Persistence in probability for general multispecies systems

Let

be a $k$-species stochastic difference equation satisfying C1--C3: the populations remain in a compact set $\mathbf{S}$, the functions $f_i(x,\omega)$ are strictly positive, continuous in $x$ and measurable in $\omega$, and $\sup_{x\in\mathbf{S}}\mathbb{E}[(\log f_i(x,\xi_t))^2]<\infty$ for every $i$. Let $\mathbf{S}_0$ denote the extinction boundary, and let $\lambda_i(\mu)$ be the invasion rate of species $i$ at an invariant measure $\mu$. **General multispecies persistence conjecture.** Under the assumptions of the persistence theorem,

is persistent in probability. The cited theorem proves almost-sure persistence when every invariant boundary measure has a species with positive invasion rate, equivalently under its stated weighted invasion-rate condition; persistence in probability is the asserted strengthening.