Poster Presentations

Does Speed Matter? Investigating the Role of Cell Speed Heterogeneity in Collective Migration

Collective cell migration is a fundamental process in embryonic development, yet most mathematical models assume a homogeneous population of identical agents. Motivated by experimental data on cranial neural crest cells, we investigate how heterogeneity in individual cell speeds influences the emergent dynamics of the group. We propose a stochastic Interacting Particle System (IPS) based on a run-and-tumble mechanism, where cells transition between "running" (active motion) and "tumbling" (reorientation) states. Unlike standard models, we introduce heterogeneity by segregating the population into "fast" and "slow" species, as well as considering a continuous spectrum of speeds.

Multidimensional opinion formation

Existing research on modelling opinion dynamics primarily examines scenarios in which people have opinions on only one topic. This poster presents a possibility of modelling opinion dynamics when people have opinions on multiple topics. I will introduce a kinetic model that describes the effect of interactions between people on their opinions, and discuss the corresponding partial differential equation, which describes that behaviour on a macroscopic level.

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Pattern selection in a PDE model of mechanochemical interactions

Classic Turing models require a fast-diffusing inhibitor to form patterns. Here, we investigate an alternative mechanism inspired by Hydra: mechanochemical coupling. We analyze a model where local morphogen production is driven by tissue stretching. Through rigorous spectral analysis and bifurcation theory, we prove that this mechanical feedback naturally selects unimodal (single-peaked) states as the only stable solutions. This result offers a mathematical explanation for the robustness of body axis polarity in biological systems, showing that mechanics can successfully substitute for chemical inhibition to prevent the formation of multiple heads.

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Mechanism-driven extensions of Turing pattern formation in enzymatic reaction–diffusion systems

Diffusion-driven (Turing) instabilities provide a classical mechanism for spatial self-organization in reaction–diffusion systems. In metabolic pathways, however, reactions are mediated by enzymes and proceed through reversible enzyme–substrate complexes, whose dynamics are often simplified using quasi-equilibrium approximations. In this work, we investigate how such quasi-equilibrium enzyme–substrate kinetics interact with diffusion to shape spatial pattern formation. We consider reaction–diffusion models of multi-step metabolic pathways in which enzyme–substrate binding is assumed to rapidly equilibrate, while substrate and intermediate concentrations evolve on slower timescales. Despite this reduction, linear stability analysis shows that quasi-equilibrium complex formation can significantly alter the conditions for diffusion-driven instabilities. In particular, equilibrium binding modifies effective reaction sensitivities and introduces diffusion-mediated feedbacks that shift or reshape Turing instability regions compared to models based solely on effective rate laws. Our results demonstrate that, even under quasi-equilibrium assumptions, enzyme–substrate interactions act as modulators of spatial instability mechanisms. This highlights the importance of retaining mechanistic enzymatic structure in spatial metabolic models and extends classical Turing theory to enzyme-mediated reaction networks.

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Resilience Optimisation of Tipping-Prone Systems Through Spatial Heterogeneities

Spatially extended multistable systems exhibit tipping and front propagation, where local perturbations can induce large-scale transitions. When interventions are spatially limited, it is crucial to determine where heterogeneity most effectively enhances resilience. We propose a constrained optimisation framework to identify optimal spatial perturbations under PDE dynamics. As a test case, we study the one-dimensional Allen–Cahn equation,

∂y/∂t = ∂²y/∂x² + y(1 − y²) + μ(t) + μ_het(x) + μ_pert(x),

where μ_pert(x) is optimised under size constraints. We show how optimal perturbations interact with the saddle-node bifurcation threshold and the Maxwell point, enabling prevention of tipping, control of front propagation, and spatial confinement of transitions.

Ecological interactions and spatial dynamics in microbial aggregates: A novel modelling framework

We present a mathematical model based on a system of partial differential equations (PDEs) with cross-diffusion and reaction terms to describe ecological interactions between multiple bacterial species and substrates within microaggregates, where bacteria proliferate in response to substrate availability and undergo passive dispersal driven by population pressure gradients. The ecological interactions include interspecific competition for shared substrates, and commensalism, whereby one species benefits from the metabolic by-products of another. The main motivation comes from individual-based models (IBMs) of microbial aggregates, where simulations reveal that substrate-limited conditions can give rise to rich spatial patterns. Our numerical experiments demonstrate that our PDE-based model captures the key qualitative features of three verification scenarios that have previously been investigated with IBMs. Moreover, we formally derive a competition system from an on-lattice biased random walk, and establish local well-posedness for a parameter-symmetric subcase of it. We then formally analyse the travelling wave behaviour of this case in one spatial dimension and compare the minimal travelling wave speed with the wave speed measured in the simulations.

Spontaneous Particle Aggregation with Memory

We present a nonlinear and nonlocal model of spontaneous biological aggregation. On the microscopic scale, it is represented as an agent-based stochastic model where each individual modulates its random movement based on the perceived local density of its neighbours. Memory is introduced via a chain of internal variables, allowing agents to retain past environmental information. With appropriate parameter settings, the model exhibits emergent formation of particle clusters. We present results of systematic stochastic simulations, showing that short-term memory promotes cluster coarsening, while long-term memory disrupts aggregation, increasing the number of outliers and instances with no clustering. Statistical analysis shows that memory inhibits the particles' responsivity to environmental cues, specifically the perceived density of their neighbours, explaining the reduced clustering tendency at higher values of the memory length. To gain deeper insights into the formation and shape of particle clusters, we derive the Fokker-Planck equation in the macroscopic limit, characterize its steady states, and provide results of numerical simulations.

