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## A robust measure of food web intervality

Venue: | Proc. Nat. Acad. Sci |

Citations: | 16 - 3 self |

### BibTeX

@ARTICLE{Stouffer_arobust,

author = {Daniel B Stouffer and Juan Camacho and Luís A Nunes Amaral},

title = {A robust measure of food web intervality},

journal = {Proc. Nat. Acad. Sci},

year = {},

pages = {19015--19020}

}

### OpenURL

### Abstract

We introduce a mathematically robust measure for food web intervality. Intervality of a food web is related to the number of trophic dimensions characterizing the niches in a community. We aim to determine the minimum number of variables required to describe the factors that influence the trophic organization of the species in a community. We find that empirical food webs are not interval in the strictest sense of the definition. However, upon comparison to suitable null hypotheses, we conclude that empirical food webs exhibit a strong bias toward contiguity of prey, that is, toward intervality. Indeed, we demonstrate that species and their diets can be mapped onto a single dimension, an insight that must guide ongoing efforts to develop dynamical models of ecosystems. Introduction In spite of their "baroque" complexity, the structure of natural food webs displays a number of remarkably simple regularities 1 Stouffer et al. 10 demonstrated that these three models share two fundamental mechanisms which account for the models' success in reproducing the empirical patterns: (i) Species form a totally ordered set in niche space, that is, species can be ordered along a single niche dimension; (ii) Each species has an exponentially-decaying probability of preying on a given fraction of the species with lower niche values In spite of the above similarities, the models do differ in significant ways. An important difference concerns how species' prey are organized along the single dimension. In the niche model, species prey upon a contiguous range of prey. In the nested-hierarchy and generalized cascade models, in contrast, the diets are not restricted to a contiguous range. This difference implies that these models give rise to networks with different intervality The idea of intervality in complex food webs was introduced by Cohen 11 , who reported, as did subsequent studies Importantly, the degree of intervality of a food web is related to the number of trophic dimensions characterizing the possible niches in a community 12 . More specifically, one may ask what is the minimum number of variables required to describe the factors that influence the trophic organization of the species in a community? Is this number the same or different for different communities? Any departure from intervality has been understood to imply additional complexity in the mechanisms responsible for the structure of the food web. Recently, the number of higher quality food web data sets has been steadily increasing and these data have been the focus of a number of recent studies on food web structure In this manuscript we address the question of how "non-interval" empirical food webs truly are. To this end, we define a novel measure of intervality that is more robust than those already in the literature. Notably, we find that while empirical food webs are indeed non-interval, their degree of "intervality" can be understood as a perturbation on an underlying interval structure. Our results provide significant support to the conjecture that ecosystem niches may be mapped onto a single dimension. Food web intervality In the studies of Cohen 11, 12 , Cohen et al. The second measure, Cy 4 , is defined as the number of chord-less cycles of length four in the consumer overlap graph. In the consumer overlap graph, two consumers are connected if they share at least one prey. An example of a cycle of length four would be when both species A and D share prey with species B and C. This is a cycle because it is possible to travel from any one of the four species to any other in the consumer overlap graph. If species A and D do not share any prey, or similarly species B and C do not, this cycle is chord-less and the four diets cannot be made contiguous simultaneously. Therefore, an interval food web will have no chordless cycles in the consumer overlap graph 12 . Using these two measures, Cattin et al. An irreducible gap can occur in graphs with as few as three multiphagous consumers. However, by concentrating on species triplets, one will inflate the resulting measure and will not be able to compare In contrast to previous studies, we determine here the intervality of an entire food web. To do this, we first find the order of species in the food web in such a way as to generate the "most interval" ordering of the food web. This process yields the best approximation to a food web where the species are organized along a single dimension. There are various related means by which one could define "most interval," so we discuss our definition and its justification in detail. In the idealized case of a fully interval food web, each consumer's diet is represented by a single contiguous range. If we consider a non-interval food web and attempt to reproduce the idealized web as closely as possible, we will want all prey of a given predator to "appear" as close together as possible on the resource axis ( For a food web graph F with S species, there are S! possible species orderings O k (F) = s k 1 s k 2 . . . s k S , with k = 1, . . . , S! . Because of the large number of possible permutations, it is computationally unfeasible to determine the best ordering through enumeration. It is for this reason that we employ simulated