Foraging biology predicts food web complexity
- Department of Animal and Plant Sciences, University of Sheffield, Western Bank, Sheffield S10 2TN, United Kingdom
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Edited by G. David Tilman, University of Minnesota, St. Paul, MN, and approved July 17, 2006 (received for review April 14, 2006)
Abstract
Food webs, the networks of feeding links between species, are central to our understanding of ecosystem structure, stability, and function. One of the key aspects of food web structure is complexity, or connectance, the number of links expressed as a proportion of the total possible number of links. Connectance (complexity) is linked to the stability of webs and is a key parameter in recent models of other aspects of web structure. However, there is still no fundamental biological explanation for connectance in food webs. Here, we propose that constraints on diet breadth, driven by optimal foraging, provide such an explanation. We show that a simple diet breadth model predicts highly constrained values of connectance as an emergent consequence of individual foraging behavior. When combined with features of real food web data, such as taxonomic and trophic aggregation and cumulative sampling of diets, the model predicts well the levels of connectance and scaling of connectance with species richness, seen in real food webs. This result is a previously undescribed synthesis of foraging theory and food web theory, in which network properties emerge from the behavior of individuals and, as such, provides a mechanistic explanation of connectance currently lacking in food web models.
Food webs are networks of feeding links between species and are central to our understanding of ecosystem structure, stability, and function (1–4). Recent models of food web structure [e.g., the cascade (5), niche (6), and phylogenetic (7) models] have predicted key topological properties of food webs by relying on two empirical measures of complexity, connectance and species richness. Connectance, the number of actual links (L) expressed as a proportion of the total number of possible feeding links (S 2 in a web of S species), typically lies between 0.02 and 0.4 in empirical data (8). By using these empirical values of connectance in simple, phenomenological sampling algorithms, these models can successfully reproduce many topological properties of real food webs. However, despite the central importance of connectance to the outcome of these models (9), connectance, or the complexity of food webs, itself has no accepted mechanistic explanation (1, 2, 5, 10–12).
Historically, hypotheses about the complexity of food webs have been linked to the stability of dynamical processes among
species. The possible instability of highly connected food webs remains the dominant explanation for limits on complexity
(1, 3, 10, 11, 13–15). We propose an alternative explanation that emerges from considering that species’ foraging in food webs is determined by
behavioral and morphological characteristics shaped over both ecological and evolutionary time. Links in a food web represent
a map of the foraging decisions being made by consumers in the specific context of that web. It follows that complexity, when
expressed as the number of links in a food web, is a function of the foraging biology of the species in the web (11, 12, 16, 17). More explicitly, if we assume that the diet breadth dj of a species j is the number of different prey species it feeds on, then the total number of links in a web (L) is:
and connectance (C) is:
or the mean proportional diet breadth of all species in the web (16, 17). If the diet breadth of individuals in a food web is determined by their foraging biology, then connectance is an emergent
consequence of individual consumer and resource traits (12, 16, 17).
Here we evaluate this idea by developing a quantitative model of connectance that uses foraging theory to predict diet breadth. We demonstrate that, when parameterized with foraging data from independent studies of feeding, our model predicts highly constrained diet breadth and, hence, connectance. The derivation of this key parameter of current food web models as an emergent consequence of well established individual-level biological traits represents a major step toward developing a mechanistic understanding of food web structure.
Because connectance is an integrated measure of species’ diet breadths across a web, we take as our starting point the problem
of modeling diet breadth for a single consumer. The contingency model of optimal foraging is a simple energy rate-maximizing
foraging model that is widely used to model prey choice by, and the diet breadth of, individual consumers (18). The model predicts the instantaneous diet of a consumer from three variables that characterize individuals of prey species
i and consumer species j: the net energy Ei gained by consumption of an individual of species i, the encounter rate λij at which individuals of species i are encountered by individuals of species j, and the handling time Hij spent by an individual of species j attacking species i. The model assumes that the most profitable species is always consumed (18) (i.e., a consumer has a minimum diet breadth of 1). Profitability is the rate of energy gain for consumer j eating an individual of prey species i (Pij = Ei/Hij). For each predator, prey species are first ranked by decreasing profitability (i.e., Pij > Pi+1j for i = 1 to S), and predator diet breadth is the value of k (i.e., number of prey added in order of profitability) that maximizes:
the rate of energy intake.
