Seminar 14.3.2024: Richard Kollar

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Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 14.3.2024 at 14:00 Lecture room M-223

Richard Kollár (KAMŠ FMFI UK):
Self-organization on multiple scales

Abstract:
We explore self-organization in a stochastic dynamical system using rigorous and numerical multiscale analysis. The system, at the microscale, describes lane-switching of individuals moving in opposite directions on a circular multi-lane track. At the mesoscale, some system variants are modeled as absorbing Markov chains, while others are not. However, at the macroscale, all variants reduce to an absorbing Markov chain model, ensuring self-organization within a finite time. Our findings show a constant upper limit for the mean time to self-organization regardless of the number of individuals in the system in variants with mesoscopic Markovian properties. That aligns with a simplified Poisson process model following the Mass action law. Conversely, non-Markovian variants on the mesoscale exhibit longer self-organization times, although numerical simulations indicate that some of them also achieve bounded self-organization times. We also identify a non-Markovian variant that demonstrates crowd formation tendencies, leading to prolonged time of self-organization. For another variant, we confirm linear growth in self-organization time observed in the literature for simple lane-switching models. Our study provides a framework for applying absorbing Markov chain theory in self-organization, particularly in systems involving conflict resolution mechanisms. This is a joint work with Katarína Boďová.