Stochastic individual-based models with power law mutation rate on a general finite trait space

Abstract

We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph G = (V, E). The evolution of the population is driven by births, deaths, competition, and mutations along the edges of G. We are interested in the large population limit under a mutation rate µK given by a negative power of the carrying capacity K of the system: µK = K −1/α , α > 0. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the log K time scale, as K tends to infinity. Using techniques developed in [8], we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises [25] to the stochastic setting, and Theorem 3.2 of [6] to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.

A stochastic adaptive dynamics model for bacterial populations with mutation, dormancy and transfer

“…In addition, the introduction of dormancy seems to allow for the system to be driven towards a state of coexistence in the following sense: At no point in time there are more than two traits with size of order K, but on the log K timescale, there exists a finite time T 1 < ∞ such that for all ε > 0 there exists a time T 0 < T 1 such that on the time interval [T 0 , T 1 ] at least three traits are of order at least K 1−ε , which means that at least three traits are simultaneously macroscopic on a suitable interval. This behaviour has been found previously by Coquille et al (2021) in the case of asymmetric competition without HGT. In the model studied in Durrett and Mayberry (2011), the set of points where the limiting piecewise linear process changes slopes may also have a finite accumulation point, see Lemma 1 therein.…”

“…In the adaptive dynamics literature, this scaling of mutations occurred before in Smadi (2017); Bovier et al (2019). From a mathematical point of view, the main novelty of the paper Champagnat et al (2021) is the systematic study of logistic birth-and-death processes with non-constant immigration, as it was also noted in Coquille et al (2021).…”

“…Indeed, the papers of Billiard et al (2016Billiard et al ( , 2018 have investigated the consequences of a simple directional HGT mechanism in stochastic individual based models with a focus on its interplay with competition, mutation, and the maintenance of polymorphic variability. In Champagnat et al (2021), the approach is transferred and extended into an adaptive dynamics setting with moderately large mutation rates (as previously considered in Durrett and Mayberry (2011), see also Coquille et al (2021)), providing a rather new and sophisticated mathematical machinery that leads to interesting scaling limits and emergent behaviour on a 'doubly logarithmic scale'. It is shown that HGT can have major consequences for the long-term behaviour of the affected systems, including coexistence, evolutionary suicide and evolutionary cyclic behaviour, depending on the strength of the transfer rate.…”

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

“…Scaling limits of individual-based models on a discrete trait space with rare mutations and large population, and allowing to deal with negligible populations and local extinction, were proposed in [13,4,9,10,3]. These references focus on population sizes of the order of K β and characterize the asymptotic dynamics of the exponent β.…”

“…The fact that the trait space is discrete allows to describe separately the dynamics of each small sub-population. However, this makes the detailed description of the asymptotic dynamics very complicated (see [9,10]). Note that mutations are assumed individually rare in these references, but they are more frequent than in the scaling limits of adaptive dynamics (see e.g.…”