We use the contingency model to predict mean diet breadth for S species in a food web. Our diet breadth model (DBM) proceeds by choosing a level of species richness. Subsequently, each species is treated in turn as a consumer, with S − 1 species acting as resources. The network of feeding interactions in the model web is created by successively applying the optimal foraging model to each species as a consumer. In doing so, we are effectively modeling feeding among a set of carnivores (because each species can act as both consumer and resource). We therefore restrict our analyses to the animal portion of real food webs. To produce a quantitative prediction of diet breadth and ultimately connectance, we first parameterize the model by using independent foraging trait data and then address two known disparities between the level of resolution at which the model is specified and that of real food web data: aggregation across individuals, life stages, and taxa, and cumulative sampling in space and time. These disparities, although well known, are rarely quantified but must be taken into account in any comparison between model and data. We derive the ecological rationale and quantitative values for these parameters from aggregation in real food web data and from basic principles of cumulative sampling (see Methods for details on parameterization, aggregation, and cumulative sampling).
Results and Discussion
The DBM, using the empirical parameterization and incorporating both aggregation and cumulative sampling, predicts diet breadths that are very similar to those in real food webs (see Fig. 1 a and b and description of statistical tests in legend). This match between the DBM and real food webs is encouraging because without the empirically derived constraint on foraging, the DBM is theoretically capable of generating diet breadths anywhere between 1 and S species (18) (Fig. 1, light gray dots). The relative importance of foraging trait data to aggregation and cumulative sampling is seen in Fig. 1. Without aggregation or cumulative sampling, the model parameterized with only foraging trait data predicts highly constrained but lower levels of diet breadth than in real food web data (Fig. 1 b and c).
Predicted (a and c) and observed (b) diet breadths. (a) Predicted mean diet breadth of food webs from the parameterized DBM with 70% aggregation and 64 cumulative samples results in highly constrained diet breadths (black ringed circles). These values are not significantly different from the mean diet breadth of carnivores in 13 real food webs (b) (t test of predicted and observed diet breadths: mean observed = 9.7; mean predicted = 10.3; t = 0.36, df = 12, P = 0.72). Without parameterization (a; light gray circles), mean diet breadths can vary greatly. Diet breadth is influenced by the product λ̄; hence, the x axis displays this product. Different asymptotes of diet breadth at low values of λ̄ result from differences in species richness in the real food webs modeled (light gray circles). (b) Real food webs. (c) Identical to a except for the absence of aggregation and cumulative samples. Here the parameterized DBM generates constrained diet breadths (black ringed circles) that are significantly lower than observed in real food webs (b) (t test of predicted and observed diet breadths: mean observed = 9.7; mean predicted = 3.8; t = 3.00, df = 12.0, P = 0.01).
Predicting connectance of the animal portion of real food webs requires one further modification of the DBM beyond parameterization and accounting for aggregation and cumulative sampling. The animal portion of real food webs may contain both detritivores and herbivores, which are resources for, but not consumers of, other animals. We first classified entities in the real webs as detritus, autotroph, detritivore, herbivore, carnivore, or parasitoid. Then, to compare effectively connectances predicted by the DBM (carnivores) and values found in real food web data, we set equivalent proportions of the species in the DBM webs to be herbivores/detritivores, that is, to have a diet breadth of zero.
The DBM predicts well the values of connectance observed in real food webs (R 2 = 0.64; C real = 0.01–0.38 vs. C predicted = 0.01–0.4; Fig. 2 a). As expected, the matches to specific webs are better in some cases than others. However, considering the simplicity of the model and its independence from the real food web data, the correspondence is very good.
Predicting the connectance of 13 real food webs. (a) Including aggregation (70%), proportion of herbivores (web-specific), and cumulative sampling (64 samples) results in accurate prediction of observed levels of connectance in the DBM. Slope does not differ from 1 with reduced major axis regression (19), the appropriate test when there is measurement error on both the x and y axes. Reduced major axis (RMA) regression: slope = 0.71; R 2 = 0.64; t test for difference from slope of 1: t = 1.8, df = 10.4, P = 0.09. Each predicted value is the mean of 100 webs ± 2 SD. Handling times, densities, attack rates, and energy values are drawn from the empirical data. The dotted line (for reference) has intercept 0 and a slope of 1. (b) Predicted connectance is lower than observed at higher levels of connectance if aggregation and temporal scale are not included in the model. RMA regression: slope = 0.21; R 2 = 0.32; t test for difference from slope of 1: t = 6.3, df = 10, P < 0.001.
These results suggest that the constraint on diet breadth produced by optimal foraging biology provides a plausible, quantitative, and mechanistic explanation for the level of complexity in natural food webs. These realistic predictions result from several properties of foraging biology. First, diet breadth depends strongly on the dimensionless product of the mean encounter rate and the mean handling times (λ̄; Fig. 1), the empirical values of which strongly constrain predicted diet breadth (Fig. 1 c) and therefore connectance. Relative differences among the energy value of prey items (Ei) also influences diet breadth. If the most energy-rich resource is very rich relative to the second most rich, the contingency model predicts a small diet breadth (diet breadth tends toward 1). The other extreme occurs if all energy values are the same, and diet breadth reaches a maximum determined purely by λ̄. Second, diet breadth is affected by aggregation and cumulative sampling. The inclusion of aggregation and cumulative sampling in the DBM increases the variance in diet breadth predicted by optimal foraging alone (Fig. 1 a vs. c) and consequently adjusts predictions about connectance (see Fig. 3 for details). The result is an increased correspondence between model and real food webs (Figs. 1–3) driven primarily by constraints on optimal foraging and modified by aggregation and cumulative sampling.
