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, (Formula presented.), where Zt is the size at discrete time (Formula presented.), 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 (Formula presented.) generations, when its density (Formula presented.) 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 (Formula presented.). 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.
Bibliographical noteFunding Information:
This research was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Jagers was further supported by the Knut och Alice Wallenberg Foundation.
© 2019, Published with license by Taylor & Francis Group, LLC. © 2019, © Pavel Chigansky, Peter Jagers, and Fima C. Klebaner.
- Branching processes
- carrying capacity
- density dependence
- population dynamics
- threshold size