Colonization times in Moran process on graphs

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. The spatial structure is represented by a graph. 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 12n3−12n2 expected steps, and for each n, we identify the slowest graph where it takes exactly that many steps. Moreover, we establish a stronger bound of roughly n2.5 steps for undirected graphs and an even stronger bound of roughly n2 steps for so-called regular graphs. 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.Author summary: Consider an invasive species that is about to colonize an otherwise empty environment. In the absence of natural enemies, the species will eventually spread everywhere, but the time until the colonization is completed will depend on the exact spatial layout of the individual sites. In this work, we analyze this colonization time for various spatial layouts. We give precise formulas for the average colonization time for several commonly studied spatial layouts (such as well-mixed populations, cycles, or stars). For each population size n, we also identify the slowest layout and show that the corresponding (slowest possible) colonization time is cubic in the population size n. Moreover, we prove an asymptotically tight general bound on the colonization time that applies to any lattice-like layout, and another bound that applies to any layout in which all connections are two-way. We conclude by discussing the implications of our results for further study of a related, well-researched quantity called the fixation time.