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22-24 Jan 2025 Rouen (France)
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AbstractsLuca Alasio Title: Mathematical modeling of tissue atrophy in Age-related Macular Degeneration Abstract: Our visual perception of the world heavily depends on sophisticated and delicate biological mechanisms, and any disruption of these processes negatively affects our lives. Age-related Macular Degeneration (AMD) impacts the center of the visual field and has become increasingly common in our society, generating significant academic and clinical interest. I will present some recent advancements in the mathematical modeling of the retinal pigment epithelium (RPE) in the retina in cases of AMD. The RPE cell layer supports the life of photoreceptors by providing nutrients and participating in the visual cycle and "cellular maintenance." Our objectives include modeling the senescence and degeneration of the RPE, as well as modeling the progression of lesions in the epithelial tissue. This work is a joint effort with researchers at Hôpital National des Quinze-Vingts.
Matthieu Alfaro Title: On an epidemic model not in divergence form Abstract: We consider an epidemic model with distributed-contacts. When the contact kernel concentrates, one formally reaches a very degenerate Fisher-KPP equation. We make an exhaustive study of its travelling waves. For every admissible speed, there exists not only a non-saturated (smooth) wave but also infinitely many saturated (sharp) ones. Furthermore their tails may differ from what is usually expected. Some first hints on the issue of the Cauchy problem (notion of solution, existence, uniqueness) are also provided...
Nessim Dhaouadi Title: Surviving in a shifting and size changing environment in presence of selection Abstract: We introduce a model to study the adaptation of a diffusing population facing two different dynamics. On one hand, the population growth is time and space dependent, thus modelling strong heterogeneities of the environment. On the other hand, the environment itself is dynamic. It can both change size and shift over time. The reasons for such moving range boundaries could be the consequences of flooding, forest fire, etc. We will first investigate the fixed domain case, in particular estimating the principal eigenvalue of the underlying periodic parabolic operator. This estimate is crucial to construct sub and supersolutions on the moving domain. We then address the problem of extinction vs. persistence, taking into account the interplay between the moving habitat and the selection. Finally, to explore these dynamics, we construct a stable space-time finite elements scheme using upwind test functions in order to gain some insight on the dynamics of this problem. These will unravel some significant differences with classical results on fixed domains. Romain Ducasse Title: From epidemiology to opinion propagation: a reaction-diffusion model for the emergence of complexity Abstract: We introduce a model designed to describe the spread and accumulation of opinions in a population. Inspired by the "social contagion" paradigm, our model is built on the classical SIR model of Kermack and McKendrick from epidemiology, and consists in a system of reaction-diffusion equations. In the scenario we consider, individuals within the population can adopt new opinions via interactions with others, following some simple rules. The individuals can gradually adopt more complex opinions over time. Our main result is the characterization of a "maximal complexity" of opinions that can persist and propagate. In addition, we show how the parameters of the model influence this maximal complexity. This is a joint work with Samuel Tréton. Thomas Giletti Title: Back and forth between moving heterogeneities and reaction-diffusion systems
Abstract: In this talk, we will be interested in the large time spreading properties of solutions of reaction-diffusion systems from population dynamics, e.g. of competition or prey-predator type. While some situations are well-understood, in particular when a comparison principle is available or when there are only two species, in the general case this remains a mostly open problem. We will get some insight from the special case of a triangular system, where the problem reduces to a scalar equation with a so-called moving heterogeneity. Some recent results in collaboration with Leo Girardin and Hiroshi Matano will highlight the intricacy of even such a simplified situation.
François Hamel
Title: Biological invasions in patchy landscapes: a reaction-diffusion model with interface conditions
Abstract: In this talk, I will discuss a one-dimensional model for biological invasions in heterogeneous patchy landscapes. Each patch has a relatively well-defined structure which is considered as homogeneous, but coupling interface conditions are imposed between adjacent patches, incorporating patch preference data. In the case of two patches, I will mention various results on spreading, blocking, or virtual blocking phenomena when the per capita growth rates in the patches are maximal or on the contrary negative at low densities. The existence of transition fronts propagating from one patch to the other one will also be reported. The talk is based on some joint works with Frithjof Lutscher and Mingmin Zhang.