annealing, a heuristic technique which significantly reduces the computational effort required to find an optimal or close-to-optimal solution (see Methods and Kirkpatrick et al. 23 for details). When attempting to find the most interval ordering, the objective is to minimize the discontinuity of all predators' prey ( Here n i is the number of gaps in the diet of species i and g k ij is the number of species in the j-th gap in the diet of species i for O k (F). Simulated annealing yields an estimateĜ for the total number of gaps } of the food web 1 . The smaller G is the more interval the food web is. 1 Note that we use G to refer to the actual minimum number of gaps for a the most interval ordering of a food web, whereas 4 Null hypotheses for food web intervality As happens in other graph and combinatorial problems the value ofĜ is of little interest 24 ; rather, one needs to assess whether the measured value ofĜ is significantly different from the expect value for specific types of graphs. To solve this problem, one must determine the expected value ofĜ under suitable null hypotheses. We have designed three complementary null hypotheses which place different restrictions upon how consumers' diets may be organized within a food web. Our first null hypothesis is a randomized version of the empirical food web. We perform this randomization using the Markov-chain Monte Carlo switching algorithm 25 and treat single, double, and cannibal links separately (see Methods for details). The randomized empirical food web stands as a food web graph with no constraints placed upon consumers' diets. That is, in the randomization there is no correlation between the prey of a given species and their organization on the resource axis. We therefore expect thatĜ for these randomized food webs will be maximal. Comparison to this null hypothesis thus provides verification of whether there are any structural regularities in the organization of species' diets within empirical food webs. Our second null hypothesis is food webs generated by the generalized cascade model with the same number of species S and linkage density z as the empirical food webs. Whereas randomization of the empirical food webs imposed no structural constraints upon consumers' diets, the generalized cascade model does. Each predator may again select their prey at random, but instead of from the entire resource axis, their selections are restricted to only those species with niche values less than or equal to their own. This mechanism leads to a smaller number of gaps for species placed lower on the resource axis. Comparison of the empirical data to this null hypothesis will provide evidence as to whether empirically observed diets exhibit additional structural constraints. To this point, our null hypotheses will provide an indication of whether empirical food webs have a larger number of gaps than would be expected for random structures with no bias toward contiguity of prey. In order to quantify any bias toward contiguity of prey in empirical food webs, we provide a third null hypothesis based upon a generalization of the niche model of Williams & Martinez 2 . Let us first recall the definition of the niche model. Each of the S species i are assigned a niche value n i drawn from a uniform distribution in the interval [0, 1]. A predator j in the niche model preys on â G refers to the estimate obtained with simulated annealing. The only case when we can be sure thatĜ = G is whenĜ = 0. 5 range r j of the resource axis; r j = n j x, where x is drawn from a beta-distribution p(x) = β(1− x) (β−1) . Here β = S 2 /2L − 1 and L is the number of trophic links in the ecosystem. The center of the range r j is selected uniformly at random in the interval [r j /2, n]. All species whose niche values n i fall within this range are considered prey of species j. To allow for a tunable bias toward prey contiguity, we generalize the niche model in the following manner. First, we reduce the range r j for a predator j to r j = c r j = c n j x, where c is a fixed parameter in the interval [0, 1]. Because species are distributed uniformly on the resource axis, a predator j with range r j has on average r j S prey. The same applies to the reduced range r j , and therefore a predator has ∆k = r j − r j S = (1 − c) j i S expected prey unaccounted for. Next, we select these ∆k preyrounded to the nearest integer value-randomly from species i with niche value n i ≤ n j that are not already a prey of species j. If c = 0 there is no pressure for contiguity (the web is strictly non-interval), while for c = 1 we recover the niche model (the web is thus fully interval). Empirical results We study 15 empirical food webs from a variety of environments: three estuarine-Chesapeake Bay We then want to be able to estimate the probability that the valueĜ e appears given each null hypothesis. To do this, we examine not just the mean ofĜ model , but the probability distribution. We employ the Kolmogorov-Smirnov test 6 We first compare the set of empirical food webs {F} to the set of randomized food webs {F R } For the remaining four food webs, the largest probability is p R = 4.4 × 10 −4 . We now compare the set of empirical food webs to the set of generalized cascade model-generated food webs {F GC } for Skipwith Pond, Coachella Valley, and Caribbean Reef, respectively-are likely due to their large directed connectances (defined as L/S 2 ) 2 . To this point, our results provide an indication that empirical food webs are significantly more interval than would be expected for food webs with no bias toward prey contiguity. We now investigate our generalized niche model to determine