Speakers

Heterogeneity in Reaction - Diffusion Systems: A Feature, Not a Perturbation

Spatial heterogeneities are often treated as a nuisance in reaction--diffusion theory: they break translation invariance, complicate spectral analysis, and can destroy the clean bifurcation pictures familiar from homogeneous media. In this talk we argue that heterogeneity is not only an obstacle, but also an opportunity providing a controlled way to create, select, and stabilize coherent structures. We present an analytical framework for existence, stability, and bifurcations of solutions in reaction--diffusion systems with spatially varying coefficients, with an emphasis on front and wave-train type dynamics. The discussion is organized around two representative examples. First, we consider heterogeneous front solutions in a FitzHugh--Nagumo equation, where spatial variability induces fronts that move with variable speed through motionless heterogeneous background states. Second, we study wave trains described by a Ginzburg--Landau amplitude equation that arise as slow modulations in a Swift--Hohenberg model, focusing on how nonuniform coefficients shape wave-number selection.

A unifying theme is the use of perturbation methods based on a small parameter - but crucially, the parameter does not measure the size of the spatial heterogeneity. Instead, it captures a scale separation or proximity to a critical regime, allowing heterogeneities of order one while retaining analytical tractability. The main novelty is an extension of established perturbative frameworks to systematically incorporate non-autonomous terms, yielding solvability conditions and reduced modulation equations in spatially varying settings.

This is joint work parly with Jolien Kamphuis and partly with Frits Veerman and Lara van Vianen.

TBA

Can we predict wavelength changes in models for banded vegetation patterns?

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.

Communicating Across Boundaries via Interactive Simulations

This talk will be a mix of "communicating" and "doing" science in different ways, aided by modern conceptual and computational tools. I will demonstrate the value of communication across disciplinary boundaries using real-time interactive simulations through VisualPDE.com. I will describe how past and ongoing work with developmental biologists, geographers, ecologists, and microbiologists can be enriched with these tools, and in particular how deep insights can be rapidly communicated without the need for vast disciplinary expertise. Among other examples, I will discuss stochasticity in 2D Rayleigh-Bénard convection, as well as the role of subcritical diffusion-driven instabilities in oncological, ecological, and embryological settings, demonstrating the ease with which nuanced and technical theoretical ideas can be illustrated and explored in real-time. I will end by discussing fundamental limitations of phenomenological dynamical systems modelling in general, giving rise to the need for better frameworks to do science. 

Segregation and instabilities in a Lotka–Volterra competition model with diffusion and avoidance

In this work we investigate a minimal reaction-diffusion system with nonlinear cross-diffusion, showing that this term alone is sufficient to trigger spontaneous pattern formation, even in a triangular configuration preserving the classical Lotka--Volterra competitive kinetics. Beyond the expected stationary Turing patterns, our analysis reveals the emergence of oscillatory non-homogeneous solutions, which is surprising, since no Hopf bifurcation, wave instability, or external periodic forcing occurs in the model.

A detailed spectral analysis reveals that these oscillations originate from spatial resonance mechanism, in which the fundamental mode interacts with its subharmonics and superharmonics. This resonance induces secondary instabilities and leads to the excitation of subharmonic oscillations. By deriving the normal form for the unstable amplitudes and applying a center--unstable manifold reduction, we characterize the amplitude dynamics on the resonant manifold and determine the critical threshold at which subharmonic excitation sets in.

These findings demonstrate that nonlinear cross-diffusion acts as the fundamental mechanism responsible for both spatial segregation and the onset of resonant oscillatory instabilities in minimal competitive systems.

TBA

Beyond water limitation in vegetation-autotoxicity patterning: a cross-diffusion model

Many mathematical models describing vegetation patterns are based on biomass-water interactions, due to the impact of this limited resource in arid and semi-arid environments. However, in recent years, a novel biological factor called autotoxicity has proved to play a key role in vegetation spatiotemporal dynamics, particularly by inhibiting biomass growth and increasing its natural mortality rate. In a standard reaction-diffusion framework, biomass-toxicity dynamics alone are unable to support the emergence of stable spatial patterns. In this talk, we derive a cross-diffusion model for biomass and toxicity dynamics as the fast-reaction limit of a three-species system involving dichotomy and different time scales. Within this general framework, in addition to growth-inhibition and extra-mortality already considered in previous studies, the additional effect of ''propagation reduction'' induced by autotoxicity on vegetation dynamics is obtained. By combining linearised analysis, simulations, and continuation, we investigate the formation of spatial patterns. Thanks to the cross-diffusion term, for the first time, a spatial model based solely on biomass-toxicity feedback without explicit water dynamics supports the formation of stable (Turing) vegetation patterns for a wide range of parameter values. We complement these findings with a preliminary analysis of such spatially periodic patterns by means of Geometric Singular Perturbation Theory.

Joint work with Francesco Giannino and Cinzia Soresina.

