  To understand the mechanisms underlying species coexistence, ecologists often
study invasion growth rates of theoretical and data-driven models. These growth
rates correspond to average per-capita growth rates of one species with respect
to an ergodic measure supporting other species. In the ecological literature,
coexistence often is equated with the invasion growth rates being positive.
Intuitively, positive invasion growth rates ensure that species recover from
being rare. To provide a mathematically rigorous framework for this approach,
we prove theorems that answer two questions: (i) When do the signs of the
invasion growth rates determine coexistence? (ii) When signs are sufficient,
which invasion growth rates need to be positive? We focus on deterministic
models and equate coexistence with permanence, i.e., a global attractor bounded
away from extinction. For models satisfying certain technical assumptions, we
introduce invasion graphs where vertices correspond to proper subsets of
species (communities) supporting an ergodic measure and directed edges
correspond to potential transitions between communities due to invasions by
missing species. These directed edges are determined by the signs of invasion
growth rates. When the invasion graph is acyclic (i.e. there is no sequence of
invasions starting and ending at the same community), we show that permanence
is determined by the signs of the invasion growth rates. In this case,
permanence is characterized by the invasibility of all -i communities, i.e.,
communities without species i where all other missing species having negative
invasion growth rates. We show that dissipative Lotka-Volterra models satisfy
our technical assumptions and computing their invasion graphs reduces to
solving systems of linear equations. We provide additional applications of the
results and discuss open problems.
