We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loopensemble
To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and datadriven models. These growth rates correspond to average percapita 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 more »
 Award ID(s):
 1716803
 Publication Date:
 NSFPAR ID:
 10375836
 Journal Name:
 Journal of Mathematical Biology
 Volume:
 85
 Issue:
 5
 ISSN:
 03036812
 Publisher:
 Springer Science + Business Media
 Sponsoring Org:
 National Science Foundation
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