Abstract:
This thesis contributes to the mathematical analysis of kinetic equations describing emergent phenomena in biological systems through asymptotic analyses. This framework connects individual-based models, characterized by microscopic interactions, with macroscopic equations formulated as partial differential equations (PDEs). The manuscript is organized into four major sections, each addressing a different question in the application of kinetic theory to biological systems. The first part focuses on a third-order stochastic differential equation (SDE) system that combines orientation alignment dynamics with angular velocity synchronization to describe the collective behavior of flagellated bacteria. From this agent-based model, we derive hydrodynamic equations using the concept of Generalized Collision Invariants, yielding a macroscopic system for the density, mean direction, and angular velocity of particles. Numerical simulations of the particle system reveal diverse emergent patterns, including rotating clusters, traveling orientation waves, and pulsating dynamics, which are compared to simulations of the macroscopic model. Motivated by biological phenomena where interactions between multiple agents occur over finite time spans and involve continuous state changes, the second part of the manuscript extends the classical kinetic theory to account for non-instantaneous, multi-particle collisions. Inspired by coagulation-fragmentation processes, we introduce a system of equations governing the evolution of cluster distribution densities fk (v1, ..., vk). We first derive a first-order correction to the instantaneous model that captures the effects of finite duration interactions. We then establish the well-posedness of the approximated model and its convergence to the classical instantaneous kinetic equation in the limit as the interaction duration tends to zero. This result confirms that the first-order correction yields a consistent approximation of the instantaneous model in the regime of short interaction times. Furthermore, the third section examines the existence of steady-state solutions to a transport-coagulation-nucleation equation that models aggregate formation in biological processes, particularly autophagy.
Under suitable decay assumptions on the transport velocity, the existence of non-trivial stationary solutions with finite mass is established via Schauder fixed-point theorem. Lastly, the thesis explores protein transport mechanisms within the cell, specifically between the endoplasmatic reticulum and the nuclear envelope. We introduce a simplified diffusion model, from which asymptotic analysis provides explicit expressions for transport rates in agreement with experimental data, illustrating how the geometry of intracellular compartments can influence the transport of some proteins within the cell.

Online:
univienna.zoom.us/j/62551307515
Meeting ID: 625 5130 7515
Kenncode: 041666