Presented by Chiun-Chuan Chen (National Taiwan University)
title: Variational Methods and Traveling Wave Solutions of Reaction-diffusion Equations
abstract: Heinze, Lucia, Muratov, Novaga, and others developed a variational method for studying traveling wave solutions of reaction-diffusion equations when they have a suitable structure. This method not only allows us to obtain traveling wave solutions but also provides a variational description of the wave speed. In this talk we will present some old and new examples of this approach, including equations with non-local terms, multi-phase waves, and a Stefan-type free boundary problem.
Presented by Wonhyung Choi (KAIST)
title: Reaction-advection-diffusion system for two strongly competitive species
abstract: In this study, we explore the behavior of two strongly competitive species within a spatially heterogeneous environment using a Lotka–Volterra-type reaction-advection-diffusion model. The model assumes that one species diffuses at a constant rate, while the other species moves toward a more favorable environment through a combination of constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection-diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The results are obtained by analyzing the stability of semitrivial solutions. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates, particularly for relatively moderate interspecific competition parameters in species with directional movement. These findings underscore the crucial role of directed movement and interspecific competition coefficients in shaping the dynamics and coexistence of strongly competing species. This is a joint work with Prof. Inkyung Ahn.
Presented by HARUNORI MONOBE (Osaka Metropolitan University)
title: Compact traveling waves to a mean-curvature flow with driving force
abstract: Mean-curvature flow with a driving force appears in various mathematical problems such as motion of soap films, grain boundaries and singular limit problems of various reaction-diffusion systems, e.g., Allen-Cahn-Nagumo equation. In this talk, we show the existence and uniqueness of traveling waves, composed of a Jordan curve (or closed surface), for an anisotropic curvature flow with a driving force. We call such a solution ``compact traveling wave" in this talk. Our aim is to investigate the condition of external driving force when the curvature flow has traveling waves. This is based on a joint work with H. Ninomiya.
Presented by Chun-Hsiung Hsia (National Taiwan University)
title: On the Synchronization Analysis of a Strong Competition Kuramoto Model
abstract: This is joint work with Chung-En Tsai. When modeling the classical Kuramoto model, one of the key features is the tendency to synchronize. Accordingly, the most well-adopted choice of the coupling function is the sine function. Due to the oddness of the sine function, the synchronized frequency would be the average of all the natural frequencies. In our study, we investigate the synchronization behaviors of the Kuramoto model with a pure competition coupling function. Namely, instead of the sine function, we choose max{0,sinθ} to be the coupling function. This indicates the relation of pure competition between oscillators. We prove asymptotical phase synchronization for identical oscillators and asymptotical frequency synchronization for non-identical oscillators under reasonable sufficient conditions. In particular, under our sufficient conditions, the synchronized frequency is the maximal frequency of all the natural frequencies. On the other hand, in the parameter regime which is out of the scope of the analysis of our theorems, it is possible that the synchronized frequency could be larger than the maximal frequency of the natural frequencies of all the oscillators. In this article, we also provide numerical experiments to support the analysis of our theorem and to demonstrate the aforementioned phenomenon.
Presented by Olivier Hénot (École Polytechnique)
title: Computer-Assisted Proofs for Nonlinear Diffusion Problems: Steady-States and Stability
abstract: In this talk, we explore computer-assisted proof (CAP) techniques for solving nonlinear diffusion problems. We begin by reviewing CAP methodologies, particularly those based on spectral methods and the contraction mapping theorem. These tools are applied to rigorously compute steady-state solutions for systems of the form $partial_t u = Delta Phi(u) + R(u)$ on bounded domains with Neumann boundary conditions. In particular, we focus on domains of the form $Pi[a_i, b_i]$ to use a Fourier series discretization of the solutions. We then introduce a new computer-assisted approach under development, to study the stability of such steady-state solutions. Our strategy consists in constructing an explicit local Lyapunov functional, based on the Lyapunov matrix equations used in control theory.
