面向年龄结构随机出生死亡过程的动力学理论
A kinetic theory for age-structured stochastic birth-death processes
AI总结：
本文提出一种全新的完全随机动力学理论，用于描述相互作用的年龄结构种群。通过定义多粒子概率密度函数，推导出年龄依赖的出生死亡过程的动力学方程，同时提供直观高效的建模方法，适用于人口学、干细胞动力学和疾病演化的研究。
摘要：
经典年龄结构质量作用模型如麦肯迪克-冯福斯特方程已被广泛研究，但它们在结构上无法描述随机波动或种群大小依赖的出生和死亡率。将半马尔可夫年龄依赖过程用例如贝尔曼-哈里斯方程处理的随机理论，无法解析种群的年龄结构，也无法量化种群大小依赖性。相反，当前包含大小依赖种群动态的理论（如包含承载能力的数学模型，如逻辑方程）难以扩展以考虑年龄依赖的出生和死亡率。在本文中，我们提出了一个新完全随机动力学理论的系统推导，用于相互作用的年龄结构种群。通过定义多粒子概率密度函数，我们推导出年龄结构种群随机演化的动力学方程的层次结构。我们证明，完全随机年龄依赖的出生死亡过程禁止相应概率密度的因子分解，因此必须通过BBGKY-like层次结构来求解。然而，在两个简单的极限情况下，得到了显式解，并与对应的平均场结果进行了比较。我们的结果通过提供直观且高效的建模方法，扩展了确定性模型和现有主方程方法，适用于人口学、干细胞动力学和疾病演化的研究。
英文摘要：
Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but they are structurally unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using e.g., the Bellman-Harris equation, do not resolve a population's age-structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the Logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an ageing population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a BBGKY-like hierarchy. However, explicit solutions are derived in two simple limits and compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution.
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