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Highlights
• We study dormancy in the ‘rare mutation’ regime of stochastic adaptive dynamics.
• We first derive the polymorphic evolution sequence, based on prior work.
• Our evolutionary branching criterion extends a result by Champagnat and Méléard.
• In a classical model dormancy can favour evolutionary branching.
• Dormancy also affects several more population characteristics.
Abstract
In this paper, we investigate the consequences of dormancy in the ‘rare mutation’ and ‘large population’ regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second ‘small mutations’ limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011).
The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the ‘speed of adaptation’ of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.
Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.
During my initial days here in Frankfurt, in October 2020 amidst the pandemic crisis, all my notes revolved around three articles by Bolthausen and Kistler, which now form the starting point of this work.
The ones introduced by Bolthausen and Kistler are abstract mean field spin glass models, reminiscent of Derrida’s Generalized Random Energy Model (GREM), which generalize the GREM while remaining rigorously solvable through large deviations methods and within a classical Boltzmann-Gibbs formalism. This allows to establish, by means of a second moment method, the associated free energy at the thermodynamic limit as an orthodox, infinite-dimensional, Boltzmann-Gibbs variational principle.
Dual Parisi formulas for the limiting free energy associated with these Hamiltonians hold, and are revealed to be the finite-dimensional (”collapsed”) versions of the classical, infinite-dimensional Boltzmann-Gibbs principles.
In the 2nd chapter of this thesis, we uncover the hidden yet essential connection between real-world spin glasses, like the Sherrington-Kirkpatrick (SK) model and the random energy models. The crucial missing element is that of TAP-free energies: integrating it with the framework introduced by Bolthausen and Kistler results in a correction to the Parisi formula for the free energy, which brings it much, much closer to the ”true” Parisi solution for the SK-model. In other words, we can identify the principles that transform the classical Boltzmann-Gibbs maximization into the unorthodox (and puzzling) Parisi minimization.
This arguably stands as the primary achievement of this work.
Using limit linear series on chains of curves, we show that closures of certain Brill-Noether loci contain a product of pointed Brill-Noether loci of small codimension. As a result, we obtain new non-containments of Brill-Noether loci, in particular that dimensionally expected non-containments hold for expected maximal Brill-Noether loci. Using these degenerations, we also give a new proof that Brill-Noether loci with expected codimension −ρ≤⌈g/2⌉ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill-Noether loci by considering the locus of the source curves of unramified double covers.
Die Arbeit befasst sich mit einer Vereinfachung des von Devroye (1999) geprägten Begriffs der random split trees und verallgemeinert diesen im Sinne von Janson (2019) auf unbeschränkten Verzweigungsgrad. Diese Verallgemeinerung deckt auch preferential attachment trees mit linearen Gewichten ab, wofür ein Beweis von Janson (2019) aufbereitet wird. Zusätzlich bleiben die von Devroye (1999) nachgewiesenen Eigenschaften über die Tiefe der hinzugefügten Knoten erhalten.