Abstract Many populations, e.g. not only of cells, bacteria, viruses, or replicating DNA molecules, but also of species invading a habitat, or physical systems of elements generating new elements, start small, from a few lndividuals, and grow large into a noticeable fraction of the environmental carrying capacity K or some corresponding regulating or system scale unit. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from Z0 = z0 in a branching process or Malthusian like, roughly exponential fashion, , where Zt is the size at discrete time , a > 1 is the offspring mean per individual (at the low starting density of elements, and large K), and W a sum of z0 i.i.d. random variables. It will, thus, become detectable (i.e. of the same order as K) only after around generations, when its density will tend to be strictly positive. Typically, this entity will be random, even if the very beginning was not at all stochastic, as indicated by lower case z0, due to variations during the early development. However, from that time onwards, law of large numbers effects will render the process deterministic, though inititiated by the random density at time log K, expressed through the variable W. Thus, W acts both as a random veil concealing the start and a stochastic initial value for later, deterministic population density development. We make such arguments precise, studying general density and also system-size dependent, processes, as . As an intrinsic size parameter, K may also be chosen to be the time unit. The fundamental ideas are to couple the initial system to a branching process and to show that late densities develop very much like iterates of a conditional expectation operator. The “random veil”, hiding the start, was first observed in the very concrete special case of finding the initial copy number in quantitative PCR under Michaelis-Menten enzyme kinetics, where the initial individual replication variance is nil if and only if the efficiency is one, i.e. all molecules replicate.

中文翻译：

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具有相互作用和环境依赖性的种群：从少数（几乎）独立的成员到高密度的确定性进化

摘要 许多种群，例如不仅是细胞、细菌、病毒或复制 DNA 分子的种群，还有侵入栖息地的物种，或产生新元素的元素物理系统，从少数个体开始，然后大到显着环境承载力 K 的一部分或一些相应的调节或系统规模单位。通常，初始稀疏集合的元素不会相互妨碍，并且它们的数量将从 Z0 = z0 在分支过程或类似马尔萨斯的类似马尔萨斯的过程中增长，大致呈指数方式，其中 Zt 是离散时间的大小，a > 1是每个个体的后代平均值（在元素的低起始密度和大 K 下），W 是 z0 个 iid 随机变量的总和。因此，只有在几代之后，它才会变得可检测（即与 K 具有相同的数量级），当它的密度趋于严格为正时。通常，由于早期开发过程中的变化，该实体将是随机的，即使一开始根本不是随机的，如小写 z0 所示。然而，从那时起，大数定律效应将使过程具有确定性，尽管由时间 log​​ K 处的随机密度启动，通过变量 W 表示。因此，W 既充当隐藏开始的随机面纱，又充当随机以后确定性人口密度发展的初始值。我们使这样的论点精确，研究一般密度和系统大小相关的过程，如 。作为固有大小参数，K 也可以选择为时间单位。基本思想是将初始系统与分支过程耦合，并表明后期密度的发展非常类似于条件期望算子的迭代。在 Michaelis-Menten 酶动力学下的定量 PCR 中寻找初始拷贝数的非常具体的特殊情况中首先观察到隐藏开始的“随机面纱”，其中初始个体复制方差为零当且仅当效率为一，即所有分子都复制。