Nonlinear PDE models in life and social sciences
- Helene Ranetbauer (University of Vienna)
- Jan Haskovec (KAUST, KSA)
In the last years, there has been a fast development in the PDE (partial differential equation) models applied to life and social sciences. The models are typically derived from microscopic approaches like individual or lattice based models, discrete network structures, or based on fundamental physical laws. They usually exhibit an underlying energy structure, being gradient flows with respect to a suitable metric, which makes them an attractive object for mathematical research. Despite many recent advances in their study, important questions concerning for instance well-posedness or efficient numerical approximation remain open. The main objective of this minisymposium is to bring together experts working in the respective areas of PDE modelling, focusing on applications in pedestrian dynamics, crowded transport phenomena, cell movement and biological network formation.
The presentations will include analytical aspects, modelling problems and numerical results. More specifically, they will range from gradient flow theory on the micro- and macroscopic level, dynamics on discrete networks, derivation of the corresponding mean-field limits to efficient numerical methods. The speakers will report on the newest progress in their fields, exchange ideas and highlight novel mathematical problems.
Motivated by the formation of fingerprint patterns we consider a class of interaction models with anisotropic interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic...
In this talk we present a mesoscopic model for natural network formation processes, acting as a bridge between a discrete and continuous network approach proposed by Hu and Cai. All models describe the pressure field and the dynamics of the conductance network under pressure force effects.
We start by presenting the different approaches and analyze their corresponding properties. We will...
After reviewing the basics about the relation between some motion flow models (pedestrians, animals, vehicles) and a class of Hamilton-Jacobi-Bellmann equations on Networks we introduce and discuss some numerical methods for the approximation of the solution. Several tests are performed to illustrate the properties previously theoretically presented.
Starting from a detailed physical model for the interplay of actin filaments, myosin motor proteins and cross-linker proteins in a contracting cell division ring, we derive a continuum model as a short filament limit of the agent based model. The model features highly nontrivial pattern formation and traveling wave solutions and explains the aggregation of actin and myosin predicted by the...