The principal spectral gap for birth--death processes with a strong Allee effect

Abstract

For birth--death processes with a strong Allee effect, long survival near a positive stable equilibrium does not by itself determine the principal spectral gap. We identify its large-population limit when the birth and death rates at population size $n$ are $n\widetildeλ(n/K)$ and $n\widetildeμ(n/K)$, where $K$ is the population scale and $\widetildeλ,\widetildeμ$ are the per-capita rates. Write $V(x)=x(\widetildeλ(x)-\widetildeμ(x))$ for the deterministic drift, with unstable threshold $x_1$ and positive stable equilibrium $x_2$. For the two smallest eigenvalues $ρ_1(K)<ρ_2(K)$ of the negative generator killed at extinction, we prove \[ \lim_{K\to\infty}\bigl(ρ_2(K)-ρ_1(K)\bigr) =\min\{\widetildeμ(0)-\widetildeλ(0),V'(x_1),-V'(x_2)\}, \] so the limit depends on the linearization rates at all three equilibria. Each term can be the unique minimum even when $x_1$ and $x_2$ are fixed. The proof combines a frozen boundary model with two local oscillator limits through discrete Ismagilov--Morgan--Simon (IMS) localization. A uniform comparison of the global and stable local ground states controls the orthogonality constraint in the variational lower bound; projected local trial vectors give the matching upper bound. The result determines the limiting $L^2$ spectral gap and optimal Poincaré constant of the $Q$-process, which describes conditioning on indefinite survival. Combined with principal-eigenvalue asymptotics from a companion work, it separates an exponentially growing quasi-stationary mean extinction time from a spectral relaxation time with a finite positive limit.