Food Web Graph of a Coevolutionarily Stable Community

Abstract

A model community, which consists of multiple trophic levels (several species of prey, and several predator species including omnivores) is considered from the viewpoint of coevolutionary stability. Some properties of community digraphs are studied and resource utilization patterns of predators are given as necessary conditions for a coevolutionarily stable community (CSC). If a member of a predator species can stimultaneously search for multiple prey species, a set of the prey species is interpreted as a single "prey trophic species." (1) For any subgraph of a CSC food web, the number of links between predator and prey trophic species is less than the sum of the number of predator species and the number of true prey species. (2) If an predator member searches for only one prey species, then for any CSC food web, the number of links connecting that predator and its prey is less than the sum of the number of predator species and that of prey species, where a species at an intermediate trophic level is doubly counted as prey and also as predator. (3) The maximum number of links in a CSC digraph is proportional to the number of species in the community. With our model we attempt to explain intervality of a niche overlap graph and lack of a hole in a common enemy graph. Finally, we examined two examples of real communities to confirm validity of our results.