## Introduction

The combination of information on species’ abundance and body mass with the traditional food web directed graph is a powerful descriptive tool to characterize an ecological community (Ulanowicz 1984; Moore, deRuiter & Hunt 1993; deRuiter, Neutel & Moore 1995; Rott & Godfray 2000; Bersier, Banasek-Richter & Cattin 2002). Cohen, Jonsson & Carpenter (2003) and Jonsson, Cohen & Carpenter 2004) used average body mass (*M*) and numerical abundance per unit of habitat (*N*), attaching an (*M*, *N*)-pair to each species in the food web (hereafter, simply web) of Tuesday Lake, Michigan, USA. Unlike studies of single or a few predator–prey relations in simplified natural, experimental or theoretical webs, the studies by Jonsson, Carpenter and the present authors investigate the multivariate relations among body mass, numerical abundance and all predator–prey interactions in a well-defined natural community. The motivation is to provide a detailed, quantitative, community-wide context for the study of predator–prey relations, first in Tuesday Lake and then as a model for empirical studies in other natural communities.

The community-wide perspective on predator–prey relations yields many new empirical regularities, to be reported below. These empirical regularities are unfamiliar because the data from Tuesday Lake are, so far, unique in combining comprehensive data on a web, body mass and numerical abundance (see Materials and methods for details). These data were used to address familiar ecological questions concerning the importance of functional taxonomic groups.

One of the numerous new patterns reported here is that the length (defined below) of trophic links, when species are plotted on log(*M*) vs. log(*N*) coordinates, is approximately normally distributed. Moreover, the mean length is about equal to the community span (the number of orders of magnitude body-mass variation in the community plus the number of orders of magnitude numerical abundance variation in the community) divided by the mean number of links in food chains in the web. We hypothesize that this rough equality will hold in other webs as well.

The hope that predation matrices generated by the cascade model or niche model combined with the *M* and *N* distributions of Tuesday Lake would give normal link length distributions failed decisively. Analysis of these models and four new models indicates that species’ average body mass and numerical abundance are not sufficient to account for the observed normal distribution of link lengths, without also taking account of trophic specialization associated with major taxonomic groups such as phytoplankton, zooplankton and fish. This specialization appears as rectangles in a body-mass-indexed predation matrix. The analysis indicates that taxonomy matters for feeding relations, beyond the very important effects of average body mass and numerical abundance. A new ‘cascade model with functional groups’ attempts to model webs with additional *M* and *N* data, and to combine functional specialization with other food web patterns to produce the observed normal distribution of link length.

Another new reported pattern is that the angles of trophic links from the positive *x*-axis when species are plotted on log(*M*) vs. log(*N*) coordinates have median slightly but significantly less than 135°. Because the species arrange themselves roughly in a linear pattern of slope about −1 when plotted on these coordinates, it is to be expected that the median angle of links is roughly 135°. However, the median angle of links is also significantly less than the angle of the best-fitting line to species on a log(*M*) vs. log(*N*) coordinates. The cascade model is sufficient to explain the observed median angles.

Normal link length distribution and many of the other statistical regularities found in Tuesday Lake, in addition to their direct biological significance, are also quantitative benchmarks to which future models of webs with *M* and *N* data should be compared.

The Tuesday Lake data cannot address questions of dynamics because the data describe only two points in time (1984 and 1986). These data also cannot identify the effects of the intervention in 1985 (removal of three fish species and introduction of another fish species) because no control lake is available.