Presented by Chang-Yuan Cheng (National Kaohsiung Normal University)
title: Intra- and inter-specific competitions between stage-structured species
abstract: Creatures have the varied abilities in their different life stages to compete for resources, space or mating, so separating a population by life stages is an essential baseline in describing an ecological population. In addition, behaviors of creatures (like competition between species and their life regulation) and the interaction of species with the environment (for example, dispersal according to the spatial heterogeneity of a habitat) are considerable features to construct mathematical models. Based on the considerations, a model with two life stages, immature and mature, incorporating both intra- and inter-competitions between two species is explored to study the invasion of species in a two-patch environment. The monotone dynamics in such a model provides us a property to explore its local and global dynamics. This is a joint work with Prof. Chih-Wen Shih and Prof. Kuang-Hui Lin.
Presented by Danielle Hilhorst (CNRS - Universite of Paris-Saclay)
title: Mathematics for Geothermy
abstract: This talk covers Adrien Beguinets PhD thesis, which focuses on developing mathematical models for geothermal energy, specifically within the ANR UPGEO project. The thesis addresses how fluid flow and ground deformation interact in porous media, using Biot system of equations to couple mechanical displacement and fluid pressure. The model is expanded to include temperature, making it relevant for geothermal applications. The research uses the Gradient Discretisation Method (GDM) for numerical solutions and derives error estimates between the true solution and its numerical approximation. Unlike traditional models with deterministic parameters, this work models permeability and elasticity as random fields, utilizing Karhunen-Loève expansions to manage uncertainties. The thesis shows that truncating these expansions still maintains accuracy while reducing computational costs. The study also incorporates a thermo-poroelastic model, which couples mechanical, fluid, and thermal effects in the subsurface. This model is solved numerically using finite elements, and the randomness in the coefficients is addressed with Monte Carlo simulations. The main result is an error estimate that combines numerical and statistical uncertainties, providing a more accurate and efficient model for geothermal energy applications. This is joint work with Adrien Beguinet, Ludovic Goudenege, and Jhe-Kuan Su.
Presented by Sébastien Codina (Embassy of France to the Republic of Korea)
title: The Embassy of France congratulates the opening of ReaDiNet 2024.
abstract:
Presented by Hirokazu Ninomiya (Meiji University)
title: Propagation and Blocking of Bistable Waves by Environment-Dependent Variable Diffusion
abstract: Biological diffusion processes are often influenced by environmental factors. In this talk, we investigate the effects of variable diffusion, which depends on a point between the departure and arrival points, on the propagation of bistable waves. This process includes neutral, repulsive, and attractive transitions. By using singular limit analysis, we derive the equation for the interface between two stable states and examine the relationship between wave propagation and variable diffusion. More precisely, when the transition probability depends on the environment at the dividing point between the departure and arrival points, we derived an expression for the wave propagation speed that includes this dividing point ratio. This shows that, asymptotically, the boundary between wave propagation and blocking in a one-dimensional space corresponds to the case where the transition probability is determined by a dividing point ratio of 3:1 between the departure and arrival points. Furthermore, as an application of this concept, we consider the Aliev-Panfilov model to explore the mechanism for generating target patterns. This is based on a joint work with K. Nakajima.
Presented by Jae Kyoung Kim (KAIST/IBS)
title: Beyond microtubules: The cellular environment at the endoplasmic reticulum attracts proteins to the nucleus, enabling nuclear transport
abstract: All proteins are translated in the cytoplasm, yet many, including transcription factors, play vital roles in the nucleus. While previous research has concentrated on molecular motors for the transport of these proteins to the nucleus, recent observations reveal perinuclear accumulation even in the absence of an energy source, hinting at alternative mechanisms. Here, we propose that structural properties of the cellular environment, specifically the endoplasmic reticulum (ER), can promote molecular transport to the perinucleus without requiring additional energy expenditure. Specifically, physical interaction between proteins and the ER impedes their diffusion and leads to their accumulation near the nucleus. This result explains why larger proteins, more frequently interacting with the ER membrane, tend to accumulate at the perinucleus. Interestingly, such diffusion in a heterogeneous environment follows Chapmans law rather than the popular Ficks law. Our findings suggest a novel protein transport mechanism arising solely from characteristics of the intracellular environment.
