Stratigraphy of a Random Acyclic Directed Graph: The Size of Trophic Levels in the Cascade Model

When an ecological food web is described by an acyclic directed graph, the trophic level of a species of plant or animal may be described by the length of the shortest (or the longest) food chain from the species to a green plant or to a top predator. Here we analyze the number of vertices in different levels in a stochastic model of acyclic directed graphs called the cascade model. This model describes several features of real food webs. For an acyclic directed graph D, define the ith lower (upper) level as the set of all vertices . of D such that the length of the shortest (longest) maximal path starting from . equals i,i=0,1.. In this article, we compute the sizes of the levels of a random digaph D(n,c) obtained from a random graph on the set {1,2,.,n} of vertices in which each edge appears independently with probability c/n, by directing all edges from a larger vertex to a smaller one. The number of edges between any two levels of D(n,c) is also found.