Food webs are more than the sum of their tritrophic parts

Abstract

Many studies have aimed to understand food webs by investigating components such as trophic links (one consumer taxon eats one resource taxon), tritrophic interactions (one consumer eats an intermediate taxon, which eats a resource), or longer chains of links. We show here that none of these components (links, tritrophic interactions, and longer chains), individually or as an ensemble, accounts fully for the properties of the next higher level of organization. As a cell is more than its molecules, as an organ is more than its cells, and as an organism is more than its organs, in a food web, new structure emerges at every organizational level up to and including the whole web. We demonstrate the emergence of properties at progressively higher levels of structure by using all of the directly observed, appropriately organized, publicly available food web datasets with relatively complete trophic link data and with average body mass and population density data for each taxon. There are only three such webs, those of Tuesday Lake, Michigan, in 1984 and 1986, and Ythan Estuary, Scotland. We make the data freely available online with this report. Differences in web patterns between Tuesday Lake and Ythan Estuary, and similarities of Tuesday Lake in 1984 and 1986 despite 50% turnover of species, suggest that the patterns we describe respond to major differences between ecosystem types.

Trophic cascades and tritrophic interactions have long been studied. Community-level trophic cascades involve trophic levels (1, 2). Species-level trophic cascades (3) involve interactions among a few species (4, 5) and have been documented in many systems (6, 7). Tritrophic interactions occur by diverse mechanisms (8–11), can cross ecosystem boundaries (12, 13), and matter practically, for example in management of marine fisheries (14) and biological control of crop pests by using consumers of pests (15).

Species-level tritrophic interactions (henceforth tritrophic interactions) have rarely been examined within a larger food web (henceforth web). Some studies that did so counted the commonness of a tritrophic interaction motif by using only web topological structure (16, 17). The stability of tritrophic interactions has been investigated as a function of ratios of consumer body mass to resource body mass in each trophic link (henceforth link) and of the numbers of predators and prey of each species (18), although the dynamical model of that study was not tested by dynamic data, and measurements of population density were not considered.

Webs with the average body mass (M) and population density (N) of each species or other taxon (henceforth M,N-webs) (19–26) reveal previously unrecognized features of web structure (27, 28) and connect body mass (29) and metabolism (30) to webs (31). We analyzed links, tritrophic interactions, and food chains in M,N-webs from Tuesday Lake, Michigan, in 1984 and 1986 and Ythan Estuary, Scotland. These three examples are all of the M,N-webs we know where the links, average body masses, and population densities are based on direct observations in situ, and the data are publicly available. This paucity of directly observed, appropriately organized, publicly available M,N-webs seems shocking after more than a century of scientific ecology, limits the possibility of empirically based generalization and indicates the great need for more such M,N-webs from varied habitats to be published.

We used statistics on link length, link angle, and between-angle of tritrophic interactions. Intuitively, the length of a link from resource R (eaten) to consumer C (eater) is the number of orders of magnitude of difference in body mass plus the number of orders of magnitude of difference in population density between R and C. The angle or slope of a link measures the rate of change in biomass, population productivity and population consumption from R to C. The between-angle of a tritrophic interaction from R to intermediate taxon I to C measures the acceleration (rate of change in the rate of change) in biomass, population productivity and population consumption from R to I to C. Definitions are specified in detail below.

Two models that treated a trophic link as independent of other trophic links in a tritrophic interaction or food chain were rejected. Our statistics and modeling revealed that links appearing in tritrophic interactions differed systematically from random links, that food chains were more than compositions of overlapping tritrophic interactions, and that some features of communities were not wholly accounted for by food chains. At each higher level of structure, new properties emerged. These results support a holistic quantitative approach to understanding food webs, however informative reductionist study of components may be. These results matter practically because they suggest that the larger food web context may affect efforts to manipulate links, tritrophic interactions, and longer chains within webs.

Results

Definitions.

Link statistics.