Presented by Yong-Jung Kim (KAIST)
title: Fractionation By Diffusion: Experiments and Two-Component Diffusion Law
abstract: The fundamental question regarding the fractionation phenomenon is whether diffusion alone is responsible for it or whether an additional advection dynamic is involved. We studied the fractionation by diffusion of particles in spatially heterogeneous environments. By experimentally observing the time-sequential fractionation patterns of dye particles diffusing across a solid–solid interface of varying polyacrylamide gel densities, we found that the two-component diffusion model accurately captures the observed fractionation dynamics. In contrast, single-component diffusion models by Fick, Wereide, and Chapman do not. Our results indicate that diffusion alone can explain the fractionation phenomenon and that additional advection dynamics are not involved. The underlying physics in the fractionation phenomenon is discussed by using a two-component random walk model.
Presented by Chih-Chiang Huang (National Chung Cheng University)
title: The effects of long-range interaction to wave propagation
abstract: In the talk, we study a nonlocal equation induced by a free energy. Due to the Turing instability, a local pattern and wave propagation can be observed. Applying a variational method, we can construct the existence of traveling waves and discuss a related wave interaction problem. This is a joint work with Chao-Nien Chen, Yung-Sze Choi and Shyuh-yaur Tzeng.
Presented by Jaewook Ahn (Dongguk University)
title: Pattern and singularity formations for chemotaxis-consumption systems
abstract: Bacillus subtilis swim toward oxygen-rich air-water interfaces in water droplets and form large clusters near the boundary. To describe such pattern formation, chemotaxis systems with signal consumption have been proposed, which, in the numerical literature, have displayed various patterns similar to those observed in actual experiments. In this talk, I will present related analytical results on chemotaxis-consumption systems, particularly those with Dirichlet boundary conditions for the signal. One of our results reflects that bacteria populations tend to aggregate near the boundary, where signals have been prescribed, in the large time limit at least when the initial mass is sufficiently small. On the other hand, for the system of chemorepulsive counterpart, it is shown that a finite time blowup can be observed whenever the diffusion effect on bacteria populations is slightly weakened. Some results on the existence of bounded solutions will also be discussed.
Presented by Sohei Tasaki (Hokkaido University)
title: Mathematical analysis of motility response to resources in bacterial cell populations
abstract: Motile bacteria regulate their motility in response to resource levels, thereby creating a robust cellular social system that can tolerate environmental fluctuations. The resource-response curve of quantified motility varies among bacterial species and resources. Here we consider the optimization problem of finding the resource-motility function that maximizes a certain growth index in a mathematical model of cellular and nutrient distribution described by a reaction-diffusion system. The results suggest that the optimal motility response is also nonmonotonic when the nutrient distribution is spatially nonmonotonic, although the optimal function differs depending on the situation. Furthermore, we report that this motility function can be applied to simulate the formation dynamics of cell aggregation patterns by incorporating multiple motilities maximized at different resource levels in a mathematical model.
Presented by Minyoo Kim (KAIST)
title: Fractionation Phenomena in Heterogeneous and Anisotropic Persistent Random Walk
abstract: Random motion of microscopic particles in heterogeneous environments leads to fractionation, with the Soret effect being one of the most representative examples. This raises a fundamental question: what characteristics of random motion give rise to such fractionation phenomena? We investigate whether the persistence of random movements has the property and show that fractionation occurs when the persistence is anisotropic. This is shown by investigating the convergence of a heterogeneous persistence random walk system and the resulting diffusion equation.
Presented by Junseong Park (KAIST)
title: Mathematical modeling with heterogeneous permeable barriers
abstract: Mathematical modeling is presented by using discrete-time random walk on a 1D lattice with a non-constant permeable barrier. We first show that the convergence of the discrete density of this model equals to the solution of the continuum diffusion equation with a heterogeneous permeability. Also, we formally show that the model can obtain permeable boundary condition.
Presented by Gaël Raoul (CNRS - Ecole Polytechnique)
title: Growing random planar network with oriented branching and fusion
abstract: Formation of biological networks and their spatial structure is of major importance in many different contexts: vascularization of an organ, growth of mycelium, etc. The spatial progression of such networks in space has been considered through Partial Differential Equation models, and in some cases, this progression of networks can be described with precision. The dynamics of networks once it is well developed (behind the front), however, remains poorly understood. In particular, it is difficult to describe the structure of the network once it is fully formed. In this talk, I will present a recent work with Vincent Bansaye and Milica Tomasevich, where we consider a very simple model for network formation. We chose to focus on a very particular model, where the branches of the network grow into lines at a constant speed, and branch on the right-hand side only. This allow us to describe the dynamics in detail and to infer some statistical properties of the resulting network. We hope some of the ideas used in this work can be useful to analyze more general models.
