arXiv subjects

Anna Kraut

Publications and source records attributed to Anna Kraut.

6 recordsLinked to original sources

arXiv2026

Stochastic individual-bases models

These are lecture notes for an advanced topics course in the master's programme at Bonn University. It aims to give a concise review of some of the work that has been done around the topic of adaptive dynamics from a rigorous stochastic point of view over the last 25 years and to make this area accessible to students with a good knowledge of probability in general, and the theory of Markov processes in particular. Our emphasis is on the issue of emerging scaling limits, where scaling parameters are time, population size, mutation rates, and mutation step size. These allow to exhibit within a fairly simple class of models a variety of biologically relevant phenomena. These notes are organised in three parts. Part 1 presents some historical background as well as the mathematical setting and main tools. Part 2 is the core of the notes and discusses the various scaling regimes and scaling limits. Part 3 looks at some extensions on the basic model, notably diploid models, phenotypic plasticity, and effects of environmental changes over time.

q-bio.PE↗

arXiv2025

Crossing a fitness valley in a changing environment: With and without pit stop

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation. Biologically motivated by the influence of seasons or the variation of drug concentration during medical treatment, the model parameters vary over time as piecewise constant and periodic functions. We study the typical evolutionary behavior of the population by looking at limits of large populations and rare mutations. An analysis of the crossing of valleys in the fitness landscape in a changing environment leads to various interesting phenomena on different time scales, which depend on the length of the valley. By carefully examining the influence of the changing environment on each time scale, we are able to determine the crossing rates of fit mutants and their ability to invade the resident population. In addition, we investigate the special scenario of pit stops, where single intermediate mutants within the valley have phases of positive fitness and can thus grow to a diverging size before going extinct again. This significantly accelerates the traversal of the valley and leads to an interesting new time scale.

math.PR↗

arXiv2024

A general multi-scale description of metastable adaptive motion across fitness valleys

We consider a stochastic individual-based model of adaptive dynamics on a finite trait graph $G=(V,E)$. The evolution is driven by a linear birth rate, a density dependent logistic death rate an the possibility of mutations along the (possibly directed) edges in $E$. We study the limit of small mutation rates for a simultaneously diverging population size. Closing the gap between the works of Bovier, Coquille and Smadi (2019) and Coquille, Kraut and Smadi (2021), we give a precise description of transitions between evolutionary stable conditions (ESC), where multiple mutations are needed to cross a valley in the fitness landscape. The system shows a metastable behaviour on several divergent time scales associated to a degree of stability. We develop the framework of a meta graph that is constituted of ESCs and possible metastable transitions between those. This allows for a concise description of the multi-scale jump chain arising from concatenating several jumps. Finally, for each of the various time scale, we prove the convergence of the population process to a Markov jump process visiting only ESCs of sufficiently high stability.

math.PR↗

arXiv2023

Effective growth rates in a periodically changing environment: From mutation to invasion

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation, with linear birth and death rates, as well as a density-dependent competition. To depict repeating changes of the environment, all of these parameters vary over time as piecewise constant and periodic functions, on an intermediate time-scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). Studying the growth of emergent mutants and their invasion of the resident population in the limit of small mutation rates for a simultaneously diverging population size, we are able to determine their effective growth rates. We describe this growth as a mesoscopic scaling-limit of the orders of population sizes, where we observe an averaging effect of the invasion fitness. Moreover, we prove a limit result for the sequence of consecutive macroscopic resident traits that is similar to the so-called trait-substitution-sequence.

q-bio.PE↗

arXiv2020

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

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 $μ_K$ given by a negative power of the carrying capacity $K$ of the system: $μ_K=K^{-1/α},α>0$. 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 [Champagnat, Méléard, Tran, 2019] 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 [Kraut, Bovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille, Smadi, 2018] 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.

math.PR↗

arXiv2020

From Adaptive Dynamics to Adaptive Walks

We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modelled as a measure-valued Markov process. Multiple variations of this system have been studied in the simultaneous limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and then let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple microscopic types are present at the same time. The limiting process resembles an adaptive walk or flight and jumps between different equilibria of coexisting types. The graph structure on the type space, determined by the possibilities to mutate, plays an important role in defining this jump process. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.

q-bio.PE↗