Taxa were plotted on (log(M), log(N)) coordinates (base-10 logarithms throughout) (Fig. 1 A–C). In all three webs, the log population density log(N) of taxa decreased approximately linearly with increasing log body mass log(M), confirming the existence of an approximately allometric (or power law) relationship between population density and body mass with negative exponent (31, 32). Linear regression coefficients and confidence intervals are given in Table S2. In Tuesday Lake 1984 and 1986 (Fig. 1 A and B), the three classes of basal, intermediate, and top taxa were clearly separated by average body mass. In Ythan Estuary (Fig. 1C), by contrast, these three groups intermingled much more as a function of body mass. Thus food chains in Ythan Estuary started or stopped in the middle of the range of M, whereas those in Tuesday Lake generally began at small M and ended at large M.

Fig. 1.

Properties of food webs in Tuesday Lake 1984 (A and D), Tuesday Lake 1986 (B and E), and Ythan Estuary (C and F). (A–C) Log population density N declines approximately linearly as a function of increasing log average body mass M of taxa. Trophic links were omitted for clarity, but food generally flowed from upper left toward lower right, because consumers were generally less abundant and had larger average body mass than the resource taxa they consumed. In Tuesday Lake (A and B), basal, intermediate, and top taxa were distinct with high, intermediate, and low population density. In Ythan Estuary (C), basal and top taxa spread to the middle of the body mass distribution. Numbers, means, and standard deviations of log(M) and log(N) of taxa in each category are in Table S1. Linear regression coefficients are in Table S2. (D–F) Upper angle A_upper was a horseshoe-shaped function of lower angle A_lower of 2-chains. Vertical and horizontal lines represent median lower and upper angles for all 2-chains (solid line), median angle of all links (dashed line), and allometric angle (dash-dotted line). Counts of data points in each quadrant refer to quadrants formed by medians of lower and upper angle [#dots(l.u.)], median angle of all links [#dots(med)], and allometric angle [#dots(allom)].

The l₁ distance (henceforth distance) between two taxa a and b was defined as ∣log(M_a) − log(M_b)∣ + ∣log(N_a) − log(N_b)∣ (27). The first term ∣log(M_a) − log(M_b)∣ = ∣log(M_a/M_b)∣ is the absolute log body mass ratio, the number of orders of magnitude of difference in body mass. The second term ∣log(N_a) − log(N_b)∣ = ∣log(N_a/N_b)∣ is the absolute log density ratio, the number of orders of magnitude of difference in population density. The distance between two taxa a and b measures the square-block or Manhattan-grid distance between a and b when they are plotted on (log(M), log(N)) coordinates. This distance is preferred to Euclidean distance because the square-block distance has a direct biological interpretation: the l₁ distance is the number of orders of magnitude of difference between two taxa in body mass and in population density.

Viewing a link as a vector (R, C) from resource R to consumer C, the length of a link (or link length) was defined as the distance from R to C (using the definition of distance in the previous paragraph). The angle of a link (or link angle) was the counterclockwise angle to the link from a horizontal arrow starting from R and pointing right parallel to the positive log(M)-axis, and took values in the interval [−180°, 180°) (Fig. 2A)]. (The angle is not defined when M_R = M_C and N_R = N_C, as in cannibalism, for example.) If the link angle equaled −45°, then the link had slope −1 because tan(−45°) = tan(−π/4 radians) = −1. In this case, the resource biomass B_R = M_RN_R equaled the consumer biomass B_C = M_CN_C because each factor of increase in average body mass from R to C was accompanied by an exactly equal factor of decrease in population density from R to C. Moreover, if population productivity and population consumption scaled allometrically with M as NM^b, 0 < b < 1, then in a link with slope −b, the population productivity and population consumption of R equaled the population productivity and population consumption of C. Often b = 2/3 or 3/4 is claimed (29, 33). If b = 2/3, then, in a link with angle −33.7° = arctan(−2/3), the value of NM^b for R equals the value of NM^b for C, and the same is true for a link with angle −36.9° = arctan(−3/4) if b = 3/4. Thus the angle or slope of a link revealed the change in biomass, population productivity and population consumption from R to C.

Fig. 2.

