Colonization times in Moran process on graphs

TL;DR

This paper analyzes colonization times in the Moran process on graphs, showing polynomial bounds for colonization duration and proposing a shift from step-counting to real-time measurement in spatially structured populations.

Contribution

It provides the first polynomial bounds on colonization times in Moran processes with zero reproduction residents and identifies the slowest spatial structures for colonization.

Findings

Colonization time is at most a polynomial function of population size.

Exact colonization time for the slowest spatial structure is identified as (n) = /2 n^3 - /2 n^2.

Proposes measuring real process duration instead of step counts.

Abstract

Moran Birth-death process is a standard stochastic process that is used to model natural selection in spatially structured populations. A newly occurring mutation that invades a population of residents can either fixate on the whole population or it can go extinct due to random drift. The duration of the process depends not only on the total population size $n$, but also on the spatial structure of the population. In this work, we consider the Moran process with a single type of individuals who invade and colonize an otherwise empty environment. Mathematically, this corresponds to the setting where the residents have zero reproduction rate, thus they never reproduce. We present two main contributions. First, in contrast to the Moran process in which residents do reproduce, we show that the colonization time is always at most a polynomial function of the population size $n$. Namely, we show that colonization always takes at most $\frac12n^3-\frac12n^2$ expected steps, and for each $n$, we exactly identify the unique slowest spatial structure where it takes exactly that many steps. Moreover, we establish a stronger bound of roughly $n^{2.5}$ steps for spatial structures that contain only two-way connections and an even stronger bound of roughly $n^2$ steps for lattice-like spatial structures. Second, we discuss various complications that one faces when attempting to measure fixation times and colonization times in spatially structured populations, and we propose to measure the real duration of the process, rather than counting the steps of the classic Moran process.