MS05 - ECOP-09

Nonlocal Models: Progress and Challenges in Analysis, Applications and Numerics

Wednesday, July 16 at 10:20am

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Valeria GIunta (Swansea University), Yurij Salmaniw - University of Oxford

Description:

Nonlocal advection-diffusion models are a rapidly growing area of research at the interface of mathematics and biology. These models, which include interactions beyond the immediate environment, capture complex phenomena such as long-range cell adhesion, bacterial signalling and swarming dynamics. Despite many recent advances, there are still many open questions that pose exciting challenges to researchers from different backgrounds. This mini-symposium aims to bring together a diverse group of speakers, ranging from Early Career Researchers (ECRs) to established experts. Our focus is not only on sharing recent advances in the theoretical and applied understanding of nonlocal models, but also on creating a space for networking and meaningful exchange of ideas. We see this symposium as a platform to support the consolidation of a cohesive community working on nonlocal PDEs from different perspectives, including analysis, numerics and applications. We will pay special attention to the discussions after each talk to encourage a relaxed and open dialogue. By facilitating interactions across career stages and expertise, we aim to inspire new collaborations and promote a sense of belonging among participants, further strengthening the global community engaged in this innovative and impactful field.

Raluca Eftimie

Université de Franche-Comté, France

"Mathematical models for non-local cell-cell interactions in health and disease"

Non-local cell-cell interactions via long cellular protrusions seem to be more and more prevalent in cell biology: from airineme-mediated inter-cellular communication between different skin cells in zebrafish, to cytoneme-mediated cell-cell interactions between keratinocytes in epidermal remodelling, and even tunnelling nanotubes-mediated interactions between cancer cells and surrounding non-tumour cells. In this talk, we will present a class of non-local mathematical models developed to investigate normal and abnormal wound healing processes such as keloids (these abnormal processes lead to tissue overgrowth, remodelling and invasion similar to those observed in benign tumours). The models account for non-local cell-cell and cell-matrix interactions via different signalling molecules as well as long-distance cell protrusions. We will discuss various analytical and numerical aspects associated with these non-local models from the perspective of biological applications.

Junping Shi

College of William & Mary, Williamsburg, USA

"Biological Aggregations from Spatial Memory and Nonlocal Advection"

We present a nonlocal single-species reaction-diffusion-advection model that integrates the spatial memory of previously visited locations and nonlocal detection in space, resulting in a coupled PDE-ODE system reflective of several existing models found in spatial ecology. We prove the existence and uniqueness of a Hölder continuous weak solution in one spatial dimension under some general conditions, allowing for discontinuous kernels such as the top-hat detection kernel. A robust spectral and bifurcation analysis is also performed, providing the rigorous analytical study not yet found in the existing literature. In particular, the essential spectrum is shown to be entirely negative, and we classify the nature of the bifurcation near the critical values obtained via a linear stability analysis.

Sara Bernardi

Politecnico di Torino, Italy

"Variations in nonlocal interaction range lead to emergent chase-and-run in heterogeneous populations"

In a chase-and-run dynamic, the interaction between two individuals is such that one moves towards the other (the chaser), while the other moves away (the runner). This interaction is observed in various biological systems, including cells and animals. In this talk, I will explore the behaviors that can emerge at the population level in a heterogeneous group containing subpopulations of chasers and runners. A wide variety of patterns can form, ranging from stationary patterns to oscillatory and population-level chase-and-run, with the latter describing a synchronized collective movement of the two populations. A key aspect of our study is the role of interaction ranges—the distances over which cells or organisms can sense one another’s presence. I will show that robust population-level chase-and-run emerges when the interaction range of the chaser is sufficiently larger than that of the runner. Our findings are contextualized with examples from cellular dynamics, specifically neural crest and placode cell populations, and offer insights into similar phenomena observed in ecological systems. This talk will aim to provide a deeper understanding of chase-and-run dynamics within nonlocal advection-diffusion models and contribute to the broader understanding of how simple individual interactions can lead to complex, coordinated behaviors at the population level.

Jun Jewell

University of Oxford, UK

"Long-ranged interactions shape populations and patterns in biology"

Movement shapes how populations distribute across space and evolve over time. Across biological scales, individuals move in response to interactions that are often long-ranged (nonlocal). Animals use scent cues to establish territorial boundaries, predators pursue prey based on sight or sound, and cells can aggregate by extending pseudopodia toward distant neighbours. We explore these processes using nonlocal advection-diffusion models, analysing their bifurcations to gain insight into emergent spatial and temporal dynamics. A key result is that, unlike in local models (e.g. Fickian diffusion), Turing bifurcations in these nonlocal systems fundamentally depend on spatial dimension. For example, purely repulsive interactions cannot generate spatial patterns in one spatial dimension, but can in two. Additionally, even simple interactions, such as attraction and logistic growth within a single species, can produce spatio-temporal oscillations that exhibit signs of chaos. This provides an example of spatio-temporal complexity of relevance to ongoing debates on how common chaos is in ecosystems. We also explore more complex mechanisms like chiral movement, which is often exhibited by cells and also used by prey to evade predators. We show how it can suppress oscillations, and instead promote stationary patterns. Finally, we highlight cautionary cases where linear stability analysis fails to predict long-term behaviour, including populations with a Turing instability that forms patterns only transiently before collapsing to extinction. These results emphasise the need for analytical tools which go beyond local linear stability analyses in order to understand complex biological systems in the long-term.

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Annual Meeting for the Society for Mathematical Biology, 2025.