Definitions of angles. (A) Link angle was negative (the typical case) when the link from R to C resulted from a clockwise turn with respect to a horizontal line (dashed line here) starting from R and pointing right parallel to the positive log M axis. (B) When the upper link (I, C) turned counterclockwise from the lower link (R, I), between-angle was positive. (C) When the upper link (I, C) turned clockwise from the lower link (R, I), between-angle was negative. R, resource taxon; I, intermediate taxon; C, consumer taxon.

Tritrophic statistics.

A 2-chain consisted of three taxa (R, intermediate taxon I, and C), and two links: the lower link (R, I) and the upper link (I, C). When plotted on (log(M), log(N)) coordinates (Fig. 2 B and C), the upper link usually appeared below and right of the lower link because as body mass M increased up the typical 2-chain, population density N typically decreased. (Exceptions to this typical case arise when, e.g., C is a parasite smaller in body mass than its host I, but such exceptions were rare in our data; Table S3.) A 2-chain depicted a tritrophic interaction. The 2-span of a 2-chain was defined as the distance from R to C. L_lower and A_lower denoted the length and angle of lower link (R, I); similarly for L_upper and A_upper. By the triangle inequality, 2-span ≤ L_upper + L_lower. When I fell outside the rectangle described by M_R ≤ M_I ≤ M_C and N_C ≤ N_I ≤ N_R, the inequality was strict: 2-span < L_upper + L_lower. The difference between the mean of L_upper + L_lower and the mean 2-span for all 2-chains measured how much 2-chains departed on average from M_R ≤ M_I ≤ M_C and N_C ≤ N_I ≤ N_R.

The between-angle of a 2-chain was defined as the angle in the interval [−180°, 180°) from (R, I) to (I, C). Positive angles (Fig. 2B) represented a counterclockwise turn from lower to upper link, negative angles (Fig. 2C) a clockwise turn. For example, if the lower link had angle −45° and the upper link had angle −30°, then the between-angle was +15°. The mean and standard deviation of between-angles over all 2-chains measured how log body mass ratios and log population density ratios varied between successive links in 2-chains. As a difference of angles, the between-angle of a 2-chain measured the acceleration (rate of change in the rate of change) in biomass, population productivity and population consumption from R to I to C. (In kinematics, acceleration is the rate of change of velocity, which is the rate of change of position.) A positive mean between-angle signified that biomass, population productivity and population consumption increased faster in the upper link (I, C) than the corresponding measures increased in the lower link (R, I).

Multitrophic statistics.

The allometric slope of a web was defined as the slope of the ordinary linear regression line of log(N) as a function of log(M) for all nonisolated taxa. The allometric angle of the web was defined to be the angle between −90° and 90° corresponding to the allometric slope, measured from horizontal pointing right. The community span of a web was defined as the range of log(M) values plus the range of log(N) values over all nonisolated taxa. The community span was the number of orders of magnitude difference in mass between the largest and the smallest nonisolated taxa plus the number of orders of magnitude difference in population density between the rarest and the commonest nonisolated taxa.

Maximal food chains, those from a basal to a top taxon, hereafter called chains, were enumerated (27) as any chain passing from resource to consumer at each link but not including the same taxon twice (cannibalistic links were not included, and cycles, if present, were not completely traversed). All 2-chains were parts of the enumerated chains. The span of a chain or chain span was defined as the distance between its top and basal taxa. The community span exceeded or equaled the span of every chain. The difference between the community span and the mean chain span measured the degree to which average chains traversed the web in (log(M), log(N)) space. The count chain length of a chain was defined as the number of links comprising the chain. The sum chain length of a food chain was defined as the sum of the lengths of links comprising the chain. The wiggling of a web was defined as the mean sum chain length divided by the mean chain span. The minimum possible value of a web's wiggling was 1, and the excess over 1 measured the average changes in direction of links in chains as links progressed from basal to top taxa.

Null hypotheses.

