LMS workshop on the Mathematics of delayed phenomena

A two-day event on the dynamics of time-delay systems
22-23 March 2023

A multidisciplinary two-day workshop on the study of delays in nonlinear dynamics. Delays, which can either be constant, time-, state- or noise-dependent, generically induces an infinite-dimensional nature for the dynamical system. Recent theoretical and numerical advances enable the detection and description of bifurcations and chaotic transitions, in particular in the presence of environmental noise. These breakthroughs have far-reaching consequences in various scientific fields where delays are ubiquitous, such as climate modelling, epidemiology, endocrinology, nonlinear optics, control theory and engineering.
The workshop will feature 12 speakers from the very active community of researchers contributing to the study of delayed phenomena, thriving in three main scientific areas of study:

Development of theoretical knowledge to characterise global and local dynamics in delayed systems
Development of numerical tools for the detection and continuation of bifurcations in multidimensional parameter spaces
Modelling physical or biological systems taking into account inherent delays and making use of theoretical and numerical tools to describe bifurcations or critical phenomena.

This meeting is made possible thanks to the support of the London Mathematical Society and the Department of Mathematics, Physics and Electrical Engineering at Northumbria University. The workshop will be held in person. Talks will be livestreamed. Let us know if you would like to participate remotely, and we will send you the information on how to access the livestream.

Speakers

Schedule

22 March (ELA 001)

1.00-1.15pm: Welcome
1.15-1.50pm: Marianna Cerasuolo (Portsmouth)
1.50-2.25pm: Kyle Wedgwood (Exeter)
2.25-3pm: Yuliya Kyrychko (Sussex)
3-3.45pm: Coffee break and poster (in ELA 003)
3.45-4.20pm: Serhiy Yanchuk (Potsdam)
4.20-4.55pm: Francesca Scarabel (Leeds)
4.55-5.30pm: Andrew Keane (Cork)
9.05-9.40am: Sarah Loos (Cambridge)
9.40-10.15am: Xuerong Mao (Strathclyde)
10.15-11am: Coffee break and discussions
11.00-11.35am: Natalia Janson (Loughborough)
11.35-12.10pm: Rod Halburd (UCL)
1.00-1.35pm: Andrew Krause (Durham)
1.35-2.10pm: Tyler Cassidy (Leeds)

Abstracts

Marianna Cerasuolo (Portsmouth): Numerics and approximations for gamma distributed delay differential equations

Prostate cancer is the fifth most common cause of death from cancer, and the second most common diagnosed cancer in men. In the last few years many mathematical models have been proposed to describe the dynamics of prostate cancer under treatment. So far one of the major challenges has been the development of mathematical models that could represent experiments in vivo conditions (experiments on individuals) and therefore be suitable for clinical applications, while being mathematically tractable.

In this talk, taking a step in this direction, I am going to propose a nonlinear distributed-delay dynamical system that explores neuroendocrine transdifferentiation in human prostate cancer in vivo. Sufficient conditions for the existence and the stability of a tumour-present equilibrium will be given, and the occurrence of a Hopf bifurcation will be proven for a uniform delay distribution. Numerical simulations will be showed to explore differences in behaviour for uniform and exponential delay distributions. The study of the dynamical system will show how the choice of the delay distribution is key in defining the dynamics of the system and in determining the conditions for the onset of oscillations following a switch in the stability of the tumour-present equilibrium.
Tyler Cassidy (Leeds): Numerics and approximations for gamma distributed delay differential equations

Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). Unfortunately, this Erlang approximation imposes an artificial relationship between the mean and variance of the delayed process. Accordingly, we develop a numerical method to numerically integrate the gamma distributed DDE without relying on the Erlang approximation. Alternatively, we derive an ODE approximation of the gamma distributed DDE that is a more accurate than the common Erlang approximation. Using our numerical method to provide reference solutions, we show that the Erlang approximation may produce qualitatively different solutions that the underlying gamma distributed DDE while our proposed approximation does not.

Full details at https://sites.google.com/view/mdp2023.