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Well-Posedness of a Free-Boundary Problem Modeling Microbial Biofilms"

We study the existence and uniqueness of solutions to a system of PDEs modeling the growth of a multispecies biofilm under minimal assumptions on the equation terms. A local existence and uniqueness result is established via the Banach Fixed Point Theorem, and global existence and qualitative behavior are obtained under a nonlimiting substrate condition and quasimonotonicity assumption on the reaction terms.

Spatiotemporal oscillations in reaction cross-diffusion models

Nonlinear reaction—cross-diffusion models, such as the Keller—Segel model of chemotaxis, have been proposed to describe spatial and spatiotemporal pattern formation observed in nature. Example applications include pigmentation on snake skin, and intricate patterns of bacteria, slime moulds, and other microorganisms. While useful to understand pattern formation in such systems, linear stability analysis of spatially homogeneous states can fail to capture the emergent dynamics of solutions to this system.  We discuss spatiotemporal oscillations, and even chaotic solutions, which emerge in these systems despite the absence of Hopf or wave bifurcations which would indicate oscillatory behaviours. We use multiple scales asymptotics to develop a quintic weakly nonlinear theory that predicts the structure of solutions to such systems near Turing bifurcations, in particular near the codimension-2 point where the bifurcation changes from supercritical to subcritical. Focusing for concreteness on a Keller—Segel model with logistic growth of the cell density, we numerically show that solutions are generically oscillatory, and provide evidence that they are likely chaotic. Motivated by this behaviour, we also consider the theory of competing modes with slowly varying amplitudes to help explain the emergence of oscillations, motivated by similar approaches in fluid dynamics on subharmonic instabilities. We conjecture that these instabilities are distinct from other routes of spatiotemporal oscillation observed in reaction—transport systems, but that they may play important roles in understanding complex and chaotic motions observed in some systems.

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Global well-posedness and stability analysis of a degenerate system for a prey-predator and natural enemy interaction

In this study, we introduce a degenerate reaction-diffusion system that models the interactions among the prey, the predator, and the predator's natural enemy. A typical scenario arises in crop-pest-natural enemy interactions in agriculture, in which the crop is a plant species that lacks diffusion; the natural enemy not only feeds on the predator but also supports the crop's growth. Firstly, the well-posedness properties, such as global existence and uniform boundedness in time of the solution, are proved. Next, we present the stability analysis of the constant equilibria, including linear and nonlinear stability.

Modelling wildfires: Model extensions of an advection-diffusion-reaction model for fire front propagation

Given the recent increase in wildfires, developing a better understanding of their dynamics is of growing importance. Advection–diffusion–reaction models provide a widely used framework for studying wildfire spread and behaviour. We investigate such a model and discuss potential extensions, as well as alternative and simplified representations of key underlying processes. Numerical simulations are employed to examine the resulting dynamics and to illustrate the capabilities and limitations of different modelling assumptions and approximations.

Spatial models of forest-savanna bistability

Empirical studies suggest that for vast tracts of land in the tropics, closed-canopy forests and savanna are alternative stable states, a proposition with far-reaching implications in the context of ongoing climate change. Consequently, numerous mathematical models, both spatially implicit and explicit, have been proposed to capture the mechanistic basis of this bistability and quantify the stability of these ecosystems. We present an analysis of a spatially extended version of the so-called Staver-Levin model of forest-savanna dynamics (a system of nonlinear partial integro-differential equations). On a homogeneous domain, we uncover various types of pattern-forming bifurcations in the presence of resource limitation, which we study as a function of the resource constraints and length scales in the problem. On larger (continental) spatial scales, heterogeneity plays a significant role, and incorporating domain heterogeneity leads to interesting phenomena such as front-pinning, complex waves, and extensive multi-stability, which we investigate analytically and numerically.

Navigating early warning signals: Bridging gaps with spatial coherence

Many systems may cross tipping points as conditions change over time, which has led to increased interest in early warning methods. Most existing approaches are based on critical slowing down, which often assume white noise and neglect spatial effects.

We developed a data-driven method that avoids these assumptions and explicitly includes spatial structure. Using linear regression, we infer a linearised reaction–diffusion model directly from data. This model describes the local dynamics and can be used for stability analysis before a collapse occurs.

The inferred linear model offers a promising new alternative for addressing a critical question that many early warning models cannot: Are we approaching a tipping point, or are we heading towards a possible Turing bifurcation?

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Stability of stationary, (spatially) periodic vegetation patterns in ecosystems

Self-organized vegetation patterns driven by scale-dependent feedbacks naturally arise in a variety of ecosystems, as ecological processes often occur on vastly different scales. In fact, it has recently been proposed that these patterns may increase the resilience of an ecosystem. The governing models, such as Klausmeier-Gray-Scott-type systems for drylands, have the structure of singularly perturbed reaction-diffusion systems. Here, we consider general two-component reaction-diffusion models (in one spatial dimension) that exhibit stationary, (spatially) periodic patterns. We use methods from geometric singular perturbation theory to construct these singular periodic patterns, through a decomposition into long- and short-range (`slow’ and `fast’) dynamics. The slow-fast decomposition also forms the foundation for the spectral stability analysis of the resulting patterns, coupled with Floquet theory and matched asymptotic expansions. We derive leading order expressions for the bands of essential spectrum of the spatially periodic patterns, thereby obtaining explicit analytic control over the far from equilibrium nature of the Busse balloon associated to these patterns. Finally, we demonstrate our results through an example Klausmeier-Gray-Scott-type system and show that it admits spectrally stable periodic patterns.

Joint work with Paul Carter, Gianne Derks, Arjen Doelman, and Daniel Shvartsman.