biological validity, but they are only explicit statements of assumptions made implicitly in other treatments of tracer data. At a minimum SOURCE and STEP provide a computationally efficient way to constrain the possible mixtures that contribute to consumers. By using the centre of mass estimates with their associated standard deviations, SOURCE and STEP can help ecologists highlight which portions of the food web are best known and which need further inquiry.
We gratefully acknowledge Washington Sea Grant (R/ES-3) and the National Science Foundation Columbia River Land-Margin Ecosystem Research grant (OCE-941202081) for their support of this project. Many, many thanks go to Bruce Peterson for his permission to use his data, and for his help and encouragement. We also benefited greatly from discussion with LMER Food Web Workshop participants Brian Fry, Linda Deegan and Robert Garritt. Several manuscript drafts were improved thanks to the comments of James Jay Anderson, Gerald Folland, Bruce Frost and Daniel E. Schindler.