Quasi-stationary distributions and sharp metastable asymptotics for birth--death processes with a strong Allee effect

Abstract: We study density-dependent birth--death processes with a strong Allee effect and absorbing extinction. The deterministic system is bistable: extinction and a positive equilibrium are locally asymptotically stable, separated by an unstable Allee threshold. Let $K$ be the population-size scaling parameter. As $K\to\infty$, we derive a sharp Eyring--Kramers-type asymptotic formula, including the leading prefactor, for the principal eigenvalue of the killed generator, and a uniform approximation of the corresponding positive eigenvector. Consequently, we obtain a sharp asymptotic formula for the mean extinction time under the quasi-stationary distribution (QSD). In the bistable regime, the initial population determines the competition between early extinction and positive metastability. After initial relaxation, the law is approximated in total variation by an initial-state-dependent convex combination of the QSD and the Dirac mass at extinction. The mixing coefficient is identified as the probability of reaching the positive stable region before absorption. We identify a probabilistic transition layer of width $O(\sqrt K)$ around the Allee threshold, where the mixing coefficient converges to an explicit Gaussian profile. Notably, a population starting at the threshold reaches the positive metastable region or becomes extinct with asymptotically equal probabilities. We establish existence and uniqueness of the QSD, quantitative convergence to the quasi-stationary regime, and a discrete Gaussian approximation of the QSD centered at the positive stable level. Our analysis combines a self-adjoint spectral realization of the killed generator, matched asymptotic analysis of the associated second-order recurrence, and discrete Laplace estimates for reversible weights.