A system-centered probabilistic formalism linking multiple equilibria and biodiversity in ecological and evolutionary models

Abstract

Ecological and evolutionary communities do not always settle into a single predictable configuration. Models such as Lotka-Volterra or replicator dynamics may predict several admissible states. The multiplicity of such states is central to understanding robustness, alternative community configurations, coexistence, and shifts in biodiversity. Yet, standard practices are often state-centered: they either focus on the properties of typical equilibria or pool states across systems, losing information about how many states can arise in a single system, including cases where no state exists. Here, we introduce a system-centered probabilistic formalism that captures this hidden structure and could be applied across a wide range of ecological and evolutionary dynamical models, drawing inspiration from hurdle and zero-inflated models. Each ecological system is viewed as generating a distribution over its possible outcomes. This allows us to jointly quantify the occurrence of a given state type, its multiplicity within each system, and the number of species co-existing within these states. The formalism is generic and can be applied to different families of dynamical models. To illustrate its scope, we consider two case studies: the generalized Lotka-Volterra and replicator dynamics with random and structured parameters. This unified perspective reveals patterns that remain invisible when equilibria are analyzed individually or pooled across systems. By turning the state space of a model into interpretable probabilistic quantities, our formalism offers a new way for studying the robustness of community structure, the likelihood of alternative outcomes under small changes in initial conditions, and the potential for shifts between states of low and high diversity.