Determinants of connectance in the DBM. Aggregation of species into trophospecies increases connectance (the dashed line is for when the number of trophospecies is 70% of the original number of species). Longer time periods over which diets are accumulated increase connectance (the dotted line is for 64 time steps). Assuming a certain proportion of herbivores, which by definition do not feed on animals, decreases connectance of the animal part of the food web (the dash–dot line is for 40% herbivores). Regardless of aggregation, cumulative sampling, and herbivore proportion, connectance declines with S (see Results and Discussion). Each line is the mean connectance of 100 model food webs at species richness S = 20–200 in steps of 10. Connectance for the 13 food webs examined in Figs. 1 and 2 are overlayed for reference.
The DBM also reproduces one of the most widely discussed patterns in food web structure, a decline in connectance with increasing species richness (2, 10–12, 20–22) (Fig. 3). The relationship between log(L) and log(S) (which avoids the problem of nonindependence in the C vs. S relationship) has a slope between 1.2 and 1.3, which lies intermediate to predictions from the fixed diet breadth hypothesis (slope = 1) (2) and the constant connectance hypothesis (slope = 2) (12, 16, 22). This pattern matches that found in real food web data (21). In our model, this negative relationship between C and S cannot be attributed to sampling bias or to stability constraints, two other explanations offered for the pattern (2, 11, 12).
Our model presents a previously undescribed synthesis of foraging behavior and food web theory in which network (food web) properties emerge from the behavior of individuals (nodes). The scaling approach of the DBM, where species (nodes) possess optimization rules (foraging biology) from which food web (network) complexity emerges, has potential application to understanding emergent properties in many types of networks. Furthermore, this previously undescribed approach to modeling food webs can be extended in a variety of ways to address a range of important ecological questions as follows:
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(i) We could include more complex trophic structure. The DBM currently predicts the diets of carnivores only and uses a very general set of constraints on foraging (18). More detailed or system-specific versions of the DBM can be developed in at least three ways. First, a priori structural constraints can be added to the DBM by including more trophic levels and basal resources. Second, foraging models specific to plants, parasites, and herbivores (23) can be included to allow exploration of a broader set of physiological constraints on foraging. Finally, distributions of parameters such as handling time and encounter rate may differ systematically between systems dominated by different types of forager. We do not expect these more specific models to alter the efficacy of the DBM, but they would enable it to make predictions that are testable by system-specific comparisons.
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(ii) The foraging traits of individuals can be linked to their body size. Here, optimal foraging produces constraints not just on numbers of links but also their pattern, particularly if combined with body size distributions. As a consequence of the allometric scaling of foraging behavior with body size (24), we hypothesize that other aspects of web structure, such as feeding hierarchies, will emerge from the DBM (O.L.P., A.P.B., and P.H.W., unpublished data).
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(iii) As species are added to or removed from communities, foraging models can reflect dynamic changes in diet breadth and hence connectance. This finding offers the potential for insight into how food web structure will respond to patterns of species turnover that may result from species invasion and extinction.
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(iv) Some models of food web dynamics now include adaptive foraging as a property of established links (25–27). Our DBM provides the opportunity to further examine the consequence of optimal foraging for the establishment of links and subsequent population dynamics and stability.
Methods
DBM Parameterization.
The DBM requires four parameters: species richness of the food web, encounter rates, handling times, and energetic value of resources. Species richness was taken from real food webs. We gathered estimates of attack rate (Aij), density (Ni), and handling time (Hij) from 123 published type 2 functional responses across a wide range of taxa (data available from the authors upon request). Encounter rate (λij) is the product of attack rate and density, λij = AijNi, where Aij is the attack rate of a consumer on species i and Ni is the density of species i (28). Density Ni was set at half the log-transformed maximum density used in the functional response study to approximate field density. We assume that naturally, densities are approximately log-normal in their distribution. Half the maximum log density (i.e., the midpoint of a symmetrical distribution of log-density that varies between 0 and the maximum) should reflect such a range of densities effectively. Moreover, because we sample from a distribution of half-maximum log densities taken from 123 different organisms, for any given species the distribution of resource densities will be quite large. These data were converted to standard units of meters squared (m2) per second (m2/s) or meters cubed per second (m3/s) (Aij), per m2 or per m3 (Ni), and seconds (Hij), although this particular choice of units does not affect diet breadth in the model.
