Abstract
Processes of Darwinian evolution are dynamic, nonlinear, and underly fluctuations. A
way to analyze systems of Darwinian evolution is by using methods well established in
statistical physics. The main mechanisms that are responsible for evolutionary changes
are reproduction, mutation, and selection. Individuals reproduce and inherit genes and
traits, such that a population evolves. Mutations occur spontaneously, e.g., by errors in
reproduction, whereby different new types of genes or traits can emerge. Selection acts
on different types.
This thesis focuses on selection in systems that underlie the principles of Darwinian
evolution, as well as fluctuations. Once there are different types, their interactions with
each other can influence their reproductive rates. One important framework to look
at such interactions is game theory. In evolutionary game theory, different types are
identified with different strategies, and the payoff of a strategy affects the reproductive
success. An important property of evolutionary games is that, in general, the evolutionary
success of a strategy varies with the composition of the population.
The event of a mutation taking over a population is called fixation. The quantities
mainly considered in this thesis are the fixation times of a mutant strategy. They are
a measure for the time a population spends reaching the state of only mutants, when
starting from a few.
The role of selection is to control the payoff differences between types, which gives
rise to several regimes of selection. In the absence of selection evolution is neutral and
fluctuations dominate. An important limit case is weak selection, which introduces a
small bias to the random evolutionary changes. In this thesis, weak selection analysis
plays an important part in the classification of different evolutionary processes. This
allows to simplify the nonlinear dynamical system and thus an analytical description.
Here, approximative formulations of the fixation times under weak selection are presented,
and the universality of the weak selection regime is addressed. On intermediate scales,
one can observe that the average fixation time of an advantageous mutation increases
with selection, although the probability of fixation also increases.
One can then move on to strong selection, such that selection dominates the dynamics
even in small systems. Here, one can observe segregation effects, where the initial
condition determines the fate of the finite population in a deterministic way.
Another important evolutionary mechanism is gene flow, e.g., caused by migration
between populations of the same species. In this context, migration can counterbalance
selection. In systems with bi-stable evolutionary dynamics, the migration-selection
equilibrium can lead to coexistence that is stable for a long time. This thesis gives a
quantitative analysis of the dynamical and statistical properties of such a system. To
this end, the extinction (fixation) times are analyzed also in the nonlinearly coupled
population system.
