An impulsive flow-kick model of influenza
Sarah Iams

The resilience of a system is often described as its ability to maintain structure and function in the face of disturbance. I will introduce the flow-kick impulsive framework for an ordinary differential equation model, which is a modeling framework where disturbance is incorporated as intermittent discrete impulses (“kicks”) to an underlying continuous system (“flow”). Incorporating kicks generates a discrete system that is closely related to the underlying continuous model. I will discuss contexts in which this framework is appropriate, and provide examples for discussion.


The massless electron limit for the Vlasov–Poisson–Landau system
Patrick Flynn

The Vlasov–Poisson–Landau system is a kinetic model for ionized plasmas. In this talk, I will describe the massless electron limit for this system. This leads to a simplified model for the positively charged ions, where ion–electron collisions can be ignored. In joint work with Yan Guo, we give a rigorous justification of this limit. A major difficulty is that the limit is singular: specifically, there is a small factor of epsilon in front of the time derivative of the electron distribution. In general, such a coefficient could cause rapid oscillations or growth.


Multifractality in the evolution of vortex filaments
Daniel Eceizabarrena

Multifractality is one of the main properties expected in fluid turbulence. It is based on the idea that the regularity of a turbulent flow, measured via Hölder regularity, strongly depends on the point in space. In this setting, the flow is called multifractal if the measured Hölder regularities make up an open interval.


From structure to dynamics in combinatorial threshold linear networks
Caitlin Lienkaemper

Neural circuits display nonlinear dynamics. A network’s structure is a key feature determining dynamics, but many questions remain as to how structure shapes activity. We study this relationship in a simple model of neural activity, combinatorial threshold linear networks (CTLNs), whose activity is governed by a system of threshold-linear ordinary differential equations determined by an underlying directed graph. Like real networks, CTLNs display the full spectrum of nonlinear behavior, including multistability, limit cycles, and chaos.


Nonlocal cell adhesion models: bifurcations and boundary conditions
Andreas Buttenschön

In both normal tissue and disease states, cells interact with one another, and other tissue components using cellular adhesion proteins. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, and cancer metastasis. Traditionally continuum models (PDEs) of tissues are based on purely local interactions. However, these models ignore important nonlocal effects in tissues, such as long-ranged adhesion forces between cells. For this reason, a mathematical description of cell adhesion had remained a challenge until 2006, when Armstrong et.