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To understand the mechanisms underlying species coexistence, ecologists oftenstudy invasion growth rates of theoretical and data-driven models. These growthrates correspond to average per-capita growth rates of one species with respectto 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 frombeing rare. To provide a mathematically rigorous framework for this approach,we prove theorems that answer two questions: (i) When do the signs of theinvasion growth rates determine coexistence? (ii) When signs are sufficient,which invasion growth rates need to be positive? We focus on deterministicmodels and equate coexistence with permanence, i.e., a global attractor boundedaway from extinction. For models satisfying certain technical assumptions, weintroduce invasion graphs where vertices correspond to proper subsets ofspecies (communities) supporting an ergodic measure and directed edgescorrespond to potential transitions between communities due to invasions bymissing species. These directed edges are determined by the signs of invasiongrowth rates. When the invasion graph is acyclic (i.e. there is no sequence ofinvasions starting and ending at the same community), we show that permanenceis 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 negativeinvasion growth rates. We show that dissipative Lotka-Volterra models satisfyour technical assumptions and computing their invasion graphs reduces tosolving systems of linear equations. We provide additional applications of theresults and discuss open problems.

Permanence via invasion graphs: Incorporating community assembly into Modern Coexistence Theory