Managing an Epidemic with real world frictions -  Insights and Spatial

Perspectives 

After an epidemic outbreak, governments introduce interventions to

reduce infections. In practice, the effectiveness of such measures is

often reduced by frictions, such as delayed reactions, imperfect

information, economic costs, and changes in population behavior. We want

to gain some insights how these frictions may affect epidemic dynamics

and management.

We present a compartmental epidemiological model with a rule-based

control mechanism. Countermeasures are activated when certain indicators

exceed given threshold values. Model components include economic losses

caused by illness and by intervention measures, as well as behavioural

responses that may reduce compliance over time.

Analytical results focus on stability properties and on the reproduction

number under dynamic interventions. Simulation results illustrate the

sensitivity of the model to key parameters and the role of the control

mechanism. For some concrete data, we use scenarios inspired by the

early COVID-19 outbreak in Italy.

Due to limited data on economic and psychological factors, the model is

not intended yet for quantitative prediction. Instead, it provides a

conceptual framework for studying trade-offs in epidemic management. We

conclude with a discussion of limitations and an outlook on extensions

including spatial dynamics. 

Asymptotic analysis of an integro-differential model from quantitative genetics

We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the  Fisher's  infinitesimal model. Considering a small variance regime, we characterize the steady states of the problem and we analyze their stability. Our method relies on a spectral analysis involving Hermite polynomials, highlighting the specific structure of the nonlinear reproduction term. This is a joint work with M. Hillairet. 

Mean-fields limits and the multiscale approach to cell movements

In this talk I focus on a quite general class of hybrid mathematical models of collective motions of cells under the influence of chemical stimuli and their macroscopic counterparts. The initial models are hybrid in the sense that cells are

discrete particles driven by ODE, while the chemoattractant is considered as a continuous signal which solves a diffusive equation. For these models it is possible to prove the mean-field limit in the Wasserstein distance to a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. This approach and results are not based on empirical measures, but rather on marginals of large number of individual densities, and we show the limit with explicit bounds,

by proving also existence and uniqueness for the limit system. In the monokinetic case we derive new pressureless nonlocal Euler-type model with chemotaxis, which will be compared with other macroscopic models of cell movement. These results have been obtained in collaboration with Thierry Paul and, for the numerical part, with Marta Menci and Tommaso Tenna.  

Phenotype-structured PDE models for chemotaxis self-organisation.

Chemotaxis is a powerful mechanism for driving the self-organisation of populations that range from cells to organisms. Recently there has been significant interest in the impact of phenotype structuring on populations, where in the context of chemotaxis this structuring could introduce variability in traits that range from their sensitivity to gradients, proliferation, or secretion of autoattractants. In this talk I will describe a nonlocal PDE framework that incorporates a continuous phenotype structuring for a chemotactic population, and show how this structuring can impact on two classic phenomena of chemotactic system: waves and aggregation behaviour. Work with Tommaso Lorenzi, Fiona Macfarlane, and Chiara Villa. 

Multi-scale modelling of pattern formation by cell populations on surfaces

I will talk about a few projects in which the behaviour of cells affects pattern formation on surfaces. First, I will show how cell-cell interactions generate emergent fluid-like material properties in cell aggregates, with implications for pattern formation during chemotactic migration. Then, I will show how cell-fluid interactions affect surface attachment of swimming bacteria in fluid flow.

How spatial patterns can lead to less resilient ecosystems

Several theoretical models predict that spatial patterning increases ecosystem resilience. However, these predictions rely on strong simplifying assumptions, such as assuming isotropic and infinitely large ecosystems, and empirical evidence directly linking spatial patterning to enhanced resilience remains scarce. We intro duce a unifying framework, encompassing existing models for vegetation pattern formation in water-stressed ecosystems, that relaxes these assumptions. This framework incorporates finite vegetated areas surrounded by desert and anisotropic environmental conditions that lead to non-reciprocal plant interactions. Under these more realistic conditions, we identify a novel desertification mechanism, known as convective instability in physics but largely overlooked in ecology. These instabilities form when non-reciprocal interactions destabilize the vegetation–desert interface and can trigger desertification fronts even under stress levels where isotropic models predict stability. Importantly, ecosystems exhibiting periodic vegetation patterns are more susceptible to convective instabilities than those with homogeneous vegetation, suggesting that spatial patterning may reduce, rather than enhance, resilience. These findings challenge the prevailing view that self-organized patterning enhances ecosystem resilience and provide a new framework for investigating how spatial dynamics shape the stability and resilience of ecological systems under changing environmental conditions 

Aggregation-diffusion in heterogeneous environments

Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical way. However, most existing studies do not account for the effect of the underlying environment on organism movement. In reality, the environment is often a key determinant of emergent space use patterns, albeit in combination with collective aspects of motion. Here, I will present work towards this end, studying aggregation-diffusion equations in a heterogeneous environment in one spatial dimension. Under certain assumptions, it is possible to find exact analytic expressions for the steady-state solutions when diffusion is quadratic. Minimising the associated energy functional across these solutions provides a rapid way of determining the likely emergent space use pattern, which can be verified via numerical simulations. This energy-minimisation procedure is applied to a simple test case, where the environment consists of a single clump of attractive resources. Here, self-attraction and resource-attraction combine to shape the emergent aggregation. Two counter-intuitive findings emerge from these analytic results: (a) a non-monotonic dependence of clump width on the aggregation width, (b) a positive correlation between self-attraction strength and aggregation width when the resource attraction is strong. 

