Understanding the mechanisms allowing interacting populations to co-occur underlies many questions in ecology, evolution, and epidemiology. One widely used metric for understanding coexistence are invasion growth rates: the average per-capita growth rates of populations when rare. These invasion growth rates are the basis of Modern Coexistence Theory (MCT). Currently, this theory assumes coexistence occurs when each species, say species i, has a positive invasion growth rate in the community determined by its absence, so called -i communities. Intuitively, coexistence occurs if each species can recover from being rare. However, there are well known examples where this heuristic fails. Moreover, even if this heuristic applies, it isn't always clear what is meant by -i communities. To address these issues, we introduce a broadly applicable mathematical theory to answer two questions (i) when do the signs of the invasion growth rates determine coexistence? (ii) When the signs are sufficient, which invasion growth rates need to be positive? To address these questions, we introduce a mathematical theory for deterministic models accounting for any number of species, any form of discrete-structure with each species (e.g. age, size, patches), and any finite-dimensional auxiliary processes (e.g. seasonality, trait evolution, plant-soil feedbacks).
Results/Conclusions
We introduce invasion graphs where vertices correspond to proper subsets of species (communities) and directed edges correspond to potential transitions between communities due to invasions by missing species. Importantly, these directed edges allow for transitions due to multispecies introductions and 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 coexistence is determined by the signs of the invasion growth rates. Furthermore, coexistence requires that each species i can invade its -i communities i.e. communities without species i where all other missing species having negative invasion growth rates. To illustrate the applicability of the results, we show that dissipative Lotka-Volterra models generically satisfy our technical assumptions and computing their invasion graphs reduces to solving systems of linear equations. We also apply our results to models of competing species with pulsed resources and sharing a predator that exhibit switching behavior. Open problems for both deterministic and stochastic models are discussed. Our results highlight the importance of using concepts about community assembly to study coexistence.