Sophie Hecht
Title: From a nonlocal mean-field to a porous medium system for heterogeneous population
Abstract:
We consider a system of PDE describing the long-range interaction between individuals. The system is quadratic, written under the form of transport equations with a nonlocal self-generated drift. We establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. The major new feature in our analysis is that we do not need diffusion to gain compactness, but rely on a full rank assumption on the interaction kernels. These theoretical results are sustained by numerical simulations. We then extend these convergence results to the case of a population structured in size.
Laura Kanzler
Title: First order non-instantaneous corrections in collisional kinetic models
This is joint work with Carmela Moschella, Christian Schmeiser and Veronica Tora.
Thomas Lepoutre
Title: Mathematical modelling of the motility regime switching in Myxobacteria
Abstract: Myxobacteria can travel with different speeds. Single bacteria can travel slower than clusters of bacteria. This lead to a model consisting in a system of reaction diffusion equations where the clusters have a speed advantage.
Exchange between clusters and single compartment correspond to fragmentation and coagluation terms. In the simplfied case of a maximal cluster size of 2 bacteria and in the limit of fast coagulation fragmentation, we show that there exists a transition threshold. Before that threshold, the speed advantage does not affect the global propagation speed whereas after it determines it. It corresponds to a transition between pulled and pushed fronts.
This is joint work with Maxime Estavoyer.
Idriss Mazaris
Title: Optimisation of space-time periodic eigenvalues
Abstract: Parabolic periodic eigenvalue problems are important in the study of reaction-diffusion equations, and so is their optimisation with respect to the potential. The main question under consideration is the following: how to choose $m$ so as to minimise the eigenvalue $\lambda$? Naturally we would need to specify the proper constraints, but, at a qualitative level, there are two main questions. The first one is the symmetry of optimisers: is it true that it always better to replace $m$ with another potential (that satisfies the same constraints) but that is also symmetric in time and in space? The second one, has to do with the monotonicity of the optimisers: provided the answer to the first question is positive, is it true that the optimiser is not only symmetric, but also monotonous? Let us emphasise that these questions are answered positively when considering the symmetry and monotonicity with respect to the space variable only.
Benoit Perthame Title: Structured equations in biology; relative entropy, Monge-Kantorovich distance
Abstract: Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals, as time elapsed after infection. During the 70s, many mathematical studies were developed when equations are structured by age, size, more generally a physiological trait. The renewal, growth-fragmentation are the more standard equations.
The talk will present structured equations, show that a universal generalized relative entropy property is available in the linear case, which imposes relaxation to a steady state under non-degeneracy conditions. In the nonlinear cases, it might be that periodic solutions occur, which can be interpreted in biological terms, e.g., as network activity in the neuroscience.
When the equations are conservation laws, a variant of the Monge-Kantorovich distance (called Fortet-Mourier distance) also gives a general non-expansion property of solutions.
