Understanding self-organisation in nature: Patterns and Bifurcations in Nonlocal Advection-Diffusion Models

ValeriaGiunta

Swansea University

"Understanding self-organisation in nature: Patterns and Bifurcations in Nonlocal Advection-Diffusion Models"

Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level emerges from individual behaviour, a mathematical approach is essential to elucidate these dynamics. In nature, individuals - whether cells or animals - inspect their environment before moving. This process is typically nonlocal, meaning that individuals gather information from a part of their environment rather than just their immediate location. Empirical research increasingly highlights nonlocality as a key aspect of movement, while mathematical models incorporating nonlocal interactions have gained attention for their ability to describe how interactions shape movement, reproduction and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movement driven by nonlocal species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide range of spatio-temporal patterns, including segregation, aggregation, time-periodic behaviour, and chase-and-run dynamics. I will also discuss the existence of parameter regions with multiple stable solutions and hysteresis phenomena. Overall, I will explore various methods for analysing the bifurcation and pattern formation properties of these models, which provide essential mathematical tools for understanding the many aggregation phenomena observed in nature.
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