The empirical energy distribution from which to sample in our simulations was estimated from four food webs (Broadstone Stream, U.K.; Eastern Wedell Sea Shelf, Antarctica; Grand Caricaie marsh, Switzerland; and Tuesday Lake, U.S.) (29), where body size M follows an approximately log-normal distribution with standard deviation in log body size between one and five. We assumed that energetic content Ei scales linearly with body mass, Ei = eMi. The diet breadth predicted by the contingency model of optimal foraging and the results of all our simulations are insensitive to the value of e. This result occurs because diet breadth depends on the maximum rate of energy intake and not absolute values of energy intake. A corollary is that diet breadth is insensitive to the units in which energy is measured.
Aggregation.
The “species” documented in food webs are often aggregations across more than one taxonomic species, life stage, or size of individuals. Such aggregation is commonly based on trophic similarity with the resulting aggregations being termed “trophic species” (2, 30). Feeding links specific to any of the component parts of such aggregated entities are attributed to the entity as a whole. The DBM, on the other hand, assumes that all individuals in a species are identical, i.e., subject to the same constraints on foraging, and that diet breadth is determined on a time scale during which parameters remain constant (i.e., fixed prey density = fixed encounter rate). We quantitatively accommodate this difference between real data and the DBM by aggregating species in much the same way as in real web data.
To create an aggregated food web containing S trophic species from a food web that contains a greater number of taxonomic species, we used the procedure of Martinez (31). Species with similar diets were clustered into trophic species by hierarchical agglomerative clustering using the average linkage method according to their trophic similarity, measured with the Jaccard index. Two trophic species were linked by a feeding interaction if at least one of the species in one of the trophic species fed upon one of the species in the other trophic species. Thus, as in real food web data, a link between two species does not necessarily imply a link common to all consumer, or resource, individuals concerned.
Appropriate, empirically justified levels of aggregation were estimated in two ways. First, across the animal portion of all of the real webs, potential taxonomic aggregation was estimated by (i) adding extra species where these were explicitly indicated in a particular taxon, or (ii) adding one additional species for each entity in the web that was not explicitly represented as a single species (e.g., a species represented as genus or above) (32). This method suggested that webs were aggregated to approximately three-quarters of their true size (mean = 72%, range = 44–100%). In these data, life-stage aggregation cannot be estimated, so even a perfect measure of taxonomic aggregation would represent an upper estimate. A second method was to estimate both types of aggregation from the Skipwith web (33), for which we have the necessary additional data. This process was done by applying the above method (which yields 91% taxonomic aggregation in this case) and also by taking account of the number of trophically distinct life stages of species recognized in the original time-specific data for this web (33) that were aggregated in the summary web. Taking this process into account increased aggregation to 69%.
Cumulative Sampling.
Diets in food web data derive from cumulative observation and sampling in both space and time over which abundances of resources vary. We accommodate this feature quantitatively in the DBM by accumulating diets over multiple model webs, in which only density is varied. At each species richness corresponding to a real web, we constructed 64 model webs in which the parameters S, Ai, and Hij were held constant, but densities (Ni) were varied. Densities were randomly sampled from the empirical distribution, and the final model web was represented by the cumulative diet breadth of the 64 model webs. In the absence of empirical data on levels of cumulative sampling in real webs, we examined the effect of increasing the number of samples (i.e., time) on cumulative diet breadth and connectance: They reach an approximate asymptote at 64 samples.
Real Food Web Data.
Thirteen food webs from Dunne et al. (8) were analyzed and used in comparisons with the DBM predictions. We excluded webs from Dunne et al. (8) that contained no interactions suitable for modeling by the contingency model of optimal foraging (for example, one web contained only herbivory and parasitism).
Acknowledgments
We thank Fred Scharf and Jennifer Dunne for providing data from their studies and Nick Worsfold, Edd Hammill, Dave Raffaelli, Oswald Schmitz, Dylan Childs, Ben Hatchwell, Jeremy Fox, Rhonda Snook, Jens Rolff, and two anonymous reviewers for helpful comments. A.P.B. was supported by Natural Environment Research Council (London, U.K.) Fellowship NER/I/S/2001/00779, and O.L.P. is a Royal Society University Research Fellow.
Footnotes
- *To whom correspondence should be addressed. E-mail: a.beckerman{at}sheffield.ac.uk
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Author contributions: A.P.B., O.L.P., and P.H.W. designed research, performed research, analyzed data, and wrote the paper.
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Conflict of interest statement: No conflicts declared.
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This paper was submitted directly (Track II) to the PNAS office.
- Abbreviation:
- DBM,
- diet breadth model.
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Freely available online through the PNAS open access option.
- © 2006 by The National Academy of Sciences of the USA
