"Fitness valleys and multi-scale analysis in changing environment"Esser, ManuelThe biological theory of adaptive dynamics aims at studying the interplay between ecology and evolution through the modeling of the basic mechanisms: heredity, mutations and competition. A rigorous derivation of the theory was achieved over the last two decades in the context of stochastic individual-based models. The typical evolutionary behaviour can be studied by looking at limits of large populations and rare mutations. While early works have shown a seperation of time scales of ecology and evolution, later articles gave conributions to eleborate the full picture of evolution in terms of a multi-scale analysis on general finite trait graphs including also fitness valleys. Despite the variety of different scenarios that have been analysed so far, all these previous works ask for the parameters of the population process to stay constant overtime. In the present work, we break with this assumption. To depict repeating changes of the environment, all of the model 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). This can biologically interpreted as the influence of seasons or the deviation of drug concentration during medical treatment. Analysing the influences of the changing environment carefully on each time scale, we are able to determine the effective growth rates of emergent mutants and their invasion of the resident population. 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 similar to the so-called trait-substitution-sequence. This is work of an ongoing collaboration with Anna Kraut (University of Minnesota). |
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