We compared statistics from each empirical web, excluding isolated taxa and cannibalistic links, to two null hypotheses. In each empirical web, we enumerated all possible ordered triples, (t_R, t_I, t_C), of three distinct taxa with M_R ≤ M_I ≤ M_C. For each triple, we computed 2-span, between-angle, A_upper and A_lower as if t_C ate t_I and t_I ate t_R, whether or not these trophic relationships existed. In this way, we produced hypothetical distributions of statistics for all triples in each web under the null hypothesis that 2-chains had only the structure inherited from the observed M and N distributions of taxa and an ordering by body mass. Links with undefined angle and 2-chains with undefined between-angle were excluded (as they were in analyzing data). We also compared webs with simulations of the cascade model, which assumes M_R ≤ M_I ≤ M_C, in Comparisons with Cascade Model Simulations Plus M and N Distributions in SI Text. More recent models allow for deviations from strict ordering but are not considered here for reasons explained in SI Text.

Several measures of a single web were not statistically independent, such as between-angles and link lengths. For example, two link lengths were not independent if a taxon was involved in both links. The lack of independence violated the assumptions of regression and some standard statistical tests. We treated the P values from such tests not as probabilities but as descriptive statistics we called nominal P values. A result was nominally significant if the nominal P value was <5%. The term “nominal” warns that the reported P value is vulnerable to the dependence of the underlying measures.

Tritrophic Interactions: Univariate Distributions.

Between-angle, 2-span, A_upper and A_lower.

In all three webs overwhelmingly, most links had M_R ≤ M_C and most 2-chains had M_R ≤ M_I ≤ M_C (Table S3). Mean between-angle was nominally significantly positive for Tuesday Lake 1984 (mean = 15.83°, t test nominal P < 0.0001) and for Tuesday Lake 1986 (mean = 31.66°, t test nominal P < 0.0001), but nominally significantly negative for Ythan Estuary (mean = −27.46°, t test nominal P < 0.0001; Table S1). The positive mean between-angle for Tuesday Lake (both years) indicated that the upper link tended to turn counterclockwise from the lower link, and oppositely for Ythan Estuary (Fig. 3). On average, moving from basal to top taxa, angles of consecutive links in a typical Tuesday Lake (Ythan Estuary) chain had progressively less negative (progressively more negative) values, leading to convexity (concavity) of typical chains on (log(M), log(N)) coordinates. Link angle increased with resource or consumer log body mass in Tuesday Lake, but decreased in Ythan Estuary (Table S2). In Tuesday Lake, biomass, population productivity and population consumption increased faster in the upper link than the corresponding measures increased in the lower link of tritrophic interactions, whereas the opposite held in Ythan Estuary.

Fig. 3.

Typical Tuesday Lake (A) and Ythan Estuary (B) chains inferred from average between-angles. Tuesday Lake upper links tended to turn counterclockwise from lower links, the opposite of Ythan Estuary.

The pronounced positive mean between-angle in Tuesday Lake and the pronounced negative mean between-angle in Ythan Estuary differed from the mean between-angles of hypothetical triples, which were slightly negative (Table S1). For all three webs, comparisons between 2-chains and triples of the distributions of between-angle, 2-span, A_upper, and A_lower rejected the null hypothesis that distributions were the same with nominal P < 0.0001 (Kolmogorov–Smirnov test, Matlab function kstest2).

Standard deviations of between-angle distributions were 52.09° for Tuesday Lake 1984, 61.81° for Tuesday Lake 1986, and 61.15° for Ythan Estuary. Between-angle histograms for webs and triples showed approximately symmetric unimodal distributions.

L_upper and L_lower.

If (hypothetically) 2-chains were pairs of links chosen randomly and independently from all links, one would expect mean(L_upper) plus mean(L_lower) to be close to twice the mean link length. However, twice the mean link length was greater than the mean of L_upper + L_lower across all 2-chains (Table 1). Thus 2-chains differed from random pairs of links. Short links were overrepresented in 2-chains. As required from the triangle inequality, mean L_upper + L_lower across all 2-chains always exceeded mean 2-span.

Table 1.

