Special Session 12: Propagation Phenomena in Reaction-Diffusion Systems

Regularity and long-time dynamics of some free boundary models: Successes and challenges

Yihong Du University of New England (Australia) Australia Co-Author(s):

Abstract: In 1937, independently, Fisher and KPP (Kolmogorov-Petrovskii-Piskunov) used a simple reaction-diffusion equation to model the spreading of a species, which is now known as the Fisher-KPP model. A striking feature of the model is that it predicts an asymptotic spreading speed (determined by an associated traveling wave solution). Such a phenomenon has been observed in many examples of species spreading in the real world, and was first rigorously proved by Aronson-Weinberger (1975), after which much further developments of the mathematical theory for propagation have been achieved along various lines. In this talk, I will focus on one aspect of some recent developments of this theory, which involves reaction-diffusion equations with free boundaries. I will discuss the longtime dynamics of these models and the associated regularity questions, including some success stories and challenges.

Propagation of opinions on a network. The Daley-Kendall model.

Romain Ducasse LJLL, universite Paris Cite France Co-Author(s):

Abstract: We study a model designed to describe the spatial spread of opinions in a population. This model is based on the Daley-Kendall model for opinion propagation. It consists in a system a nonlinear integral equations, and shares some ressemblances with the SIR model from epidemiology, in that it treats the transmissions of the opinions similarly to those of a virus. The crucial difference is that the Daley-Kendall has a nonlinear saturation terms, that accounts for the stiffling of the opinion. We consider the model set on graph, so that the transmission and the stiffling depend on the neighbour nodes. We study how the opinion propagates through the graph (convergence toward steady states, speed of propagation). We also show that, in some situations, the model converges toward pattern-like steady states.

Biological invasions in patchy landscapes: a reaction-diffusion model with interface conditions

Francois Hamel Aix-Marseille University France Co-Author(s):    Quentin Griette, Frithjof Lutscher, Mingmin Zhang, Min Zhao

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. I will also report on some propagation or extinction phenomena in the case of periodic patches involving bistable or monostable reactions. The talk is based on some joint works with Quentin Griette, Frithjof Lutscher, Mingmin Zhang and Min Zhao.

Rigorous Construction of Stop-and-Go Waves in the Optimal Velocity Model via a differential-difference equation

Kota Ikeda Meiji University Japan Co-Author(s):    Tomoyuki Miyaji

Abstract: We investigate nonlinear wave structures in the Optimal Velocity (OV) model, a fundamental microscopic traffic flow model describing car-following dynamics on a circuit. By introducing a traveling-wave representation of vehicle headways, the original ordinary differential system is reduced to a difference-differential equation. We focus on steep optimal velocity functions close to a step function, which generate sharp transition layers in the headway profile. In the singular limit, explicit heteroclinic transition layer solutions connecting two uniform traffic states can be constructed. Motivated by related exactly solvable queue models in the literature, we rigorously prove the existence of heteroclinic traveling waves for sufficiently steep optimal velocity functions. We further establish the existence of homoclinic solutions formed by the interaction of increasing and decreasing transition layers and derive a necessary condition on the amplitude parameter for their existence. To construct periodic stop-and-go waves on a circuit, we impose a global constraint reflecting conservation of total road length. Within this constrained framework, we prove the existence of large-period periodic solutions consisting of alternating transition layers and quasi-uniform states. These results provide a rigorous foundation for nonlinear congestion wave patterns beyond local bifurcation analysis.

Spot solutions to a neural field equation on the sphere and spheroid

Hiroshi Ishii Hokkaido University Japan Co-Author(s):    Riku Watanabe

Abstract: This talk deals with the Amari model, an integro-differential equation that is one of the fundamental neural field equations. Neural field equations are mathematical models used to analyze the time evolution of excitation patterns in neural fields and to understand the dynamics of neuronal populations at the tissue level. In recent years, increasing attention has been paid to the influence of the geometry of organs, such as the brain, on pattern formation, and to the constraints on solution behavior imposed by curved surfaces. To investigate such geometric effects, we consider the sphere and a slightly deformed sphere (spheroid) as model surfaces, and analyze how the structure and stability of spot solutions depend on the geometry of the surface. First, we construct spot solutions on the sphere and characterize their linear stability through spectral analysis of the linearized operator. Next, we perform a perturbation analysis from the sphere to construct spot solutions at the north pole of the spheroid and derive the eigenvalues that determine stability. Finally, numerical simulations based on the theoretical results are carried out to examine the behavior of spot solutions.

Classification and bifurcation structures of cross-diffusion limits in the SKT model

Kousuke Kuto Waseda University Japan Co-Author(s):    Yaping Wu

Abstract: This talk concerns asymptotic regimes of positive steady states in the Shigesada-Kawasaki-Teramoto competition model under large cross diffusion limits. First, we present a systematic classification of limiting regimes according to unilateral or full cross diffusion limits together with boundary conditions including Neumann and Dirichlet cases. We explain how these four settings lead to distinct limiting shadow systems and determine qualitative structures of coexistence states such as segregation, extinction, or concentration profiles. This classification provides a unified framework that organizes previously known results and clarifies structural relations between different limiting procedures. Second, we analyze the unilateral cross diffusion limit with Neumann boundary conditions and describe bifurcation structures of large amplitude steady states near critical spectral thresholds. Combining spectral analysis, singular perturbation techniques, and reduction methods, we establish existence of solution branches and obtain precise descriptions of their asymptotic profiles and instability properties.