Presented by Kunwoo Kim (POSTECH)
title: Phase analysis for a family of stochastic reaction-diffusion equations
abstract: In this talk, we consider reaction-diffusion equations of the Fisher-KPP type, which are perturbed by a multiplicative type space-time white noise. We show that if the strength of the effect of the noise is large, the solution decays to 0 uniformly as time goes on, which implies that there is a unique invariant measure. On the other hand, if the noise strength is small enough, there are infinitely many invariant measures. This is based on joint work with Davar Khoshnevisan, Carl Mueller and Shang-Yuan Shiu.
Presented by Kentaro Nagahara (Institute of Science Tokyo)
title: Constructing global maximizer without bang-bang property on asymmetric network graphs
abstract: This talk addresses the monostable reaction-diffusion logistic equation related to mathematical biology. This equation, also known as the Fisher-KPP equation, was proposed by J.G. Skellam in 1951 as a model equation describing the population dynamics of organisms, and it has been extensively studied. In this talk, we focus on the total population of species inhabiting a spatially heterogeneous network of patches and discuss the properties of the global maximizer of the total population under some constraints. We will introduce the global maximizer that possesses a property known as the bang-bang property, which has been revealed in recent studies, and show that a global maximizer without the bang-bang property can be constructed on asymmetric network graphs.
Presented by John Meng-Kai Hong (National Central University)
title: Shock wave solutions to the interaction-dominated types of generalized Keller-Segel equations and their viscous shock profiles
abstract: In this talk, we study the shock wave solutions to a type of one -dimensional generalized Keller-Segel equations whose interaction term dominates the diffusion term. The limiting system of the K-S equations, in which the diffusion coefficients are vanished, can be written as a 3 by 3 hyperbolic system of balance laws. We construct an approximate solution to the generalized Riemann problem of this balance laws as the building block of the generalized Glimm scheme for Cauchy problem. We also study the viscous shock profiles of the shock wave solutions by using the geometric singular perturbations method.
Presented by Yoshitaro Tanaka (Future University Hakodate)
title: Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain
abstract: Motivated by various phenomena such as cell migration, cell adhesion and collective motion, many evolutional equations with convolution-type interactions are proposed. This type of the interaction is called the nonlocal interaction, and it is imposed as a velocity of the advection term in the models that target the above phenomena. In these evolution equations, by changing the shape of the integral kernel, it is possible to change the phenomenon being described and also to reproduce the various patterns as the solutions. In this talk, motivated by converting the nonlocality in the advection term by the spatially local effect, we approximate the nonlocal Fokker-Planck equation by the Keller-Segel system with multiple chemotactic factors in a singular limit analysis. In particular, by controlling the parameters in the Keller-Segel system, we show that any even integral kernels can be approximated by the linear sum of the fundamental solutions for an elliptic equation. From this result, we explain the mathematical relationship between the processes by the aggregation-diffusion with Haptotaxis and chemotaxis. This talk is based on the result of the collaboration with Prof. Hideki Murakawa from Ryukoku University.
Presented by Changwook Yoon (Chungnam National University)
title: Ratio-dependent motility in biological diffusion models
abstract: In this study, we explore the application of ratio-dependent motility to biological diffusion models, providing a framework for understanding how organisms adjust their movement based on the ratio of available resources to population density. This approach reflects a more realistic perspective on biological systems, where resource competition, rather than absolute abundance, determines motility. We apply this motility concept in both chemotaxis and prey-predator models, demonstrating its potential to enhance global solvability and provide deeper insights into the dynamics of resource-limited environments.
Presented by Hyunjoon Park (Meiji University)
title: Adjustment of the interface of the shadow wave-pinning model
abstract: In this talk we study the behavior of the shadow wave-pinning model. The model composed of a non-local reaction diffusion equation and was constructed to approximate the original wave-pinning model which describes the biochemical cell polarization. Such solution is expected to generate a steep transition layer within a relatively short time and then propagates according to the volume preserving motion. And, in between it is expected that there is an adjustment period connecting these two movement. Our aim is to underrstand the behavior of the solution of the adjustment period. This is based on the joint work with Professor H.Matano, Y.Mori and Dr. R.Mori.
