letters to nature180 NATURE|VOL 404|9 MARCH 2000|www.nature.com18. Gilinsky, N. L. in Analytical Paleobiology (eds Gilinsky, N. L. & Signor, P. W.) 157±174 (ThePaleontological Society, Knoxville, Tennessee, 1991).19. Cleveland, W. S. & McGill, R. The many faces of a scatterplot. J. Am. Stat. Assoc. 79, 807±822 (1984).Supplementary information is available on Nature's World-Wide Web site(http://www.nature.com) or as paper copy from the London editorial of®ce of Nature.AcknowledgementsWe are indebted to the late J. Sepkoski for his fossil databases and his encouragement, andwe thank M. Foote and D. Erwin for comments on the manuscript. Our work wassupported by grants from the University of California and the NSF to J.W.K.Correspondence and requests for materials should be addressed to J.W.K.(e-mail: [email protected]).................................................................Simple rules yield complex food websRichard J. Williams & Neo D. MartinezRomberg Tiburon Center, Department of Biology, San Francisco State University,PO Box 855, Tiburon, California 94920, USA..............................................................................................................................................Several of the most ambitious theories in ecology1±14describe foodwebs that document the structure of strong and weak trophiclinks9that is responsible for ecological dynamics among diverseassemblages of species4,11±13. Early mechanism-based theoryasserted that food webs have little omnivory and several proper-ties that are independent of species richness1±4,6. This theory wasoverturned by empirical studies that found food webs to be muchmore complex5,7±9,14±18, but these studies did not provide mechan-istic explanations for the complexity9. Here we show that aremarkably simple model ®lls this scienti®c void by successfullypredicting key structural properties of the most complex andcomprehensive food webs in the primary literature. These proper-ties include the fractions of species at top, intermediate and basaltrophic levels, the means and variabilities of generality, vulner-ability and food-chain length, and the degrees of cannibalism,omnivory, looping and trophic similarity. Using only two empiri-cal parameters, species number and connectance, our `nichemodel' extends the existing `cascade model'3,19and improves its®t ten-fold by constraining species to consume a contiguoussequence of prey in a one-dimensional trophic niche20.We compare the abilities of two earlier models, the random andcascade models3,19, and our new niche model to predict a dozenproperties for each of seven food webs. The parameters of all modelsare set to synthesize webs with the empirically observed speciesnumber and connectance level. We compare model predictions withthe largest and highest-quality empirical food webs that includeautotrophs and were originally documented to study food webstructure comprehensively (Table 1). Three are from freshwaterhabitats: Skipwith Pond, Little Rock Lake and Bridge Brook Lake;two are from habitats at freshwater-marine interfaces: ChesapeakeBay and Ythan Estuary; and two are from terrestrial habitats:Coachella Valley and the island of St Martin.Throughout this work, `species' refers to trophic species, whichare functional groups of taxa that share the same predators and preyin a food web3. `Trophic species' is a widely accepted3,4,8,14,17,18andsometimes criticized convention5,14within structural food-webstudies that reduces methodological biases in the data3,4,8.Amatrix with S rows and columns represents a food web with Sspecies. Element aijis 1 if species j consumes species i and 0 if not.There are S2possible and L actual links. Directed connectance17(C)equals L/S2.In the random model3,19, any link among S species occurs with thesame probability (P) equal to C of the empirical web. This createswebs as free as possible from biological structuring while maintain-ing the observed S and C. The cascade model3,19assigns each speciesa random value drawn uniformly from the interval [0,1] and eachspecies has probability P =2CS/(S - 1) of consuming only specieswith values less than its own. This pecking order helps to explainspecies richness among trophic levels3but underestimates inter-speci®c trophic similarity19and overestimates food-chain lengthand number in larger webs3,18. The niche model (Fig. 1) similarlyassigns each species a randomly drawn `niche value'. The species arethen constrained to consume all prey species within one range ofvalues whose randomly chosen centre is less than the consumer'sniche value. The single range adds a previously discussed20com-munity-level contiguity of niche space to the cascade model bycausing species with similar niche values to share consumersfrequently within the community. The placement of the nichepartially relaxes the cascade hierarchy by allowing up to half aconsumer's range to include species with niche values higher thanthe consumer's value. All three models incorporate substantialstochastic variability along with dependence on S and C.Twelve properties of each empirical and model web are measured(see Methods):(i±iii) Species types1±8,14±18,21: the fractions of top (T, species with nopredators), intermediate (I, species with both predators and prey)and basal (B, species with no prey) species.(iv, v) The standard deviations (s.d.) of generality14(GenSD) andvulnerability14(VulSD) quantify the respective variabilities of spe-cies' normalized prey (Gi) and predator (Vi) counts:Gi1L=S^Sj1ajiVi1L=S^Sj1aijNormalizing with L/S makes s.d. comparable across different websby forcing mean Giand Vito equal 1.Table 1 Basic properties of empirical food websName Taxa SL/SC(L/S2)Skipwith Pond 35 25 7.9 0.32Little Rock Lake 181 92 10.8 0.12Bridge Brook Lake 75 25 4.3 0.17Chesapeake Bay 33 31 2.2 0.072Ythan Estuary 92 78 4.8 0.061Coachella Valley 30 29 9.0 0.31St Martin Island 44 42 4.9 0.12.............................................................................................................................................................................`Taxa' refers to the original names for groups of organisms found in the primary reference. S refers totrophic species3. The seven food webs address (1) primarily invertebrates in Skipwith Pond15; (2)pelagic and benthic species in Little Rock Lake17, the largest food web in the primary literature; (3)Bridge Brook Lake, the largest among a recent set of