Day 2 (April 7) @ 11:00–12:30

Stefanie Sonner (Radboud Universiteit)

Continuum Models for Spatially Heterogeneous Biofilm Communities

Biofilms are dense aggregations of bacterial cells attached to a surface and held together by a self-produced matrix of extracellular polymeric substances. Such multi-cellular communities are a very successful life form and are able to tolerate harmful environmental impacts that would eradicate free floating individual cells.

Biofilms affect many aspects of human life and play a crucial role in natural, medical and industrial settings. Mathematical models for their growth have been developed for several decades. We focus on deterministic continuum models for spatially heterogeneous biofilm communities that are formulated as quasilinear reaction-diffusion-systems. Their characteristic and challenging feature are the two-fold degenerate diffusion coefficients for the biomass fractions comprising a power-law degeneracy (known from the porous medium equation) and a super diffusion singularity.

The prototype growth model is discussed as well as recent variations and extensions that take additional biofilm processes into account. Analytical results are shown and numerical simulations presented to illustrate the model behavior.

Frits Veerman (University of Leiden)

Toll roads and freeways: the role of cholesterol on the stability of bilayer interfaces.

We study a multi-component extension of the functionalised Cahn-Hilliard equation, which provides a framework for the formation of patterns in fluid systems with multiple amphiphilic molecules. The assumption of a length scale dichotomy between two amphiphilic molecules allows the application of geometric techniques for the analysis of patterns in singularly perturbed reaction-diffusion systems. For a generic two-component system, we show that solutions to the four-dimensional connection problem provide the leading order approximation for solutions to the full eight-dimensional barrier problem, which can be obtained through a perturbative expansion in the layer width. In this general context, we show that the presence of a weakly hydrophilic molecule such as cholesterol can stabilise phospholipid bilayer patterns. Moreover, we show that a saddle-node bifurcation of bilayer solutions in the four-dimensional connection problem acts as a source of so-called defect solutions, i.e. solutions to the barrier problem that are not also solutions to the connection problem. The analysis combines geometric singular perturbation theory with centre manifold theory in an infinite-dimensional context.

A short article giving an overview of the model context can be found on DSWeb (“Short amphiphilic molecules: curve lengthening, defects, and the role of cholesterol”).

Daniele Avitabile (VU Amsterdam and Inria MathNeuro Team, France)

Canards and Slow Passage Through Bifurcations in Infinite-Dimensional Dynamical Systems 

Ordinary Differential Equations (ODEs) in which state variables evolve according to disparate time scales are known to support solutions exhibiting slow passages through bifurcations and canard segments. These solutions are indeed considered to be footprints of time-scale separation, and they have been studied extensively in the past decades to explain a vast repertoire of temporal patterns including mixed-mode oscillations, bursting, and excitable dynamics. 

The literature on the topic deals primarily with finite-, and usually low-, dimensional dynamical systems. In the mathematical neuroscience community, for instance, there exists a well-defined methodology for single-cell models with time-scale separation, but not for spatially-extended or network models. 

I will present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a large class of infinite-dimensional dynamical systems with time-scale separation. The framework, which relies on a centre-manifold reduction proposed by Iooss and coworkers, is applicable to models where an  infinite-dimensional dynamical system for fast variables is coupled to  a finite-dimensional dynamical system for slow variables. I will  discuss examples where the fast variables evolve according to systems of local and nonlocal reaction-diffusion PDEs, integro-differential equations, or delay-differential equations. This approach opens up the possibility of studying spatio-temporal canards and slow passages through bifurcations in spatially-extended systems, and it provides an analytical foundation for several numerical observations recently reported in literature.