Peter Jagers: How might a population have started and where is it heading?
Tid: On 2018-11-21 kl 14.00
Föreläsare: Peter Jagers, Chalmers/University of Gothenburg
Plats: Seminar Hall Kuskvillan, Institut Mittag-Leffler
Many populations, e.g. 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 in a branching process or Malthusian like, roughly exponential fashion multiplied by a random factor W mirroring variations during the early development. Thus, they will first be observed around time log K. However, from then onwards, density and law of large numbers effects will render the process deterministic, though initiated by a random size, expressed through the variable W, acting 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 K tends to infinity. The fundamental proof ideas are to couple the initial system to a branching process and to show that late population 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.