A practical guide to characterising ecological coexistence.

Coexistence is simultaneously one of the most fundamental concepts of ecology, and one of the most difficult to define. A particular challenge is that, despite a well-developed body of research, several different schools of thought have developed over the past century, leading to multiple independent, and largely isolated, branches of literature with distinct methodologies. Here, we provide a broad overview of the most common concepts and metrics currently used to detect and characterise ecological coexistence. We first introduce four classes of behaviour, which jointly describe the ways in which community dynamics can unfold: (i) the existence of a feasible steady state (or invariant set), i.e. where all coexisting species retain positive abundances in the long-term in the absence of interference by external forces; (ii) the existence of a local attractor that draws the community towards a feasible steady state from within a restricted set of starting conditions; (iii) the existence of a global attractor that draws the community towards feasible steady states from any non-zero starting condition; and (o) a null transient state, where species abundances vary over time irrespective of steady states and attractors. Next, we explain how these classes of behaviour relate to commonly used metrics for identifying and characterising coexistence, including analyses of parameter sensitivity, asymptotic return rates, invasion growth rates, and time to extinction. We then discuss the scope and limitations of each of these behavioural classes and corresponding metrics, with a particular focus on applications in empirical systems. Finally, we provide a potential workflow for matching empirical questions to theoretical tools, and present a brief prospectus looking forward to opportunities for advancing and integrating research on coexistence.