Speakers

Dynamics of exotic front solutions in reaction-diffusion systems

We give an overview of the complicated dynamics exhibited by front solutions in reaction-diffusion systems. Along the example of an Allen-Cahn-type multi-component system, we show how the dynamics of a front path can be described by a set of ODEs, which - under certain conditions - can even support chaotic motion emerging from nil-potent singularities. We also touch on the dynamics of front solutions with heterogeneous tails that arise for reaction-diffusion systems with spatially varying coefficients. Of particular interest are such traveling fronts which cannot be captured via a co-moving frame.

Hyperbolic reaction-transport equations in dryland ecological modeling

In the last centuries, climate changes and human actions have been bringing Earth to undergo abrupt shifts versus catastrophic scenarios, such as rising of global temperatures, desertification processes, more and more frequent periods of drought [1]. As a consequence of that, it is believed that dryland areas will greatly increase their extension in the very next future. To this aim, scientists have been focusing their attention on the description of those phenomena that are considered as the precursors of the desertification of a given area, such as the emergence of vegetation patterns. Identifying the complex mechanisms which rule the formation and the stability of these structures becomes, thus, crucial to predict the evolution of dryland ecosystems [2],[3],[4].

However, the geographical remoteness and the long timescale over which such phenomena occur make in-situ experiments quite prohibitive, so that the possibility of acquiring a better understanding on the involved mechanisms is generally relied on mathematical models [5]-[7].

In this talk, we will discuss some of our recent results on the dynamics of stationary and non-stationary vegetation patterns emerging in 1D and 2D hyperbolic reaction-transport systems where patterns arise from the interaction between water and vegetation biomass. Differently from classical parabolic models, inertia exhibited by both vegetation and water is here explicitly taken into account into balance equations for the dissipative fluxes. It will be shown the role of inertia is manifold and not limited to the transient regime. Indeed, it may: enlarge the region of the parameter plane where patterns may be observed; modify the critical parameters at onset (wavenumber, migration speed); alter the (supercritical or subcritical) dynamical regime in which patterns emerge; affect the transitions between different patterned states; mask future deterioration in ecosystem condition; favor the persistence of a patterned state even after a perturbation is expired and improve the resilience of ecosystems, leading to a more favorable configuration [3],[4],[8].

[1] L. Kemp, C. Xu, J. Depledge, and T.M. Lenton. Climate endgame: Exploring catastrophic climate change scenarios. Proceedings of the National Academy of Sciences, 119(34):e2108146119, 2022. doi: 10.1073/pnas.2108146119.

[2] Getzin S. et al.: Discovery of fairy circles in Australia supports self-organization theory. Proc. Natl.Acad. Sci. U.S.A. 113, 3551 (2016).

[3] Gowda K. et al.: Assessing the robustness of spatial pattern sequences in a dryland vegetation model.Proc. R. Soc. A Math. Phys. Eng. Sci. 472, (2016).

[4] Rietkerk M. et al.: Self-Organized Patchiness and Catastrophic Shifts in Ecosystems, Science 80, 374 (2021).

[5] Klausmeier K.A.: Regular and Irregular Patterns in Semiarid Vegetation, Science 284, 1826-1828 (1999).

[6] Consolo G. et al.: Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models. Phys. Rev. E 105, 034206 (2022).

[7] Grifo’ G. et al.: Rhombic and hexagonal pattern formation in 2D hyperbolic reaction–transport systems in the context of dryland ecology. Physica D 449, 1333745 (2023).

[8] Deblauwe V. et al.: The global biogeography of semi-arid periodic vegetation patterns. Glob. Ecol.Biogeogr. 17, 715 (2008).

Geometry and topology in collective dynamics models

Collective dynamics arises in systems of self-propelled particles and plays an important role in life sciences, from collectively migrating cells in an embryo to flocking birds or schooling fish. It has stimulated intense mathematical research in the last decade. Many different models have been proposed but most of them rely on point particles. In practice, particles often have more complex geometrical structures. Here, we will consider particles as rigid bodies whose body attitude is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. A hydrodynamic model will be derived when the number of particles is large. It will be used to exhibit solutions having non-trivial topology. We will investigate whether topology provides enhanced stability against perturbations, as observed in other systems such as topological insulators. 

This talk is based on recent results issued from collaborations with Antoine Diez, Amic Frouvelle, Sara Merino-Aceituno, Mingye Na and Ariane Trescases.

Pattern migration (or not?) of dryland vegetation stripes

Striped vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. Stripe formation is driven by short-range facilitation among plants and long-range competition for water on a sloped terrain. Field data from different pattern sites show that stripes are either stationary, or slowly moving upslope.

In this talk, I show how mathematical modelling and analysis comprising analytical and numerical tools for investigating periodic travelling wave existence and stability proposes a possible resolution for these contrasting field data.

Metaecosystems – network approaches in population dynamics

An ecosystem can be considered as a food web living in a certain region in space (habitat) reflecting specific environmental conditions. Instead of considering continuous space and time leading to models in form of reaction-diffusion systems, we look at ecosystems from a large-scale perspective and describe it as a set of different habitats which are interconnected by migration corridors for the dispersal of species among the habitats. This leads to a network description in which each habitat is assigned to a node, while the dispersal of species or migration along the edges of the network is often modelled as proportional to a gradient in abundances between the different habitats similar to diffusion. Such meta-ecosystems can help to understand the role of connectivity between the different habitats for the survival of species. We discuss effects like synchronization and desynchronization of species between habitats and dispersal-induced rescue of species at risk. We show how complex dynamics of species can contribute to the survival of species even in cases in which poor environmental conditions in a habitat would favor extinction.

One-dimensional short-range nearest-neighbour interaction and its nonlinear diffusion limit

Repulsion between individuals within a finite radius is encountered in numerous applications, including cell exclusion, i.e. an overlap of cells to be avoided, bird flocks, or microscopic pedestrian models. We define such individual-based particle dynamics in one spatial dimension with minimal assumptions of the repulsion force f and prove their characteristic properties. Moreover, we are able to perform a rigorous limit from the microscopic- to the macroscopic scale, where we could recover the finite interaction radius as a density threshold. Specific choices for the repulsion force f lead to well-known nonlinear diffusion equations on the macroscopic scale, as e.g. the porous medium equation. At both scaling levels numerical simulations are presented and compared to underline the analytical results. 

Pattern formation via blackboards and web browsers: New and old tools for theoretical biology

Motivated by a range of problems in embryology and ecology, I will present recent extensions to Turing's classical reaction-diffusion paradigm for pattern formation. This will start by reviewing reaction-diffusion systems and their analysis via classical linear instability theory, followed by a range of generalizations to more realistic scenarios of reaction-transport models in complex domains. Such extensions are motivated by the evolving and heterogeneous landscapes of pattern formation in nature. Throughout this discussion, numerical simulations will play key roles in validating and extending the near-equilibrium theory. To drive home this last point, I will present VisualPDE, a new web-based simulator for lightning-fast interactive explorations of these systems. Such accessible numerical tools are invaluable for rapidly prototyping models of complex biological phenomena. Importantly, accessible simulations underscore the need for sound theory which goes beyond phenomenological modelling in biology.

Who is there, when, where and why? How diffusion helps bacteria to learn more about their environment

Bacterial species employ the mechanism of Quorum Sensing (QS), a communication system, to effectively coordinate their behavior, particularly in relation to pathogenicity and significant alterations in their lifestyle. Gene regulation systems are the basis for such systems, often containing positive and negative feedback loops. This may allow under certain conditions e.g. for bistability. The diffusion of signal molecules produced by bacterial cells in the surrounding environment introduces a crucial interplay between nonlinear dynamics and spatial inhomogeneity. To gain a deeper understanding from both mathematical and biological perspectives, numerical simulations serve as valuable tools. Finally, a short view on treatments like classical antibiotics but also Quorum quenching - a kind of inhibition of Quorum sensing is given.

Pattern formation in systems coupling diffusing and non-diffusive components

Ecological dynamics with long-range individual interactions: from stochastic individual based models to nonlocal partial differential equations

A lot of ecological theory relies on ordinary-differential-equation models that assume well-mixed systems and do not incorporate any information about the spatial distribution of organisms. However, ecosystems present spatial heterogeneities at different scales that can impact individual fitness and, ultimately, population dynamics. I will present an alternative approach to describe the spatiotemporal dynamics of a population of interacting agents. To this end, I will consider a system with nonlinear birth-death rates and positive and negative inter-individual interactions acting at different spatial ranges. I will first describe the stochastic, individual-level rules that govern the reproduction and death of each individual. Then, using field-theory techniques, I will derive a non-local partial differential equation for the population density and compare its predictions with those obtained assuming well-mixed populations. Finally, I will discuss the ecological relevance of our results and how this approach can be extended to more complex scenarios.

Pattern formation by living droplets in chemoattractant gradients

Mass diffusion in non-homogenous materials with application to drug delivery

The advance in material design has led to the rapid development of novel polymers with increasing complexity and functions in bioengineering. These materials have also been introduced for the development of drug releasing devices and systems. By today’s micro-engineering potential, it is possible to fabricate and control the properties of the substrate to have the desired smart release properties and tailoring them for optimal drug administration including customizability. The effect of non-homogeneity represents an important feature that can influence greatly the release characteristics. Layer-by-layer and functionally graded materials are examples of composite materials in which properties vary from one region to another that are already currently and successfully used in a wide range of applications. In this talk we present a space-dependent diffusion-reaction model for drug releasing systems that extends the multi-layer configuration. Several possible space dependent forms for the diffusion and reaction shape-material functions are proposed. In a few cases a semi-analytical solution expressed in terms of Fourier series is possible. The drug concentration and release profiles show important differences with the uniform homogenous material case, providing guidance for design and development of micro-structure of polymer platforms for drug delivery.

Nonlocal advection-diffusion models for modelling organism movement and space use

How do mobile organisms situate themselves in space? This is a fundamental question in both ecology and cell biology but, since space use is an emergent feature of movement processes operating on small spatio-temporal scales, it requires a mathematical approach to answer. In recent years, increasing empirical research has shown that nonlocality is a key aspect of movement processes, whilst mathematical models have demonstrated its importance for understanding emergent space use patterns. In this talk, I will describe a broad class of models for modelling the space use of interacting populations, whereby directed movement is in the form of non-local advection. I will detail various methods for ascertaining pattern formation properties of these models, fundamental for answering the question of how organisms situate themselves in space, and describe some of the rich variety of patterns that emerge. Finally, I will show how to connect these models to empirical data on animal movement via efficient parametrisation techniques.

Multiscale analysis for cross-diffusion models in environments with periodic and random microstructures

A mathematical path to a hybrid 3-D multi-organ cancer invasion framework and virtual patient environment

The ability to locally degrade the Extracellular Matrix (ECM) and invade in the neighbouring tissue is a key process that distinguishes cancer from normal cells. It is also a critical step in the tumour progression and cancer metastasis. The tissue invasion involves the coordinated action of the cancer cells, the ECM, the Matrix Degrading Enzymes, and the Epithelial-to-Mesenchymal transition (EMT); a cellular (re-)programming mechanism through which cancer cells switch from an Epithelial-like proliferative phenotype to acquire Mesenchymal-like invasive properties.

In this presentation, we introduce a series of 2- and 3D mathematical models that describe the growth of the Epithelial-like (ECs) and the invasion strategy of the Mesenchymal-like cancer cells (MCs). We first address simpler model versions where we discuss the existence of classical solutions, and proceed with a more elaborate multiscale and hybrid SDE-PDE model and its predictive capacity of realistic experimental situations. We conclude with some findings from our most recent model extensions to multi-organ conformation and our first steps towards the mathematical development of a virtual cancer patient.

The material of this presentation is based on joint works with M. Chaplain, A. Madzvamuse, L. Franssen, T. Williams, A. Wilson, L. Fu, N. Harbour, J. Giesselman, Chr. Surulescu, D. Katsaounis, and N. Kolbe .

Diffusion in Deserts: Explaining Patterned Vegetation

In many arid parts of the world, vegetation self-organises into spatial patterns on length scales of hundreds of metres. Attempts to reproduce this phenomenon in the laboratory have been uniformly unsuccessful, and thus mathematical modelling provides the primary means of understanding these patterns. The key ingredients are competition for water and the gradual spread of plants through seed dispersal, which is typically modelled by diffusion. I will discuss the types of patterns that are predicted by these diffusion-based models, and the ways in which they can studied mathematically. In particular I will show that detailed mathematical study can be used to infer the historical origin of vegetation patterns in the Sahel region of Africa.

Cross diffusive aggregation and local sensing

Numerical Bifurcation Analysis for differential geometric PDEs

We describe how some differential geometric bifurcation problems can be treated with the Matlab continuation and bifurcation toolbox pde2path. The setup consists of updating a base manifold in each continuation step, and considering the pertinent PDEs for the normal displacement. Examples treated include some minimal surfaces such as Enneper's surface and a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid bridges, and some 4th-order biomembrane models. 

Joint work with Alexander Meiners, Carl von Ossietzky Universität Oldenburg

Mechanochemical pattern formation: far-from-equilibrium patterns on a deforming surface

The appearance of Turing patterns is generally believed to depend on an underlying activator-inhibitor mechanism. However, in a number of biological applications, the experimental identification of these components has been problematic. The hypothesis of mechano-chemical interaction, where the morphogen and the surface dynamically interact, provides an alternative to the activator-inhibitor paradigm. We present a mechano-chemical model, where the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replaces the inhibitor. We show how existing ideas and techniques for the rigorous analysis of far-from-equilibrium patterns can be extended to the mechano-chemical context, and demonstrate the use of geometric singular perturbation theory in the construction of patterns on (and of) a planar curve. We highlight and discuss mathematical challenges posed by this particular interplay of partial differential equations and differential geometry. 

Joint work with Anna Marciniak-Czochra, Moritz Mercker (U. Heidelberg), and Daphne Nesenberend (U. Leiden). 

Active Crowds

In this talk I will discuss the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross diffusion or degenerate mobilities. We then take the diversity of macroscopic models to a uniform structure and work out potential similarities and differences. Moreover, we discuss boundary effects and possible applications in life and social sciences. This is complemented by numerical simulations that highlight the effects of different boundary conditions. 

Joint work with M. Bruna, M. Burger and J.F. Pietschmann.

Analysis of a numerical scheme for a nonlocal cross-diffusion system

In this talk, I will consider a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system, which arises in population dynamics. This system has entropy dissipation properties on which one can rely to design a robust and convergent numerical scheme for its numerical simulation. In terms of numerical analysis, I will present discrete compactness techniques, entropy-dissipation estimates and a new adaptation of the so-called duality estimates for parabolic equations in Laplacian form. I will finally present some numerical experiments. 

This is a joint work with Maxime Herda (Inria Lille).