Alexandre Poulain
Title: Modelling the impairment of the glymphatic system due to glioma growth
Abstract: The brain lacks a lymphatic system and it was previously unclear how clearance of metabolic waste occurs. In 2012, a new theory emerged and described the glymphatic system that we will present in this talk. In many diseases affecting the brain, an impairment of this glymphatic system is observed. While many studies focused on the changes in fluid flow and clearance of toxic waste or tracer molecules in the context of many neurological diseases, the impact on the glymphatic system due to glioma growth is less studied. To solve this issue, we propose a mathematical model to study how the clearance pathways are affected by the presence of a tumor in the brain parenchyma, and how the peri-tumoral edema could impact fluid movement in the brain. Our model is a multicompartment porous medium system and represents both the fluid movement in the brain and the clearance of solutes that are transported and diffuse in the extra-cellular space. Our preliminaries results indicate that the blockage of peri-vascular routes, due for example to migratory tumor cells, results in fluid stagnation and local increase of pressure in the tumor and peri-tumor regions. Furthermore, clearance of solutes is slowed in the tumor region due to local changes of mechanical properties. Michèle Romanos
Title: Unraveling tissue interplay during morphogenesis: a viscous multiparametric model reveals the crucial role of differential growth in vetebrate embryo development
Abstract: Axial extension is a morphogenetic process that results in the acquisition of the elongated shape of the vertebrate embryonic body. This process involves several adjacent embryonic tissues, including the neural tube and the paraxial mesoderm, which will later form functional organs such as the central nervous system and the musculature. While we have made significant progress in understanding how these tissues elongate individually, the full morphogenetic implications of their growth and mechanical interaction remain elusive. In this talk, we introduce a novel 2D PDE-based mathematical model, inspired by fluid mechanics and incorporating tissue viscosities through the Brinkman law, inter-tissue friction, pressure and differential tissue growth. We use live imaging and image analysis of transgenic quail embryos to measure the model parameters and faithfully calibrate it. We show some interesting mathematical properties inherent to this model, then explore it numerically to study the influence of differential growth, tissue biophysical properties, and tissue interactions on the embryonic body’s morphogenesis during axial extension. Our model shows that it can capture the long-term tissue dynamics between the paraxial mesoderm, neural tube, and notochord extension that has been observed in vivo. The model also reveals the underestimated influence of differential tissue proliferation in inter-tissue shaping by capturing the relative impact of this process on tissue dynamics. We validate the predictions of our model in vivo by showing that long-term inhibition of the paraxial mesoderm's higher rate of proliferation significantly influences tissue dynamics and shaping of both the paraxial mesoderm and the adjacent neural tube. Overall, our work provides a new theoretical framework to understand the long-term consequences of tissue differential growth and mechanical interactions on morphogenesis. Samuel Treton
Title: Stability of the trivial equilibrium in degenerate monostable reaction-diffusion equations Abstract: This talk adresses the long-term behavior of reaction-diffusion equations ∂tu =Δu + f(u) in RN, where the growth function f behaves as u1+p when u is near the origin. Specifically, we are interested in the persistance versus extinction phenomena in a population dynamics context, where the function u represents a density of individuals distributed in space. The degenerated behavior f(u) ∼ u1+p near the null equilibrium models the so-called Allee effect, which penalizes the growth of the population when the density is low. This effect simulates factors such as inbreeding, mating difficulties, or reduced resistance to extreme climatic events. We will begin the presentation by discussing a result linking the questions of persistence and extinction with the dimension N and the intensity of the Allee effect p, as established in the classical paper by Aronson and Weinberger (1978). This result is closely related to the seminal work of Fujita (1966) on blow-up versus global existence of solutions to the superlinear equation ∂tu = Δu + u1+p. Following these preliminary results, we will focus on a reaction-diffusion system involving a “heat exchanger”, where the unknowns are coupled through the diffusion process, integrating super-linear and non-coupling reactions. An analysis of the solution frequencies for the purely diffusive heat exchanger will allow us to estimate its “dispersal intensity”, which is a key information for addressing blow-up versus global existence in such semi-linear problems. This work represents a first step toward Fujita-type results for systems coupled by diffusion and raises several open questions, particularly regarding the exploration of more intricate diffusion mechanisms. Nicolas Vauchelet
Title: Mathematical analysis of a ‘rolling carpet’ strategy for SIT
Abstract: The use of the Sterile Insect Technique (SIT) for mosquitoes populations aims at locally eliminate the population of mosquitoes vector of several diseases. To extend spatially such a technique, an idea may be to use the so-called ‘rolling carpet’ strategy which consists in moving the zone of intervention. Of course this movement should be done with care to avoid reinfestation. In this talk, I will present a mathematical approach of this strategy. More precisely, we will prove that it is always possible to generate a wave of extinction thanks to this strategy. This work has been done in collaboration with Luis Almeida, Alexis Leculier, and Nga Nguyen.
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