具有消失适应度的准临界生灭过程的入侵动力学
Invasion dynamics with vanishing fitness for a quasi-critical birth-death process
- MERGE, INRIA, École polytechnique, Institut Polytechnique de Paris（MERGE，INRIA，巴黎综合理工学院，巴黎理工学院）
- SAMOVAR, Télécom SudParis, Institut Polytechnique de Paris（SAMOVAR，南巴黎电信学院，巴黎理工学院）
- Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，LJAD）
- CMAP, École polytechnique, Institut Polytechnique de Paris（CMAP，巴黎综合理工学院，巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究正密度依赖种群入侵动力学，证明达到宏观尺度的概率按1/√K衰减，并将入侵分为逃逸零、中间扩散和宏观ODE三个阶段。
AI中文摘要：
我们研究了表现出正密度依赖效应的种群的入侵动力学。我们从单个个体出发，考虑一个单类型生灭过程。初始个体增长率消失，但随种群密度增加而增加，与个体数除以缩放参数$K$成正比。在达到宏观尺度$K$之前，种群过程几乎是临界的。我们证明了种群达到宏观水平$K$的概率随$K$趋于无穷而按$1/\sqrt{K}$递减。我们还描述了相关的轨迹，并表明入侵可分为三个时间段。首先，过程需要从零逃逸，并且在存活条件下，它线性增长直到$\sqrt{K}$量级。缩放过程由扩散近似，如同临界分支过程，但带有来自合作的额外漂移项，这打破了分支性质。其次，在中间尺度$\sqrt{K}$上，我们观察到另一个扩散，以正概率存活，无需条件。最后，在$\sqrt{K}$尺度之外，过程可由经典宏观ODE极限近似。第一阶段的证明涉及概率测度变换和鞅的一致可积性刻画，而另外两个阶段依赖于多项式时间尺度上的一致近似。
英文摘要：
We study the invasion dynamics of populations exhibiting positive density-dependent effects. We start with a single individual and consider a single-type birth and death process. The initial individual growth rate vanishes but it increases with the population density, proportionally to the number of individuals divided by a scaling parameter $K$. Before reaching the macroscopic scale~$K$, the population process is almost critical. %{\color{red} Although the process remains asymptotically critical throughout the invasion phase, three distinct dynamical regimes emerge.} We prove that the probability for the population to reach the macroscopic level $K$ decreases as $1/\sqrt{K}$ as $K$ goes to infinity. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape from zero, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The scaled process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation, which breaks the branching property. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, surviving with positive probability, without conditioning. Finally, beyond $\sqrt{K}$ scale, the process can be approximated by a classical macroscopic ODE limit. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases rely on uniform approximations on polynomial time scales.
