Title: Self-organisation in nature: patterns and bifurcations in nonlocal advection-diffusion models
Abstract: Understanding the mechanisms behind self-organisation of mobile organisms is a central question in ecology and biology. In nature, individuals - whether cells or animals - sense their environment before moving. This process is typically nonlocal, as information is gathered from a wider region rather than just the immediate surroundings. Empirical studies highlight nonlocality as a key feature of movement, and mathematical models that incorporate it are increasingly used to describe self-organisation in biological systems. Beyond capturing real-world phenomena more faithfully, nonlocal models also display richer dynamical behaviours, making them of interest to both modellers and analysts.
In this talk, I will present a class of advection-diffusion equations modelling population movement driven by nonlocal species interactions. Using analytical and numerical techniques, I will show that these models support a wide range of spatio-temporal behaviours, including segregation, aggregation, time-periodic dynamics, and chase-and-run phenomena. I will also discuss parameter regimes with multiple stable solutions and hysteresis effects.
Overall, I will highlight methods for analysing bifurcations and pattern formation in these models, which provide essential mathematical tools for understanding self-organisation and emergent behaviours in nature.
Title: Modelling collective cell motion in development
Abstract: Collective cell movement is observed in many areas in biology, including embryonic development, wound healing and disease. Models to address these phenomena range from hybrid agent-based to coupled systems of partial differential equations. I present a range of models of the latter type proposed for cancer cell invasion. I will then present a very simple agent-based model that we have used, in collaboration with experimentalists, to generate new insights into cell migration in embryonic development, specifically the chick cranial neural crest.
Title: Understanding Spatiotemporal Patterns Using Absolute Stability
Abstract: TBC
(It is a funder requirement for the organiser to give a talk)
Title: Can we predict wavelength changes of banded vegetation patterns?
Abstract: Banded vegetation patterns are a common feature in drylands. The ability to self-organise into alternating stripes of vegetated and bare soil areas is thought to be a resilience mechanism that prevents catastrophic tipping of dryland plant ecosystems. Several mathematical models exist that describe the dynamics of dryland vegetation bands as periodic travelling waves (PTWs). Models predict that if environmental stress increases, dryland vegetation bands undergo cascades of wavelength changes that progressively increase the characteristic distances between stripes before a transition to desert. It is thus of crucial importance to understand when (i.e., at what parameter values) and how (i.e., to which new wavelength) PTW wavelength changes occur.
In this talk, I show that the traditionally used method of using Busse balloon boundaries to predict parameter values at which wavelength changes occur is often insufficient. Instead, I show that model solutions enter a (potentially long) transient after crossing a stability boundary and present a method to estimate the order of magnitude of the length of this transient. I further review our current knowledge of PTW wavelength selection, a problem that remains unsolved except for special cases, and will present new numerical evidence of selection principles in the context of dryland vegetation patterns.