Statistics of links, tritrophic interactions, chains, and webs

Statistic	TL 1984	TL 1986	Ythan
Links and tritrophic interactions
Mean link length	6.33	5.90	7.29
Mean Lupper	5.41	3.43	5.06
Mean Llower	5.99	5.69	6.15
2 × mean link length	12.67	11.79	14.57
Mean 2-span	11.02	8.65	10.51
Mean Lupper+ Llower	11.40	9.12	11.20
2 × mean link length/mean 2-span	1.15	1.36	1.39
Mean Lupper + Llower/mean 2-span	1.03	1.05	1.07
Community span < mean(count chain length) × mean(link length)
Mean count chain length	4.84	4.84	4.43
Mean count chain length × mean link length	30.62	28.56	32.31
Community span	20.78	22.66	21.98
Mean count chain length × mean link length/community span	1.47	1.26	1.47
Wiggling of chains
Mean sum chain lengths	19.96	23.33	16.88
Mean chain span	18.71	20.62	13.18
Mean chain span/community span	0.90	0.91	0.60
Mean sum chain lengths/mean chain span	1.07	1.13	1.28
Mean sum chain lengths/community span	0.96	1.03	0.77
Connectance
No. of noncannibalistic links	264	236	379
(No. of taxa)2	2500	2601	8281
No. of trophic links/(number of taxa)2	0.1056	0.0907	0.0458
No. of trophic links/taxa	5.28	4.63	4.16

TL, Tuesday Lake.

In all three webs, mean(L_lower) > mean(L_upper). Histograms of L_upper and L_lower revealed major differences between these two distributions in Tuesday Lake, but not in Ythan Estuary (Fig. S1). In Ythan Estuary, L_upper and L_lower distributions were unimodal and similar in shape and location to each other. Tuesday Lake L_lower distributions were not radically different from that of Ythan Estuary, but Tuesday Lake L_upper distributions were strongly bimodal. Upper links in Tuesday Lake were either very short or very long, for good biological reasons: in (log(M), log(N)) space, zooplankton were close to each other, so upper links where zooplankton ate zooplankton were short; fish were far from zooplankton, so fish eating zooplankton formed long upper links.

Tritrophic Interactions: Bivariate Distributions, Upper Angle Versus Lower Angle.

A_upper was plotted against A_lower and the coordinate plane was divided into quadrants (Methods). The general but not universal tendency was for data to fall preferentially in the upper left and lower right quadrants, compared with the other two quadrants (Fig. 1 D–F). Data always showed a horseshoe pattern. A_upper and A_lower were not independent. A lower angle less than the median or allometric angle was, on average, followed by an upper angle greater than the median or allometric angle (Fig. 4A), and vice versa (Fig. 4B).

Fig. 4.

Upper and lower angle correlated negatively. A lower angle less than the median angle, which was approximately −40° in Tuesday Lake and approximately −52° in Ythan Estuary (for a very rough overall central tendency around −45°), was often followed by an upper angle greater than the median angle (A), and vice versa (B).

The dependence of link angles was partly a consequence of M and N distributions, because the triples reproduced horseshoe patterns in A_upper versus A_lower (Fig. S2) similar to those in webs (Fig. 1 D–F). However, A_lower and A_upper distributions differed between webs and triples nominally highly significantly (Kolmogorov–Smirnov test; see Between-angle, 2-span, A_upper and A_lower above).

Food Chains.

In general, short links were overrepresented in chains: In all three webs, the product of mean count chain length times mean link length exceeded community span (Table 1). On average, chains traversed much more of the range in (log(M), log(N)) space in Tuesday Lake than in Ythan Estuary: for Tuesday Lake, mean chain span was 0.90 (in 1984) and 0.91 (in 1986) times community span, whereas Ythan Estuary's mean chain span was 0.60 times community span.

Ythan Estuary chains wiggled slightly more than Tuesday Lake chains: mean sum chain lengths were 1.07 in 1984 and 1.13 in 1986 times mean chain spans in Tuesday Lake and 1.28 times mean chain spans in Ythan Estuary (Table 1). Wiggling compensated for the degree to which chains failed to span the community in Tuesday Lake, but not in Ythan Estuary: in Tuesday Lake, mean sum chain lengths were 0.96 times community span in 1984 and 1.03 times the community span in 1986, whereas in Ythan Estuary, mean sum chain lengths were only 0.77 times community span.

