All logarithms in this paper are base 10. Cohen et al. (2003) and Jonsson et al. (2004) suggested placing all species and trophic links in a web on axes with ordinate log(M) and abscissa log(N). Then the l1 length of a link (r, c) from prey (resource) r to predator (consumer) c is:
- l1 = | log(Mc) − log(Mr) | + | log(Nc) − log(Nr) |= | log(Mc/Mr) | + | log(Nc/Nr) |, (eqn 1)
the order of magnitude of the body size ratio between the predator and prey, plus the order of magnitude of the numerical abundance ratio between the predator and prey. The slope α of a link (r, c) is:
- (eqn 2)
the order of magnitude of the body size ratio between predator and prey, over the order of magnitude of the numerical abundance ratio between predator and prey (which is typically negative). The angle θ of a link (r, c) is the angle in the counter-clockwise direction from a right-pointing horizontal line to the link considered as a vector from prey r to predator c in the (log(N), log(M)) plane.
A trophic link that has an angle of 135° (equal to slope −1) has predator and prey of equal biomass abundance. If the angle is less than 135° but greater than −45°, then the biomass of the predator exceeds that of the prey. If the angle is greater than 135° but less than 315°, the opposite holds.
Taking population production and ingestion of a species to be proportional to NM0·75 (Peters 1983), one can show that a trophic link of slope −4/3 has predator and prey species with equal production and ingestion. A slope that is less than −4/3 with heavier predator indicates that the predator has greater ingestion and production than the prey, and a slope that is greater than −4/3 with heavier predator indicates the opposite. Slope and angle are not defined for cannibalistic links where r = c.
Slope and angle are conceptually interchangeable, but there were statistical and mathematical circumstances under which each was most appropriate, so both were used. For instance, angle interacts better with median and mean. Two links with angles 89° and 91° have mean angle 90°, but mean slope 0 because slope has a vertical asymptote at angle 90°. Slope is more useful than angle for certain linear regressions because the homoskedasticity assumptions of linear regression are more nearly satisfied for slope.
The l2 length of a link is the Euclidean length in the log(M) vs. log(N) plane, and l1/l2 = (| α | + 1)/(α2 + 1)1/2. For Tuesday Lake, where α is usually close to −1, l1/l2 will be roughly 21/2. A constant factor will not affect the trends examined, so l1 only is used because it, unlike l2, has a clear biological interpretation. The intuitive understanding of the Euclidean distance applies to l1 because it differs from l2 only by a (very nearly) constant factor.
Each link in a web may be represented by a point in a 1-, 2-, 3-, or higher dimensional Euclidean space, depending on whether 1, 2, 3 or more quantitative attributes of the link are to be studied. The set of all links is represented by a cloud of such points. For example, below we study the one-dimensional distributions of each link's length and each link's angle (being careful of cannibalistic links).
Length, angle and other distributions are also well-defined for any ordered pair of species (a, b) in the (log(N), log(M)) plane if one replaces c (‘predator’) with b and r (‘prey’) with a in eqn 1 and eqn 2[again excepting the angle of ‘cannibalistic’ ordered pairs (a, a)]. When discussing an ordered pair (a, b), species a will be called the prey and b the predator, even though there may have been no trophic relationship between a and b, or even if in reality a ate b. The set of links is contained in the set of ordered pairs, and there is no implied relationship between the body mass or numerical abundance of a and that of b in the ordered pair (a, b).
Species were divided into basal, intermediate and top (B, I and T) groups, allowing a division of ordered pairs into (B, B) (B, I) (B, T) (I, B) (I, I) (I, T) (T, B) (T, I) and (T, T) groups. This and several other divisions of species into groups permitted us to investigate whether these classifications of species were involved in statistical regularities in length and angle distributions. Any distribution of ordered pairs was thereby divided into subdistributions, one for each of these groups. Some (but not all) of these groups of ordered pairs contained links, and so subdistributions of links were also broken into groups. This procedure was carried out in other ways by starting with different initial groupings of species. The following groupings of species were used: the above BIT-grouping; a grouping that put species of similar average body mass M together (called the M-grouping); a grouping that put species of similar numerical abundance together (called the N-grouping); and the functional grouping into phytoplankton, zooplankton and fish species (called the PZF-grouping). In Tuesday Lake, the PZF-grouping was almost identical to the N-grouping.
All groupings are listed in Appendix S1 (see Supplementary material; also available on request from the authors), Table A1 and Table A2. The groups of the M-grouping were called the H group, the S group and the L group, representing heavy, standard weight and light species. The groups of the N grouping were called the R group, the U group and the C group, representing rare, uncommon and common species.
The trophic position of a species in a food chain is the number of species below it in the chain (so a species with no species below it has position 0). This definition is a slight modification of the definition of Jonsson et al. The trophic height of a species in a web is computed by collecting all chains that begin at the given species going strictly down through the web (i.e. from predator to prey at each step), but that do not contain more than one copy of the same species. One then takes the mean of the species’ trophic position in all these chains. No chains including cannibalistic links are included in the mean, and no chains that go all the way around a cycle are included, although it is acceptable to go any part of the way around a cycle. The algorithm used by Jonsson et al. treated cycles in a different way, but produced results similar to this definition, other than a uniform difference of 1.