Persistence for structured population models without monotonicity

Consider the structured stochastic difference equation

with state space $\mathbf{S}$ and extinction set $\mathbf{S}_0$, and let $\gamma$ be the Lyapunov exponent of its linearization at extinction. Assume D1--D2 and the hypotheses of the boundedness theorem. **Structured-population persistence conjecture.** If $\gamma>0$, then

is almost surely persistent and persistent in probability. A theorem establishes this conclusion under the stronger monotonicity assumptions D3--D4; the conjecture proposes that those assumptions are unnecessary.