An exact coexistence criterion for stochastic Lotka--Volterra systems with Volterra--Lyapunov stable interactions

Abstract

We consider stochastic Lotka-Volterra systems $dX=\mathrm{Diag}(X)(α+AX)dt+\mathrm{Diag}(X)ΓdB$ with nondegenerate noise and a Volterra-Lyapunov stable interaction matrix $A$, that is, $DA+A^\top D\prec 0$ for some positive diagonal matrix $D$. This class covers every self-limited predator-prey food web that is conservative in the sense of Volterra, in particular every such web whose interaction graph is a forest. We give a necessary and sufficient condition for invariant coexistence: the process has an invariant probability measure on $(0,\infty)^n$, that is, a stationary distribution under which all species have positive densities, if and only if $m=-A^{-1}β$ is positive, where $β=α-\frac{1}{2}\mathrm{diag}(ΓΓ^\top)$ is the vector of noise-corrected growth rates. This measure is then unique and ergodic, has mean $m$, and attracts every interior initial condition exponentially fast in total variation. Thus environmental noise affects coexistence only through the Itô shift $α\mapstoβ$, and the criterion is the exact counterpart of the feasibility criterion for the deterministic system $\dot x=\mathrm{Diag}(x)(β+Ax)$. Both directions rest on the fact that, for Lotka-Volterra systems, invasion rates are affine in the mean of the underlying measure. Necessity follows from an exact formula for the means of invariant measures supported on faces of the orthant. Sufficiency follows from the persistence theorem of Hening and Nguyen, for which the explicit weights $Dm$ give a stochastic counterpart of Hofbauer's average Lyapunov function, so that no enumeration of the invariant measures on the boundary is required. Applied to subcommunities, the result also characterizes every stationary regime in which some species are absent.