Discussion

In Tuesday Lake, Michigan, in 1984 and 1986, and Ythan Estuary, Scotland, the properties of links did not account for some properties of tritrophic interactions, and the properties of links and tritrophic interactions did not account for some properties of longer chains. Food chains did not account for some properties of webs. Studies of individual links (24, 26–28, 34, 35), tritrophic interactions (14, 16, 18, 38), and food chains remain valuable but insufficient to understand more complex trophic structures.

Comparing observed 2-chains with triples of taxa ordered by average body mass rejected the null hypothesis that 2-chains showed no structure other than that inherited from links, ordering by average body mass, and the distributions of M and N. For triples, the expected between-angle was close to zero, whereas the mean between-angles of 2-chains showed an average positive acceleration of biomass, population productivity and population consumption in Tuesday Lake and an average negative acceleration in Ythan Estuary.

Furthermore, links did not account fully for 2-chains because, in all three webs, mean(2-span) < mean(L_upper + L_lower) < 2 × mean link length (Table 1). The first inequality means that 2-chains wiggled so strongly that the intermediate taxon I fell outside the rectangle described by M_R ≤ M_I ≤ M_C and N_C ≤ N_I ≤ N_R. Strict inequality held in all three webs. A strong deviation in the direction of a lower link of a 2-chain from the central value of slope tended to be followed by a compensatory deviation in the direction of the adjacent upper link. The second inequality, mean(L_upper + L_lower) < 2 × mean link length, arose because, in all three webs, both lower and upper links in 2-chains were on average shorter than the mean link length. This unanticipated finding indicated that links that occurred within fewer 2-chains were longer than those that occurred in more 2-chains. (We clarify this point. Each link occurred in some number of 2-chains. That number may, a priori, be zero, if the link occurred in no 2-chains, or one 2-chain, or two, etc. Some links occurred in fewer 2-chains than other links. The finding suggests that links that occurred in fewer 2-chains had greater length than links that occurred in relatively more 2-chains.) An unusually long link was found in fewer 2-chains than a short link because a long link had a consumer many orders of magnitude larger and rarer than its resource. Consequently, consumer and resource of that long link were near the outer extremes of the range of average body size and population density, and fewer other taxa were available to form further links.

Despite a 50% turnover of taxa between 1984 and 1986, the mean between-angle remained positive in Tuesday Lake, in contrast to the negative mean between-angle in Ythan Estuary, a qualitatively different ecosystem with respect to type of habitat, size, included taxa, and data resolution. More empirical M,N-webs are needed to examine consistencies or differences in 2-chain structure in relation to habitat.

Links and 2-chains did not account for all properties of chains. For example (Table 1), mean count chain length × mean link length was longer than the community span by 26–47%. These consistent differences challenged the guess of Reuman and Cohen (27) that mean link length approximately equals community span divided by mean count chain length, although the discrepancy is less than an order of magnitude. In addition, mean sum chain lengths exceeded mean chain span by 7–28%. Such inequalities reflected both wiggling and underrepresentation of very long links in chains.

Some differences between webs went beyond what could readily be explained by the differences between communities in links or 2-chains. For example, 2 × mean link length was longer than mean 2-span by 15–39% (Table 1), so the excess of 2 × mean link length over mean 2-span accounted for much or all of the excess of mean count chain length × mean link length over community span (26–47%). By contrast, in 2-chains, mean(L_upper + L_lower) (the 2-chain version of mean sum chain length) exceeded mean 2-span by 3%, 5%, and 7%, respectively, perhaps not enough to account for the excess (7–28%, Table 1) of mean sum chain lengths over mean chain span.

