弱竞争空间逻辑分支过程的非平衡波动

Thomas Tendron
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摘要

空间对数分支过程是一个种群动力学模型,其中粒子按照独立的简单对称随机行走在晶格上移动,每个粒子以速率1分裂成随机数目的个体,同一位置的粒子对以速率c竞争。我们考虑了弱竞争机制,其中c趋于零,对应于趋于无穷大的局部承载能力,如1/c。我们证明,空间逻辑分支过程的流体力学极限是由 Fisher-Kolmogorov-Petrovsky-Piskunov 方程给出的。然后,我们证明了它的非平衡波动收敛于一个具有确定但异质系数的广义奥恩斯坦-乌伦贝克过程。证明依赖于对 Boldrighini 等人 1992 年提出的 v 函数方法的调整。一个令人感兴趣的中间结果显示了后代分布的尾部和 c 趋于零的精确机制如何影响空间对数分支过程的预期种群数量向流体力学极限的收敛速度。
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Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition
The spatial logistic branching process is a population dynamics model in which particles move on a lattice according to independent simple symmetric random walks, each particle splits into a random number of individuals at rate one, and pairs of particles at the same location compete at rate c. We consider the weak competition regime in which c tends to zero, corresponding to a local carrying capacity tending to infinity like 1/c. We show that the hydrodynamic limit of the spatial logistic branching process is given by the Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck process with deterministic but heterogeneous coefficients. The proofs rely on an adaptation of the method of v-functions developed in Boldrighini et al. 1992. An intermediate result of independent interest shows how the tail of the offspring distribution and the precise regime in which c tends to zero affect the convergence rate of the expected population size of the spatial logistic branching process to the hydrodynamic limit.
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