Speakers

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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.

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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.

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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. 

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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. 

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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. 

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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.  

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