Leírás
A system of Poissonian interacting trajectories (PIT) was recently
introduced in our joint paper with Felix Hermann, Adrián González
Casanova, Renato Soares dos Santos, and Anton Wakolbinger. Such a
system of $[0,1]$-valued piecewise linear trajectories arises as a
scaling limit of the system of logarithmic subpopulation sizes in a
Moran model with mutation and selection. The Moran model is one of the
classical population-genetic models with fixed total population size;
in the simplest case without mutation and selection it is a
continuous-time variant of the Wright-Fisher model, which corresponds
to the voter model on a complete graph. In the setting that we are
interested in, selection is strong and the rate of beneficial
mutations is in the so-called Gerrish-Lenski regime, where the
inter-arrival times between consecutive mutations are of the same
order as the durations of mutant invasions (selective sweeps). Changes
of the resident population yield kinks (slope changes) in the resident
population.
We show that the PIT exhibits an almost surely positive asymptotic
rate of increase of the fitness of the resident population (called the
speed of adaptation), which turns out to be finite if and only if
fitness increments have a finite expectation. I will sketch the proof
of this assertion, which is based on a renewal argument. Together with
results in earlier work by other authors, this argument leads easily
to a functional central limit theorem for the resident fitness in case
the fitness increments have a finite second moment.
A modification of the renewal argument implies that the time-average
of the number of kinks of the PIT converges almost surely to a
deterministic limit. This limit turns out to be positive and finite
for any fitness increment distribution (unlike the speed of
adaptation). This assertion is included in our joint follow-up paper
with Katalin Friedl and Viktória Nemkin, where we study algorithmic
aspects of interacting trajectories. If time permits, I will also
mention our algorithmic results and some interesting (and apparently
difficult) open problems related to the speed of adaptation.