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 nλ̃(n/K) and nμ̃(n/K), where K is the population scale and λ̃,μ̃ are the per-capita rates. Write V(x)=x(λ̃(x)−μ̃(x)) for the deterministic drift, with unstable threshold x1 and positive stable equilibrium x2. For the two smallest eigenvalues ρ1(K)<ρ2(K) of the negative generator killed at extinction, we prove limK→∞(ρ2(K)−ρ1(K)) =min{μ̃(0)−λ̃(0),V'(x1),−V'(x2)}, so the limit depends on the linearization rates at all three equilibria. Each term can be the unique minimum even when x1 and x2 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 L2 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.
Related
- Quasi-stationary distributions and sharp metastable asymptotics for birth--death processes with a strong Allee effect
- Cumulative Rewards for Birth–Death Processes with Catastrophes
- Darboux transformations of linear birth-death processes
- Space-time large deviations for continuum birth-and-death processes
- Random walk in a null recurrent birth-and-death dynamical environment
BibTeX
@misc{zhou2026principal,
title = {{The principal spectral gap for birth--death processes with a strong Allee effect}},
author = {Dun Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31808},
eprint = {2609.31808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31808},
}
RIS
TY - UNPB
TI - The principal spectral gap for birth--death processes with a strong Allee effect
AU - Dun Zhou
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31808
UR - https://arxiv.org/abs/2609.31808
AB - 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 nλ̃(n/K) and nμ̃(n/K), where K is the population scale and λ̃,μ̃ are the per-capita rates. Write V(x)=x(λ̃(x)−μ̃(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→∞(ρ_2(K)−ρ_1(K)) =min{μ̃(0)−λ̃(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.
ER -