发表机构
School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对具有强Allee效应的生灭过程，推导了主特征值和平均灭绝时间的尖锐Eyring-Kramers渐近公式，刻画了双稳态下的拟平稳分布及阈值处的过渡层行为。
AI 中文摘要
我们研究了具有强Allee效应和吸收性灭绝的密度依赖生灭过程。确定性系统是双稳态的：灭绝和正平衡态是局部渐近稳定的，二者被一个不稳定的Allee阈值分隔。令$K$为种群规模缩放参数。当$K\to\infty$时，我们推导出被杀灭生成元主特征值的尖锐Eyring--Kramers型渐近公式（包括前导预因子），以及相应正特征向量的均匀近似。由此，我们获得了拟平稳分布（QSD）下平均灭绝时间的尖锐渐近公式。在双稳态区域，初始种群决定了早期灭绝与正亚稳态之间的竞争。初始松弛后，该分布在全变差范数下由依赖于初始状态的QSD与灭绝处Dirac质量的凸组合近似。混合系数被识别为在吸收前到达正稳定区域的概率。我们确定了Allee阈值附近宽度为$O(\sqrt K)$的概率过渡层，其中混合系数收敛到显式高斯剖面。值得注意的是，从阈值处开始的种群以渐近相等的概率到达正亚稳态区域或灭绝。我们建立了QSD的存在性和唯一性、向拟平稳状态的定量收敛性，以及以正稳定水平为中心的QSD的离散高斯近似。我们的分析结合了被杀灭生成元的自伴谱实现、相关二阶递推的匹配渐近分析，以及可逆权重的离散Laplace估计。
英文摘要
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.
Comments52 pages
