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.