具有强阿利效应的生灭过程的主谱隙
The principal spectral gap for birth--death processes with a strong Allee effect
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中文总结 AI 辅助
本文针对具有强阿利效应的生灭过程，确定了主谱隙的大种群极限，该极限由三个平衡点处的线性化速率决定，并由此得到Q过程的谱隙和Poincaré常数，区分了拟平稳灭绝时间与谱弛豫时间。
中文摘要 AI 辅助
对于具有强阿利效应的生灭过程，在正稳定平衡点附近的长期存活本身并不能确定主谱隙。当种群规模为$n$时的出生率和死亡率分别为$n\widetilde\lambda(n/K)$和$n\widetilde\mu(n/K)$时，我们确定了其大种群极限，其中$K$为种群规模尺度，$\widetilde\lambda,\widetilde\mu$为每 capita 速率。记$V(x)=x(\widetilde\lambda(x)-\widetilde\mu(x))$为确定性漂移，其中$x_1$为不稳定阈值，$x_2$为正稳定平衡点。对于灭绝时杀死的负生成元的两个最小特征值$\rho_1(K)<\rho_2(K)$，我们证明\\[ \lim_{K\to\infty}\bigl(\rho_2(K)-\rho_1(K)\bigr) =\min\{\widetilde\mu(0)-\widetilde\lambda(0),V'(x_1),-V'(x_2)\}, \\] 因此该极限取决于所有三个平衡点处的线性化速率。即使当$x_1$和$x_2$固定时，每一项都可能成为唯一最小值。证明结合了一个冻结边界模型和两个局部振荡器极限，通过离散的 Ismagilov--Morgan--Simon (IMS) 局部化实现。全局和稳定的局部基态的一致比较控制了变分下界中的正交性约束；投影局部试探向量给出了匹配的上界。该结果确定了$Q$-过程的极限$L^2$谱隙和最优 Poincaré 常数，其中$Q$-过程描述了对无限期存活的调节。结合来自配套工作的主特征值渐近性，它将指数增长的拟平稳平均灭绝时间与具有有限正极限的谱弛豫时间区分开来。
英文摘要
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.
发表机构
- Nanjing University of Science and Technology（南京理工大学）
- School of Mathematics and Statistics（数学与统计学院）
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