一类具有单类型种群规模依赖的多类型马尔可夫分支过程
A multitype Markovian branching process with one-type population size dependence
AI总结：
本文提出一类繁殖参数仅依赖单类型个体数量的多类型马尔可夫分支过程，证明在软承载能力下灭绝几乎必然发生，并通过标度极限刻画亚稳态行为。
AI中文摘要：
受免疫调节寄生虫病宿主内框架的启发，我们构建了一个多类型马尔可夫分支过程，其繁殖参数仅依赖于单一类型的个体数量。我们对该控制类型施加一个软承载能力 $K$，作为亚临界与超临界动力学之间的阈值。我们证明，当每个非控制类型的后代批次以正概率包含一个控制类型个体时，灭绝几乎必然发生，且灭绝时间具有所有阶的有限矩。随后，我们构造了一列以 $K$ 为索引的密度依赖过程，并利用典范泛函大数定律（FLLN）和中心极限定理研究该过程的标度极限，旨在刻画在渐近稳定平衡点或FLLN的稳定极限环附近出现亚稳态行为的特征。
英文摘要：
Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity $K$ with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by $K$ and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.
