Assembly rules and a theory for invasion and extinction in minimal food webs

We propose a theory of the evolution of a minimal food web by sequential invasion of new species [1]. Our theory is based on the standard generalized Lotka-Volterra equations, where basal species compete through resource depletion [2]. The considered food webs are “minimal”, as each species only feeds on a single resource, leading to a hierarchical, tree-like food web [1,3]. We prove that at each invasion step there is one uniquely determined outcome: either the invader peacefully coexists with the residents and resources are re-distributed; the invader is eliminated; or one or several of the resident species are removed in a uniquely defined extinction cascade.
At the end of either of these processes the resulting food web relaxes to a globally stable (and feasible) steady state. We break down the essence of our theory in the conceptual “invasion extinction model” (IEM), which allows us to analytically compute the persistence time and the extinction size distribution.