Variation in Food-Web Structure: The Determinants of Connectance

Although there is considerable evidence for a number of patterns in the structure of food webs, there is a conspicuous lack of agreement among explanations for such patterns (Pimm 1982; DeAngelis et al. 1983; May 1986; Paine 1988; Strong 1988; Lawton 1989). One pattern that has received considerable attention is the observation that connectance of a food web generally decreases with increasing species richness (Pimm 1982; Strong 1988; Lawton 1989). Connectance is a parameter of central importance in both static and dynamic theories of community structure (May 1986; Lawton 1989), and, hence, understanding the determinants of connectance in food webs is a significant problem. Connectance (C) is defined, in general terms, as the number of actual direct interactions in a web divided by the total number of possible interactions. It can be calculated (excluding intraspecific feeding) either in terms of trophic links in the food-web matrix (where each link is represented by a single element of the matrix) as C = L/[S(S 1)] or in terms of the community matrix (where each link is represented by two elements, aij and aji) as C = 2L/[S(S 1)], where S is the number of species in the web and L is the number of trophic links. Calculations in this study are all based on the former measure. In various collections of food-web data, C declines approximately hyperbolically with increasing species richness (Rejm'anek and Star' 1979; Yodzis 1980; Pimm 1982; Auerbach 1984); in other words, the product SC is approximately constant. More or less equivalently, the relationship between L and S in these data is roughly linear (Cohen and Briand 1984). Throughout this note I treat these two ways of looking at the density of links in a food web as equivalent, but for clarity I concentrate discussion on the links-to-species (L-S) relationship. Three explanations have been proposed to account for the observed patterns in this relationship, and a brief review of these is worthwhile. The first hypothesis suggests that C is determined by stability. Studies of certain general models of communities indicate that increased complexity reduces the probability of such systems' being stable; the transition from predominantly stable to predominantly unstable models is marked by the switch from i(SC)?05 < 1 to i(SC)05 > 1 (where i is the average interaction strength) (May 1972, 1973; McMurtie 1975). That is, for a given interaction strength, C must decline with increasing S for a system to remain stable. Although the models make many