# Colloquium Prof. Peter Jagers, Chalmers University Göteborg: "How might a population have started and where is it heading?"

Oct 26, 2018 from 04:15 PM to 05:15 PM

LMS 4 - Rau, 424 - Kleiner Hörsaal

Abstract:

Many populations, e.g. of cells, bacteria, viruses, or replicating DNA molecules, but also of species invading a habitat, or physical systemsof elements generating new elements, start small, from a few lndividuals,and grow large into a noticeable fraction of the environmentalcarrying 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 roughly at time log*K.*However, from that time onwards, density and law of large numbers effects will render the process deterministic, though initiated by the 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.