Excluded Volume Effects and Anisotropic Interactions in a System of Hard Needles  

In this talk, we will present recent work on the macroscopic dynamics of a system of hard Brownian needles in two dimensions, focusing on the role of anisotropic steric interactions. Although the needles themselves have zero volume, their mutual exclusion creates a non-trivial effective volume in configuration space, significantly influencing the system’s collective behaviour. Starting from the underlying stochastic hard needle dynamics, we apply the method of matched asymptotic expansions to rigorously derive a nonlinear, non-local partial differential equation governing the evolution of the needle density in position and orientation. In the regime of strong rotational diffusion, we obtain a reduced equation for the spatial density alone and use it to compare the effective excluded volume of needle systems with that of hard spheres. Finally, we investigate spatially homogeneous solutions and demonstrate an isotropic-to-nematic transition at increasing densities, in agreement with Onsager’s theory. Our work highlights the interplay between geometry, hardcore interactions, and the emergence of order in anisotropic systems. 

Mathematical modeling of meniscus tissue regeneration in an artificial scaffold 

We present a modeling framework for the dynamics of human mesenchymal stem cells (hMSCs) and chondrocytes evolving in a nonwoven polyethylene terephtalate (PET) scaffold supplied with a differentiation medium. The differentiation of hMSCs into chondrocytes favors the production of extracellular matrix (ECM) and is influenced by fluid stress. The model takes deformations of ECM and PET scaffold into account. The scaffold structure is explicitly included by statistical assessment of the fibre distribution from CT images. The effective macroscopic equations are obtained by appropriate upscaling from the dynamics on lower (microscopic and mesoscopic) scales.

In a simplified setting of the cell and tissue cross-diffusion dynamics we also address the global existence of solutions. 

Singular limits for PDE in biology

In this talk, I will address some questions concerning singular limits in systems of PDEs arising from biology. More precisely, the analysis of diffusive enzyme reactions and plant-growth dynamics with autotoxicity will be presented. By proving and using certain results such as duality-, energy method and some bootstrap argument, we derive suitable a-priori estimates which allow to pass to the limit to obtain some cross diffusion systems. Moreover, by exploiting some projection method, we also derive the convergence rate of the singular limits. 

Mechanochemical pattern formation

Mechanochemical interaction, defined as the active interplay between a dynamical surface and a morphogen diffusing on that surface, can lead to spontaneous pattern formation. As the substrate surface plays the role of the long-range inhibitor, mechanochemical interaction can serve as an alternative to the dominant Turing paradigm when the identification of an inhibitor is problematic. I will present a paradigmatic model class for mechanochemical interaction, and will show the emergence of far-from-equilibrium patterns for a range of parameter values when specific interaction functions are chosen. These analytical results are obtained using Geometric Singular Perturbation Theory, a technique that uses time scale separation in higher-dimensional dynamical systems to reduce the system complexity, and thereby allowing us to constructively prove the existence of specific orbit types (homoclinic, heteroclinic, periodic). In addition, I will discuss recent developments in a mechanochemical model for septin binding on GUVs (giant unilamellar vesicles), where the local septin orientation can determine and predict experimentally observed surface patterns.

This work is in collaboration with Daphne Nesenberend.

Surviving the winds through pattern formation

In the Scottish Highlands, heather (Calluna vulgaris) sometimes forms strikingly regular patterns. Bands of vegetation alternate with bands of bare ground, with the whole pattern slowly moving perpendicular to the bands. I present a model where simple interactions between heather, wind, and soil are sufficient to generate periodic travelling waves similar to the real-life patterns. Heather and soil diffusion are essential components of the model, revealing how heather acts as an ecosystem engineer, extending its habitat into harsh, wind-blasted environments through pattern formation.  

Decision making in heterogeneous interacting particle systems 

In this talk we discuss the role of uninformed individuals in consensus formation within decision- making swarming models for self-propelled particles. The proposed models are inspired by empirical observations of schooling fish. We propose a coupled model that integrates spatial swarming dynamics with the evolution of individual spatial preferences. Each individual is described by its position, velocity, and a continuous preference variable; it interacts through self-propulsion, alignment, attraction-repulsion forces, and preference-based mechanisms. Building on

classical bounded-confidence models, we introduce a three-population framework that distinguishes between leaders, followers, and uninformed individuals. Our analysis reveals that uninformed individuals,

despite lacking any spatial preference bias, significantly influence group dynamics by diluting the effect of leaders and promoting more democratic decision-making. Numerical simulations demonstrate a variety of emergent behaviours, including flocking and milling. These findings support the role of uninformed agents in collective decision making and provide first analytical insights to understand leadership and consensus in heterogeneous crowds.