Hematopoiesis is a biological phenomenon (process) of production of mature blood cells by cellular differentiation. It is based on amplification steps due to an interplay between renewal and differentiation in the successive cell types from stem cells to mature blood cells. We will study this mechanism with a stochastic point of view to explain unexpected fluctuations on the mature blood cell number, as surprisingly observed by biologists and medical doctors in a rest hematopoiesis. We consider three cell types: stem cells, progenitors and mature blood cells. Each cell type is characterized by its own dynamics parameters, the division rate and the renewal and differentiation probabilities at each division event. We model the global population dynamics by a three-dimensional stochastic decomposable branching process. We show that the amplification mechanism is given by the inverse of the small difference between the differentiation and renewal probabilities. Introducing a parameter K which scales simultaneously the size of the first component, the differentiation and renewal probabilities and the mature blood cell death rate, we describe the asymptotic behavior of the process for large K. We show that each cell type has its own size and time scales. Focusing on the third component, we prove that the mature blood cell population size, conveniently renormalized (in time and size), is expanded in an unusual way inducing large fluctuations. The proofs are based on a fine study of the different scales involved in the model and on the use of different convergence and average techniques in the proofs.

造血是细胞分化产生成熟血细胞的生物学现象（过程）。由于从干细胞到成熟血细胞的连续细胞类型的更新和分化之间的相互作用，它基于扩增步骤。我们将从随机的角度研究这种机制，以解释成熟血细胞数量的意外波动，正如生物学家和医生在静息造血中所观察到的那样。我们考虑三种细胞类型：干细胞、祖细胞和成熟血细胞。每种细胞类型的特征在于其自身的动力学参数、分裂率以及每次分裂事件的更新和分化概率。我们通过三维随机可分解分支过程对全球人口动态进行建模。我们表明放大机制是由分化和更新概率之间的小差异的倒数给出的。引入一个参数K同时缩放第一分量的大小、分化和更新概率以及成熟血细胞死亡率，我们描述了大K过程的渐近行为。我们展示了每种细胞类型都有自己的大小和时间尺度。关注第三个组成部分，我们证明了成熟血细胞群体的大小，方便地重新归一化（在时间和大小上），以一种不寻常的方式扩大，导致大的波动。证明基于对模型中涉及的不同尺度的精细研究以及在证明中使用不同的收敛和平均技术。