Presented by Lionel Roques (INRAE)
title: Fickian and Fokker-Planck diffusions and their generalizations in KPP equations: How do they affect persistence and spreading behavior?
abstract: In reaction-diffusion systems, both Fokker-Planck and Fickian diffusion operators are commonly employed to model various processes. Despite their distinct interpretations, these operators are sometimes used interchangeably in ecological models. After clarifying the microscopic assumptions behind Fickian diffusion (D(x) u_x)_x, Fokker-Planck diffusion (D(x) u)_{xx}, and their generalizations [D(x)^{1-q} (D(x)^q u)_x]_x with q in (-infty,infty), this presentation aims to highlight the implications of this choice, focusing on how population dynamics may differ depending on whether Fickian or Fokker-Planck diffusion is used. Specifically, we analyze the persistence and spreading behavior in KPP-like reaction-diffusion models with periodic coefficients. We establish under which conditions these persistence and spreading properties depend on the choice of the diffusion term. Our results rely on a careful analysis of the dependence of eigenvalue problems on q. This is a joint work with Nathanaël Boutillon and Yong-Jung Kim, and it also incorporates new results on the principal eigenvalues of Schrödinger operators with advection, obtained in collaboration with François Hamel.
Presented by Ken-Ichi Nakamura (Meiji University)
title: The sign of the speed of bistable traveling fronts in multi-species models
abstract: We are concerned with the sign of the propagating speed of traveling fronts in bistable dynamics. In the competition model, the sign of front speed gives us information on which species becomes dominant and eventually occupies the whole habitat. Therefore, it is important to determine the sign of the unique front speed in bistable dynamics. In this talk, we first give some results for bistable 2-species Lotka-Volterra competition models and then discuss the signs of bistable traveling fronts in a 3-species competition model. This talk is based on joint work with Toshiko Ogiwara (Josai University).
Presented by Jimmy Garnier (CNRS Univ Savoie Mont-Blanc)
title: Propagation phenomenon in mutualistic systems structured by traits
abstract: The evolution of mutualism between host and symbiont communities plays an essential role in maintaining ecosystem function and should therefore have a profound effect on their range expansion dynamics. In particular, the presence of mutualistic symbionts at the leading edge of a host-symbiont community should enhance its propagation in space. I will present a theoretical framework that captures the eco-evolutionary dynamics of host-symbiont communities, and allow us to investigate how the evolution of resource exchange may shape community structure during range expansion.
Presented by Jenn-Nan Wang (National Taiwan University)
title: The estimation of an unknown potential in a subdiffusion equation using the Bayesian approach
abstract: In this talk, I will discuss the inverse problem of determining an unknown potential in a subdiffusion equation from its solution using a nonparametric Bayesian approach. Our aim is to establish the consistency of the posterior distribution with Gaussian priors. To do so, we need some key estimates of the forward problem. For the forward problem, we have to overcome the fact that the solution of the subdiffusion equation is less regular than that of the classical heat equation. The main ingredient is the maximum principle for the subdiffusion equation. In view of the mild ill-posedness of the inverse problem, we show that the posterior concentrates around the ground true with a polynomial rate.
Presented by Junsik Bae (KAIST)
title: Emergence of peaked singularities in the Euler-Poisson system
abstract: We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation and provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the peaked solitary waves with the sharp range of Hölder exponents. Furthermore, we provide numerical evidence for C1 blow-up solutions to the pressureless Euler-Poisson system, whose blow-up profiles are asymptotically similar to its peaked solitary waves and exhibit a different form of blow-up compared to the Burgers-type (shock-like) blow-up. This is a joint work with Sang-Hyuck Moon(UNIST) and Kwan Woo(SNU).
Presented by Thomas Giletti (Université Clermont-Auvergne)
title: From moving heterogeneities to 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.
Presented by Min-Gi Lee (Kyungpook National University)
title:
abstract:
Presented by Seungmin Kang (National Center for Theoretical Sciences)
title:
abstract: