Speakers

Title: About the many equilibria of a cross-diffusion system in population dynamics

Abstract: Cross-diffusion is a mechanism that can be used in population dynamics to model a repulsive effect between individuals. Mathematically, this corresponds to adding nonlinear diffusion terms to classical reaction-diffusion systems. Cross-diffusion can generate a rich variety of solutions, whose qualitative behavior seems to better fit observations (including spatial segregation phenomenon), but it also complicates the mathematical study of these solutions. In this talk, I will explain how this problem can be tackled by combining numerical simulations with a posteriori estimates, to obtain computer-assisted proofs, and I will present some recent results about the steady states of the SKT system.

Title: Phase separation in active Brownian particles

Abstract: In this talk, I will discuss models for active matter systems consisting of many self-propelled particles. These can be used to describe biological systems such as bird flocks, fish schools, and bacterial suspensions. In contrast to passive particles, these systems can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation. Starting with a microscopic model for active Brownian particles with repulsive interactions, I will discuss four possible macroscopic PDEs (ranging from a nonlocal model to a local cross-diffusion system). I will then present recent work concerning the stability and analysis of such models.

Title: Embedding and Approximation Theorems for Echo State Networks

Abstract: Echo State Networks (ESNs) are simple, mathematically attractive, and surprisingly successful examples of recurrent neural networks that can learn the behaviour of a dynamical system. We prove three fundamental and reassuring aspects of ESN construction and behaviour that between them help to explain this success. Along the way we make contact with a number of well-known results including Takens’ theorem and the Universal Approximation theorem.

References:

A.G. Hart et al, Neural Networks, 128:234-247 (2020)

A.G. Hart, et al, Physica D 421:132882 (2021)

Title: Body orientation dynamics

Abstract: Collective dynamics has stimulated intense mathematical research in the last decade. Many different models have been proposed but most of them rely on describing agents as point particles in position-velocity space. We propose a model where the particles carry more complex geometric structure. Specifically, the particles are rigid bodies whose attitude (or body orientation) is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. In this talk we will review recent results on this model which are issued from collaborations with Antoine Diez, Amic Frouvelle, Sara Merino-Aceituno, Mingye Na and Ariane Trescases.

Title: Rolls and domain walls in the Rayleigh-Bénard convection

Abstract: We investigate pattern formation in the classical Rayleigh-Bénard convection problem. We focus on rolls which are regular patterns and domain walls which are line defects arising between rolls with different orientations. The mathematical problem consists in solving the Navier-Stokes equations for the fluid velocity coupled with an additional equation for the deviation of the temperature from the conduction profile in a cylindrical domain. Our analysis relies upon a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, rolls and domain walls are found as equilibria and heteroclinic orbits, respectively, of a reduced system of ODEs.

Title: Recent Progress and Open Challenges in Turing-type Morphogenesis

Abstract:

Motivated by recent work with biologists, I will showcase some mathematical results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in a variety of settings such as mouse whiskers, human fingerprints, bat teeth, and more generally pattern formation on multiple scales and evolving domains. Such phenomena are typically modelled using reaction-diffusion systems of morphogens, and one is often interested in emergent spatial and spatiotemporal patterns resulting from instabilities of a homogeneous equilibrium, which have been well-studied. In comparison to the well-known effects of how advection or manifold structure impacts unstable modes in such systems, I will present results on instabilities in heterogeneous systems, as well as those arising in the setting of evolving manifolds. These contexts require novel formulations of classical dispersion relations, which have applications beyond developmental biology, such as in population dynamics (e.g. understanding colony or niche formation of populations in heterogeneous environments). These approaches also help close the vast gap between the simple theory of diffusion-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in validating even complex theories against the rich dynamics observed in nature. I will close by discussing a range of open questions, many of which fall well beyond the extensions I will discuss.

Title: Exploring Bifurcations of Stochastic Partial Differential Equations

Abstract: In this talk, I shall introduce several classical models of SPDEs of reaction-diffusion type. We shall see that local existence theory is quite well-understood by now for relatively large classes of noise arising in applications but that analyzing pattern formation in the context of noise is still a massive challenge. In particular, I shall explain recent results on trying to understand local bifurcations of SPDEs for small noise. The results include analytical proofs for stochastic fluctuations near bifurcations utilizing projections as well as numerical techniques to efficiently compute these fluctuations over large ranges of parameters via numerical continuation.

Title: Cross-diffusion effects on pattern formation in a PDE morpho-chemical model for electrodeposition

Abstract: We analyze the effect of cross-diffusion on pattern formation in a PDE system for chemistry and morphology describing metal growth in an electrodeposition process. The reaction-diffusion system was recently shown to experience both diffusion-driven and Turing-Hopf instabilities. The case of cross-diffusion in one component only as well as the case of cross-diffusion in both the components is considered and negative cross-diffusion is also allowed. By using linear stability analysis, we obtain conditions - expressed in terms of the system parameters - for the existence of stationary Turing patterns and discuss the features of the pattern-forming regions in the bifurcation parameter space. We also derive analytical conditions for the destabilization à la Turing of the Hopf limit cycle in the neighbour of the codimension-two Turing-Hopf bifurcation point and discuss the impact of cross-diffusion for the emergence of Turing-Hopf patterns. The above theoretical findings are illustrated through an extensive gallery of numerical simulations by which the role of cross-diffusion on the Turing region as well as on the pattern selection is elucidated and discussed. An experimental validation of the obtained results is also presented.

Joint work with B. Bozzini and I. Sgura.

References

[1] B. Bozzini, D. Lacitignola, I. Sgura, “Spatio-Temporal Organisation in Alloy Electrodeposition:

a Morphochemical Mathematical Model and its Experimental Validation”, Journal of Solid State Electrochemistry 17(2), 467 - 479, 2013.

[2] D. Lacitignola, B. Bozzini, I. Sgura, “Spatio-temporal organization in a morphochemical electrodeposition model: analysis and numerical simulation of spiral waves”, Acta Applicandae Matematicae 132, 377 - 389, 2014.

[3] D. Lacitignola, B. Bozzini, I. Sgura, “Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay”, European Journal of Applied Mathematics, vol. 26(2), 143-173, 2015.

[4] B. Bozzini, G. Gambino, D. Lacitignola, S. Lupo, M. Sammartino, I. Sgura, “Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth”, Computers & Mathematics with Applications 70 (8), 1948-1969, 2015.

[5] D. Lacitignola, B. Bozzini, M. Frittelli, I. Sgura, “Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition”, Commun. Nonlinear. Sci. Numer. Simulat. 48, 484–508, 2017.

[6] D. Lacitignola, B.Bozzini, R. Peipmann, I Sgura, “Cross-diffusion effects on a morphochemical model for electrodeposition with cross-diffusion”, Applied Mathematical Modelling 57, 492-513, 2018.

[7] D.Lacitignola, I. Sgura, B.Bozzini, “Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion”, Applications in Engineering Science 5, 100034, (2021).

Title: Emergence of small amplitude localised cellular patches in reaction-diffusion systems

Abstract: Localised patches of cellular pattern (of which typically hexagons are most commonly observed) occur in a wide range of applications from vegetation and crime hotspot patterns. In this talk, I will present some results on the analysis of small amplitude localised cellular patches and numerical results.

Title: A free boundary problem modeling cell polarization

Abstract: We are interested in a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study the evolution of this model for several reaction rates on the membrane as well as the diffusion coefficient inside the cell being large. It turns out that, in suitable scaling limits, solutions converge to solutions of obstacle type problems, for which we show uniqueness. Further, we prove the global stability of steady states. For the steady states case the occurrence of polarization has been shown by Niethammer, Röger and Velázquez (2020) provided that the total mass of protein is sufficiently small.

Title: Coherent structures in chemotaxis models of inflammatory diseases

Abstract: The aim of this talk is to introduce and study two reaction-diffusion-chemotaxis models that describe the initial stages of a wide class of inflammatory diseases.

Inflammation is the response to outside insults, aimed at eliminating the threat and promote tissue repair and healing. It is a highly complex process, characterized by the action of both pro- and anti-inflammatory agents that work synergistically to ensure a quick restoration of tissue health. Inflammation is also believed to play a central role in the pathophysiology of many common disorders, including some degenerative patologies, such as Multiple Sclerosis, Alzheimer's and atherosclerosis.

In the last years several mathematical modeling approaches have been adopted to provide insights on the major pathological processes involved in inflammation [7, 9]. Despite the increasing interest in this area, there are only few models that incorporate spatial aspects in the description of inflammation-driven diseases [6, 8, 2].

We shall first design a model describing the spatio-temporal dynamics of a population of immune cells and of two different types of signaling molecules: a pro-inflammatory chemokine, which is the chemoattractant for the immune cells, and an anti-inflammatory cytokine, which acts, on a longer time scale, as an inhibitor of the inflammatory state [1, 3]. We are interested in the model capability of reproducing aggregation phenomena leading to the formation of localized patches of inflammatory activity. To this end, using a combination of analytical and numerical approaches, we shall investigate the conditions on the system parameters that determine the excitation of Turing and wave instabilities. The study is conducted by considering biologically realistic values of the introduced parameters, all of which are

taken from the existing literature. We shall show that, varying the control parameters, the model is able to reproduce qualitatively different pathological scenarios: a diffused inflammatory state of the type observed in many cutaneous rashes; the formation of stationary patches of inflammation; the ring-shaped skin rashes observed in Erythema Annulare Centrifugum (EAC), a very aggressive form of cutaneous eruption; oscillating-in-time spatial patterns, that qualitatively reproduce the periodic appearance of localized skin eruptions characteristic of the Recurrent Erythema Multiforme (REM) [4]. The proposed system also displays a cascade of successive bifurcations leading to chaotic behavior, already observed in existing chemotaxis models. The mechanism of the observed route-to-chaos and its relationship with self-organized criticality of macrophages will be discussed.

In the second part of the talk we shall present a recently derived mathematical model [5] describing the primary mechanisms that drive Multiple Sclerosis (MS), an immune-mediated inflammatory disease that attacks myelinated axons in the central nervous system, destroying the myelin and the axon. The model describes the spatio-temporal evolution of the early phases, characterized by activated local microglia, with the recruitment of a systemically activated immune response and by oligodendrocytes apoptosis.

We shall show that the proposed system, within numerical values of the parameters extracted by medical data, supports the formation of different demyelinating patterns, namely: the appearance of areas of apoptotic oligodendrocytes, which closely fit existing MRI findings on the active MS lesion acute relapses; concentric rings, typical of Balo's sclerosis; small clusters of activated microglia in of oligodendrocytes apoptosis, observed in the pathology of preactive lesions.

References

[1] Bilotta, E., Gargano, F., Giunta, V., Lombardo, M.C., Pantano, P., Sammartino, M. Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis (2019) Ricerche di Matematica, 68 281-294.

[2] Chalmers, A.D., Cohen, A., Bursill, C.A., Myerscough, M.R. Bifurcation and dynamics in model of early atherosclerosis: How acute inflammation drives lesion development Journal of Mathematical Biology, 71 (6-7), pp. 1451-1480.

[3] Giunta, V., Lombardo, M.C., Sammartino, M. Pattern formation and transition to chaos in model of acute inflammation (2021) in press on SIAM Journal on Applied Dynamical Systems.

[4] Lerch, M., Mainetti, C., Terziroli Beretta-Piccoli, B., Harr, T. Current Perspectives on Erythema Multiforme (2018) Clinical Reviews in Allergy and Immunology, 54 (1), pp. 177-184.

[5] Lombardo, M.C., Barresi, R., Bilotta, E., Gargano, F., Pantano, P., Sammartino, M. Demyelination patterns in a mathematical model of multiple sclerosis (2017) Journal of Mathematical Biology, (2), pp. 373-417.

[6] Luca, M., Chavez-Ross, A., Edelstein-Keshet, L., Mogilner, A. Chemotactic signaling, microglia, Alzheimer's disease senile plaques: Is there a connection? (2003) Bulletin of Mathematical 65 (4), pp. 693-730.

[7] Painter, K.J. Mathematical models for chemotaxis and their applications in self-organisation (2019) Journal of Theoretical Biology, 481, pp. 162-182.

[8] Penner, K., Ermentrout, B., Swigon, D. Pattern formation in a model of acute inflammation SIAM Journal on Applied Dynamical Systems, 11 (2), pp. 629-660.

[9] Ramirez-Zuniga, I., Rubin, J.E., Swigon, D., Clermont, G. Mathematical modeling of energy in the acute inflammatory response (2019) Journal of Theoretical Biology, 460, pp. 101-114.

Title: Stability results for jump-discontinuous patterns in reaction-diffusion-ODE models

Abstract: Motivated by applications in developmental biology, we explore pattern formation ability of models coupling reaction-diffusion equations with ordinary differential equations. We proved that in such models there exists no stable close-to-equilibrium patterns. Consequently no Turing patterns can be observed and the stationary structures that are relevant in applications must be far-from-equlibrium patterns which exhibit jump-discontinuities. Low-regularity of the solutions and lack of the spectral mapping theorem yield difficulties in stability analysis. We provide stability and instability criteria for such solutions by relating the spectral properties of the corresponding linearised operator to the semigroup properties of the model solutions.

Title: The role of nonlocal competition in ecological systems: from pattern formation to biodiversity maintenance.

Abstract: From microbial colonies to entire landscapes, biological systems often self-organize into regular spatial patterns, which might have significant ecological consequences. Several models have been proposed to explain the emergence of these patterns. Most of them rely on a Turing-like activation-inhibition scale-dependent feedback whereby interactions favoring growth dominate at short distances and inhibitory, competitive interactions dominate in the long-range. However, the importance of short-range positive interactions for pattern formation remains disputable. Alternative theories predict their emergence from long-range inhibition alone. In this presentation, I will explain how self-organized patterns might emerge in purely competitive models. I will first present in which conditions long-range competition alone can generate regular patterns of population density in systems with one and two species. Then, I will discuss the ecological implications of those patterns both for population persistence and species coexistence.

Title: Emergent robustness of bacterial quorum sensing in fluid flow

Abstract: Bacteria use intercellular signalling, or quorum sensing (QS), to share information and respond collectively to aspects of their surroundings. The autoinducers that carry this information are exposed to the external environment; consequently, they are affected by factors such as removal through fluid flow, a ubiquitous feature of bacterial habitats ranging from the gut and lungs to lakes and oceans. Here, we develop and apply a general theory that identifies the conditions required for QS activation in fluid flow by linking cell- and population-level genetic and physical processes. By accounting for a dynamic flow in our theory, we predict that positive feedback in cells acts as a low-pass filter at the population level in oscillatory flow, allowing a population to respond only to changes in flow that occur over slow enough timescales. Our theory is readily extendable and provides a framework for assessing the functional roles of diverse QS network architectures in realistic flow conditions.

Title: Multispecies models with non-local advection: linking animal movement to spatial population dynamics

Abstract: The spatial arrangement of species is a core concern in ecology. Mathematical modelling studies have given a lot of attention to understanding how birth and death processes, often mediated by competition or predation, interface with diffusive processes to give rise to spatial patterns. However, spatial patterning can also occur on much shorter timescales, where births and deaths are negligible. In this situation, organism movement is the principal process that drives pattern formation. This is partially via diffusion, but also via advection in response to both the static landscape and the presence of other individuals. This talk will focus on the effect of between-population advective movement responses on the spatial arrangement of coexistent populations of organisms. Organism movement is modelled using a non-local advection-diffusion formalism, and we show that a range of complex spatio-temporal dynamics may emerge, including stationary aggregations, segregations, oscillatory patterns, period-doubling bifurcations, and irregular spatio-temporal patterns suggestive of chaos. Formal mathematical properties of the model will be explored, as well as a demonstration that the patterning properties of this PDE system also carry over to more realistic (but more analytically tricky) stochastic individual based models, which are formally related to the PDE via a limiting process. Overall, this gives a general framework for studying the spatial patterns that may emerge from systems of multiple coexistent populations moving in response to the presence of each other.

Title: From individual-based models to continuum descriptions: Modelling and analysis of interactions between different populations

Abstract: First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities. Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Title: On the Incompressible Limit for a Tumour Growth Model Incorporating Convective Effects

Abstrafct: We present a porous medium model with applications to tissue movement and tumour growth. The model is based on the standard fluid mechanics approach to living tissues. We extend the analysis proposed in 2014 by Perthame, Quirós, and Vázquez, by incorporating the advective effects caused, for instance, by the presence of nutrients, oxygen, or a chemo-attractant. Passing to the singular limit for a stiff pressure law (incompressible limit), it is possible to connect a density- based model and a free-boundary problem of Hele-Shaw type. Our result extends known results due to weaker assumptions and a more general setting. In particular, we are able to recover the so- called complementarity relation, which allows to derive the pressure through an elliptic equation. To this end, we prove the strong compactness of the pressure gradient, blending two different techniques : an extension of the usual Aronson-Bénilan estimate in an L3-setting and an L4-uniform

bound of the pressure gradient. — Based on joint work with Noemi David .

Title: A genuinely hybrid 3-D single- and multi-organ cancer invasion framework and virtual environment

Abstract: The ability to locally degrade the Extracellular Matrix (ECM) and interact with the tumour microenvironment is a key process distinguishing cancer from normal cells, and is a critical step in the tumour 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).

In this talk, we present a 2- and 3D mathematical model which describes the transition from an epithelial invasion strategy of the Epithelial-like Cancer cells (ECs) to an individual invasion strategy for the Mesenchymal-like Cancer cells (MCs); this is a genuinely multiscale and hybrid model of PDEs and SDEs. In the second part of the talk we present the most recent extension of our model to a multi-organ (and connected via the circulatory system) conformation of the organism and investigate the principles of cancer metastasis.

The material of this presentation is based on joint works with: M. Chaplain, A. Madzvamuse, L. Franssen, T. Williams, A. Wilson, L. Fu, and N. Kolbe.

Title: Models for chemotaxis with local sensing

Abstract: We present a model for chemotaxis that is based on local sensing, that is, the cells respond to a certain concentration of chemoattractant perceived locally (as opposed to gradient sensing, when the cells are able to perceive a gradient of concentration). We will study the specificity of this model, compared for example to the minimal Keller-Segel system, with a focus on well-posedness and long-time behaviour.

Title: Optimal spatial patterns in feeding, fishing, and pollution

Abstract: Infinite time horizon spatially distributed optimal control problems may show so-called optimal diffusion induced instabilities (Brock&Xepapadeas, JEDC 2008), which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples, from (Uecker, DCDS-S, 2021): optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package pde2path to first compute bifurcation diagrams of canonical steady states, and then time-dependent optimal controls to control the systems from some initial states to a target steady state as time goes to infinity. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.

Title: Mechanochemical pattern formation: a crossover between analysis, geometry and biology

Abstract: Mechanochemical models present a new paradigm for biological pattern formation, where the interaction between domain curvature and pattern shape replaces the activator-inhibitor mechanism. Numerical simulations of a mechanochemical model formulated by M. Mercker & A. Marciniak-Czochra (U. Heidelberg) reveal a wide spectrum of novel patterning phenomena, which are as yet poorly understood from an analytical point of view. Our aim is to develop more analytical insight into the pattern formation process in mechanochemical models of this type. As a first step towards this goal, we show that one can employ methods from geometric singular perturbation theory to construct nonlinear, far-from-equilibrium patterns in a general class of mechanochemical models. This analysis reveals a direct relation between the biology --as encoded in the nonlinear interaction of model components-- and the type of (multiscale) patterns that can arise.

Title: Nonlinear diffusion models for segregation

Abstract: In this talk I will introduce two different microscopic models, which lead to aggregation and segregation in single or multi-species settings. In both approaches, particles move randomly but modulate the amplitude of their random walk depending on the local or global density. The resulting mean-field models are nonlinear diffusion equations with complex steady states. I will discuss the structure and stability of these aggregated and segregated states and illustrate the results with computational experiments.

Title: The Maths of Life and Death: Understanding the maths behind epidemics

Abstract: In this lecture, Dr. Kit Yates will look at some of the basic models and mathematics underlying the understanding of disease spread, and pick apart the meanings behind some of the terms we hear about in the news: from exponential growth and R to critical immunisation threshold and herd immunity.

Title: Design and analysis of finite-volume schemes for a class of cross-diffusion systems

Abstract: In this talk we will present an implicit Euler finite-volume scheme for an n-species population cross-diffusion system of Shigesada-Kawasaki-Teramoto (SKT) type. At the continuous level this system admits a formal gradient-flow or entropy structure which allows to prove the existence of nonnegative global weak solutions in time. Our main goal is to preserve this structure at the discrete level. In this aim our key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. Then we are able to pove the existence of nonnegative solutions to the scheme and its convergence. Furthermore we will explain how our approach can be extended to a more general class of cross-diffusion systems satisfying some structural conditions.