Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities

TL;DR

This paper analyzes how small populations grow into large, deterministic systems, highlighting the transition from stochastic early stages to predictable large-scale behavior as the environment's capacity increases.

Contribution

It provides a rigorous framework connecting initial stochastic growth to deterministic evolution in large populations, emphasizing the role of the variable W as a stochastic initial condition.

Findings

Population growth becomes deterministic at large sizes

Early stochastic effects are encapsulated by the variable W

Growth dynamics resemble iterates of a conditional expectation operator

Abstract

Many populations, e.g. of cells, bacteria, viruses, or replicating DNA molecules, start small, from a few individuals, and grow large into a noticeable fraction of the environmental carrying capacity $K$. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from $Z_0=z_0$ in a branching process or Malthusian like, roughly exponential fashion, $Z_t \sim a^tW$, where $Z_t$ is the size at discrete time $t\to\infty$, $a>1$ is the offspring mean per individual (at the low starting density of elements, and large $K$), and $W$ a sum of $z_0$ i.i.d. random variables. It will, thus, become detectable (i.e. of the same order as $K$) only after around $\log K$ generations, when its density $X_t:=Z_t/K$ 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 $z_0$, due to variations during the early development. However, from that time onwards, law of large numbers effects will render the process deterministic, though initiated 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 $K\to\infty$. 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.