Abstract:
This thesis investigates emergent phenomena associated with biological systems through the modelling, derivation, analysis, and simulation of multiscale dynamics. A central challenge is to develop models that both capture microscopic particle interactions and remain mathematically and computationally tractable at larger scales. To address this, we employ kinetic theory, coarse-graining and numerical simulations by investigation of three complementary projects.
First, we introduce a particle model for anisotropic particles interacting via a Gaussian-type
repulsive potential that encodes volume exclusion. We formally derive the corresponding kinetic and macroscopic equations via the mean-field and hydrodynamic limits and demonstrate that this repulsion naturally induces nematic alignment and non-trivial spatial effects. Moreover, we perform numerical simulations of the particle system to identify parameter regimes supporting the continuum description. The macroscopic derivation reveals a nonlinear diffusion of the particle density, modulated by particle anisotropy, and a coupled transport–diffusion dynamics for the mean orientation of the particles.
Next, we propose and study an FFT (Fast Fourier Transformation)-based spectral method to simulate a continuum model of biological network formation under periodic boundary conditions. The method is simpler to implement than implicit schemes and enables efficient computations. We reproduce the previously documented influence of the activation, diffusion and metabolic exponent parameters on the network morphology and report grid-convergence results.
Lastly, we extend a fiber network model with linking and unlinking dynamics by incorporating a repulsive potential. We provide a formal derivation of the corresponding
kinetic equation and identify two assumptions that are critical for establishing a rigorous mean-field limit. Hence, this derivation clarifies key analytical challenges for future rigorous proofs.