how it compares to the empirical data for different values of c and therefore different levels of bias toward prey contiguity. The same considerations for applicability that were discussed for the generalized cascade model hold for the generalized niche model 10 . Because of the computational effort required, we have selected only six of the eleven food webs to compare to this null model. They are Benguela, Bridge Brook Lake, Chesapeake Bay, Coachella Valley, Skipwith Pond, and St. Marks. It is worth noting that this list includes two of the three food webs with p GC > 0.05: Coachella Valley and Skipwith Pond. For each of these six food webs, we compare the empirical food webĜ e to the model Ĝ GN for c ∈ [0.5, 1.0]. We compare the model and empirical data as before, but focus particularly upon the zscore, where z =Ĝ e− Ĝ model σĜ model . Using the z-score, we can determine 95% confidence intervals on the value of c for which the empiricalĜ is likely to be observed in the generalized niche model For the six food webs we investigated, we find that the largest values of c which provide statistical 2 For densely connected food webs, predators typically have greater numbers of prey. Because these prey are constrained to have a niche value less than or equal to the predators, the greater the directed connectance the greater the probability that these prey are contiguous, despite the random predation. 7 agreement with the empirical data are remarkably close to one, 0.85 < c max < 1.00 3 . This finding enables us to quantify in a statistically sound manner the intervality of a food web; specifically, where {F GN } is the ensemble to model food webs generated according to the generalized niche model and with the same number of species and connectance of the real food web F i . Our empirical finding that I for the six empirical food webs considered is so large indicates that natural ecosystems are significantly interval and consequently there is a strong bias toward contiguity in prey selection. Discussion The concept of "niche theory" or "niche space" is a fundamental concept in study of ecosystems. Niche space was classically defined as an "n-dimensional hyperspace" with n given by the innumerable ecological and environmental characteristics The more recent "interpretation" of niche theory, however, relates to the niche providing species an ordering or hierarchy Recently, however, discussions as to how interval food webs truly are, were renewed by the stark contrasts between the niche model-and its contiguous range of prey-and the generalized cascade and nested-hierarchy models-and their random predation 10 . Our results allow us to conclusively demonstrate that natural ecosystems, while not fully interval, are significantly more interval than would be expected when compared to suitable random null hypotheses. Moreover, we find the empirical food webs to be statistically indistinguishable from model food webs whose diets are, on average, at a minimum 85% contiguous. The idea that species and their diets can be so closely mapped to a single dimension 3 It should be noted that our results may exhibit some under-estimation of c, in particular as noted earlier for densely connected food webs such as Coachella Valley and Skipwith Pond. 8 represents a tremendous insight that can guide us on how best to go about developing dynamic ecosystem models. A number of future questions must be answered before the question of food web intervality can come to a close. First and foremost is getting a better understanding of exactly what processes are behind the deviations from truly interval behavior. While some of the gaps within species diets may be due to interactions not observed during field sampling, we find it unlikely that all gaps may be attributed to this factor. It has earlier been noted, albeit on different food webs from those studied here, that ecosystems with multiple habitats, for example an estuary, were less likely to be interval than single-habitat food webs 1,11 . Indeed, once cannot expect food webs containing several habitats to be interval since each habitat will have its own independent resource axis. It would likewise be very interesting to examine additional properties of the "most-interval" ordering or orderings, {O k }. Studies which compared these orderings to those obtained when comparing species' masses, or related properties 6 , would be particularly intriguing. It is known empirically, for example, that as predator mass increases so does average prey mass Methods Simulated annealing Simulated annealing is a stochastic optimization technique that enables one to find a "low-cost" configuration while still broadly exploring the space of possibilities 23 . This is achieved by introducing a computational "temperature" T . When T is high, the system can explore configurations of high cost whereas at low T the system can only explore low-cost regions. By starting at high T and slowly decreasing T , the system descends gradually toward deep minima. For each iteration in the simulated annealing algorithm, we attempt to swap the position of two randomly selected species to go from the initial ordering O i (F) to the proposed ordering O f (F). This 9 updated ordering O f (F) is then accepted with probability where G (O f ) is the cost after the update and G (O i ) is the cost before the update. For each value of T , we attempt qS 2 random swaps with q ≥ 250. After the movements are evaluated at a certain T , the system is "cooled down" to T = cT , with c = 0.99. Generating randomized networks To generate an ensemble of random networks, one must first define the constraints of the randomiza-