Page 534 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 534
PDE MODELS IN LIFE AND SOCIAL SCIENCES (MS-71)
derived using a Lagrangian approach on the mean-field level. Based on these conditions we
propose a gradient descent method to identify relevant parameters such as angle of rotation and
force scaling which may be spatially inhomogeneous. We discretize the first-order optimality
conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate
for the convergence of the controls as the number of particles used for the discretization tends
to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility
of the approach. This is joint work with M. Burger (FAU Erlangen-Nürnberg) and C. Totzeck
(Wuppertal).
A Hamilton-Jacobi formalism for the study of propagation in
reaction-subdiffusion systems
Álvaro Mateos González, agm6@nyu.edu
New York University Shanghai, China
Coauthors: Vincent Calvez, Pierre Gabriel
Certain intracellular protein exhibit random motion that deviates from standard diffusion due
to trapping phenomena. These systems may be described by a probability density function
n(t, x, a) depending on time t, space x, and also on a structural memory or ‘age’ variable a that
allows to account for the trapping. Roughly speaking, n is governed by a renewal equation in
(t, a) (with a heavy-tailed waiting times distribution) coupled with spatial relocation at renewal.
I will motivate and give an overview of certain results obtained at the end of my PhD the-
sis on the hyperbolic space-time asymptotics of those equations, how they tend to a limiting
Hamilton-Jacobi equation, and what this means. The interesting features of our work lie in
how we dealt with complications in the limit procedure due to the memory effects being ‘non-
Markovian’ in a certain sense.
Classification and stability analysis of polarising and depolarising
travelling wave solutions for a model of collective cell migration.
Dietmar Oelz, d.oelz@uq.edu.au
University of Queensland, Australia
We study travelling wave solutions of a 1D continuum model for collective cell migration in
which cells are characterised by position and polarity. Four different types of travelling wave
solutions are identified which represent polarisation and depolarisation waves resulting from
either colliding or departing cell sheets as observed in model wound experiments. We study
the linear stability of the travelling wave solutions numerically and using spectral theory. This
involves the computation of the Evans function most of which we are able to carry out explicitly,
with one final step left to numerical simulation.
532
derived using a Lagrangian approach on the mean-field level. Based on these conditions we
propose a gradient descent method to identify relevant parameters such as angle of rotation and
force scaling which may be spatially inhomogeneous. We discretize the first-order optimality
conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate
for the convergence of the controls as the number of particles used for the discretization tends
to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility
of the approach. This is joint work with M. Burger (FAU Erlangen-Nürnberg) and C. Totzeck
(Wuppertal).
A Hamilton-Jacobi formalism for the study of propagation in
reaction-subdiffusion systems
Álvaro Mateos González, agm6@nyu.edu
New York University Shanghai, China
Coauthors: Vincent Calvez, Pierre Gabriel
Certain intracellular protein exhibit random motion that deviates from standard diffusion due
to trapping phenomena. These systems may be described by a probability density function
n(t, x, a) depending on time t, space x, and also on a structural memory or ‘age’ variable a that
allows to account for the trapping. Roughly speaking, n is governed by a renewal equation in
(t, a) (with a heavy-tailed waiting times distribution) coupled with spatial relocation at renewal.
I will motivate and give an overview of certain results obtained at the end of my PhD the-
sis on the hyperbolic space-time asymptotics of those equations, how they tend to a limiting
Hamilton-Jacobi equation, and what this means. The interesting features of our work lie in
how we dealt with complications in the limit procedure due to the memory effects being ‘non-
Markovian’ in a certain sense.
Classification and stability analysis of polarising and depolarising
travelling wave solutions for a model of collective cell migration.
Dietmar Oelz, d.oelz@uq.edu.au
University of Queensland, Australia
We study travelling wave solutions of a 1D continuum model for collective cell migration in
which cells are characterised by position and polarity. Four different types of travelling wave
solutions are identified which represent polarisation and depolarisation waves resulting from
either colliding or departing cell sheets as observed in model wound experiments. We study
the linear stability of the travelling wave solutions numerically and using spectral theory. This
involves the computation of the Evans function most of which we are able to carry out explicitly,
with one final step left to numerical simulation.
532