An unexpected, striking difference between Tuesday Lake and Ythan Estuary was that in Tuesday Lake, mean chain spans covered 90–91% of the respective community spans, whereas in Ythan Estuary, mean chain spans covered 60% of the community span (Table 1). Similarly, the Tuesday Lake mean sum chain lengths were within 4% of the community span, whereas in Ythan Estuary, the mean sum chain length was 77% of the community span. Several factors might contribute to these findings. First, the connectance (links/taxa²) of Ythan Estuary was approximately half that of either Tuesday Lake web. With fewer links per pair of taxa, mean chain span in Ythan Estuary might be expected to be relatively smaller than community span. This factor may not be the whole story because the number of noncannibalistic links per taxon of Tuesday Lake in 1986 (namely, 4.63 links per taxon) was intermediate between that of Tuesday Lake in 1984 (5.28) and that of Ythan Estuary (4.16), whereas Tuesday Lake was similar in 1984 and 1986 in the characteristics reviewed above. Another contributing factor may be the clear segregation by average body mass among the three classes of basal, intermediate and top taxa in Tuesday Lake (Fig. 1 A and B), contrasted to the much greater intermingling of these three groups with respect to body mass in Ythan Estuary (Fig. 1 C). It seems difficult to deduce this difference between webs from the differences at the level of single links or 2-chains. Furthermore, community span is sensitive to investigators' choices about which taxa to include in the web at the extremes of body size. Such choices may contribute to the differences involving community span between the Tuesday Lake and Ythan Estuary webs.

When likelihood methods (36) and foraging models (37, 38) for the topology of webs are extended to consider data on body mass and population density such as were analyzed here, they will offer additional modeling approaches that may help explain some of the patterns we report. One optimal foraging model (38) used data on body masses and population densities (allometrically predicted from body masses) to predict correctly 5–65% of links in 15 webs. It remains to be seen whether, in general, using measured population densities would improve the predictive ability of the optimal foraging model and whether a refinement of it can reproduce the empirical patterns we report.

Three ecological conclusions from this analysis are, first, that models that treat links as independent will not accurately represent tritrophic interactions or longer chains. This finding is consistent with the observation in a marine web (14) that strong interactions occur in both links of a 2-chain less often than expected by chance. Second, models that splice overlapping tritrophic interactions to compose chains will fail to reproduce some properties of chains. Third, webs differ in the acceleration of biomass, population productivity and population consumption along typical chains. The reasons for such differences remain to be determined. The multiple levels of structure and heterogeneity among webs observed here constrain and may guide future development of models of M,N-web structure.

Materials and Methods

Data.

Webs and M,N-data of Tuesday Lake, Michigan (28) and Ythan Estuary, Scotland (from David G. Raffaelli) are in Dataset S1. In Tuesday Lake, all three fish species in 1984 were removed and replaced by a different fish species in 1985. Data from 1984 and 1986 were analyzed separately. The main connected components of the Tuesday Lake webs had 50 species in 1984 and 51 species in 1986. The Ythan Estuary web had 91 taxa: one mammal, 26 birds, 18 fish, 44 invertebrates, phytoplankton, and macroalgae. Of these, 73 were species; most remaining taxa were genera.

Methods.

In bivariate distributions of y versus x where an approximate linear relationship between y and x values appeared plausible (e.g., Fig. 1 A–C), ordinary-least-squares regression was performed by using Matlab regress (results are in Table S2). The complete set of standard assumptions of linear models was not tested because independence assumptions of the models were already known to be violated. Bivariate distributions were compared visually.

The center of the distribution of angles of all links in a web was measured by the median [denoted median(A)] and by the allometric angle. Mean angle was not used because angle distributions were not symmetric. Median upper (lower) angle [i.e., median(A_upper), median(A_lower)] was used as a central value for the distribution of upper (lower) angles.

For each web, A_upper was plotted against A_lower for all 2-chains. Plots were divided into quadrants in three ways: with a horizontal line at median(A) and a vertical line at median(A); with a horizontal line at the allometric angle and a vertical line at the allometric angle; and with a horizontal line at median(A_upper) and a vertical line at median(A_lower). Data points in each of the four quadrants were counted for each method of defining quadrants. When points lay directly on quadrant dividing lines, the points were omitted from quadrant counts.