  The classification of the long-term behavior of dynamical systems is a
fundamental problem in mathematics. For both deterministic and stochastic
dynamics specific classes of models verify Palis' conjecture: the long-term
behavior is determined by a finite number of stationary distributions. In this
paper we consider the classification problem for stochastic models of
interacting species. For a large class of three-species, stochastic
differential equation models, we prove a variant of Palis' conjecture: the
long-term statistical behavior is determined by a finite number of stationary
distributions and, generically, three general types of behavior are possible:
1) convergence to a unique stationary distribution that supports all species,
2) convergence to one of a finite number of stationary distributions supporting
two or fewer species, 3) convergence to convex combinations of single species,
stationary distributions due to a rock-paper-scissors type of dynamic.
Moreover, we prove that the classification reduces to computing Lyapunov
exponents (external Lyapunov exponents) that correspond to the average
per-capita growth rate of species when rare. Our results stand in contrast to
the deterministic setting where the classification is incomplete even for
three-dimensional, competitive Lotka--Volterra systems. For these SDE models,
our results also provide a rigorous foundation for ecology's modern coexistence
theory (MCT) which assumes the external Lyapunov exponents determine long-term
ecological outcomes.