On the Shape of Expansion of a Population in the Presence of a Fast Road

King Yeung Lam The Ohio State University USA Co-Author(s):    Chris Henderson and King-Yeung Lam

Abstract: We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a thin-front limit, we deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road that governs the front location in the road-field model. Our main motivation is to apply the theory of strong (flux-limited) viscosity solutions to obtain a control-theoretic interpretation of the front location. We then formulate the ecological invasion problem as one of finding optimal paths that balance the positive growth rate in the field with the fast diffusion on the road. Along the way, we extend the results of Berestycki et al. to conical domains and exhibit non-convex expansion shapes. We also provide a new proof of known results for the one-road half-space problem using our approach. This is joint work with Chris Henderson.

Traveling Waves for Burgers-Fisher-KPP Equations with Singularity

Ming Mei Jiangxi Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present a recent study on Burgers-Fisher-KPP equation with singular slow/fast diffusion and singular/regular convection in the form of $u_t-D\Delta u^m+\alpha(u^p)_x=f(u)$ with $m,\,p>0$, focusing on the existence, non-existence, regularity and stability of traveling waves. The values of m and p essentially affect the existence/non-existence of regular/sharp traveling waves as well as their regularity. By combining phase-plane analysis and variational techniques, we obtain a complete classification of existence/non-existence of regular/sharp traveling waves related to m and p. In the singular regimes with 0

Existence and stability of nontrivial solutions for bistable reaction-diffusion equations on graphs of finite length

Harunori Monobe Osaka Metropolitan University Japan Co-Author(s):    Yoshihisa Morita

Abstract: In this study, we consider the existence and stability of steady-state solutions to reaction-diffusion equations with a bistable nonlinearity on a graph of finite length. In this presentation, we first consider a ``star-graph'' consisting of a single junction and show the existence and stability of non-constant steady-state solutions when the edge lengths are sufficiently large. We then use these results to show the existence and stability of non-constant steady-state solutions on more complex graphs. This is a joint work with Professor Yoshihisa Morita.

Propagation and blocking of driven curvature flow in a strip domain with periodic obstacles

Ryunosuke Mori Meiji University Japan Co-Author(s):

Abstract: We study the long-time behavior of driven curvature flow in a strip domain with periodic obstacles. In this setting, solutions exhibit two distinct behaviors: propagation and blocking. This talk characterizes how geometric features of the domain determine which behavior occurs. In a strip domain with periodically undulating boundaries, Matano, Nakamura, and Lou (2006) characterized the long-time behavior of driven curvature flow in terms of the boundary geometry. They introduced the notion of a maximal opening angle to describe the conditions for propagation, blocking, and the propagation speed, under which global-in-time graph-like classical solutions exist. Later, Matano and Mori extended this analysis to more general settings where such classical solutions may fail to exist, and proposed the concept of an effective opening angle to characterize propagation and blocking phenomena. In a strip domain with periodic obstacles, we introduce the notions of left and right opening angles, which are analogous to the effective opening angle. A key difference from previous studies lies in the propagation speed: while it is uniquely determined in earlier works, it is no longer unique in our setting.

An eco-epidemiological model with prey taxis and slow diffusion

Rana Parshad Iowa State University USA Co-Author(s):

Abstract: In this work we analyze the effects of prey-taxis on the existence of global-in-time solutions and dynamics of an eco-epidemiological model. In particular, we consider the influence of slow dispersal characterized by the $p$-Laplacian operator, and enhanced mortality of the infected prey. Our results have large scale applications to biological invasions and biological control of pests.

Traveling waves and propagating terraces in gravitational fingering phenomena

Iuliia Petrova University of Sao Paulo (USP) Brazil Co-Author(s):    Sergey Tikhomirov, Yalchin Efendiev

Abstract: In the talk we will discuss the gravitational fingering phenomenon - the unstable displacement of miscible liquids in porous media with the speed determined by Darcy`s law. Laboratory and numerical experiments show the linear growth of the mixing zone, and we are interested in determining the exact speed of propagation of fingers. We consider two models: Incompressible Porous Medium equation (IPM) and Transverse Flow Equilibrium (TFE). The existing theoretical upper bounds for the growth rate of the mixing zone (see e.g. Otto-Menon, 2005) are higher than the observed speed from the numerical simulations. We believe that one of the possible mechanisms of slowing down the fingers` growth is due to convection in the transversal direction, and explain it by introducing a semi-discrete model of IPM and TFE. The model consists of a system of advection-reaction-diffusion equations on concentration, velocity and pressure in several vertical tubes (real lines) and interflow between them. In the simplest setting of two tubes we show the structure of gravitational fingers - the profile of propagation is characterized by two consecutive travelling waves which we call a terrace. We prove the existence of such a propagating terrace for the parameters corresponding to small distances between the tubes. The main tool used in the proof is geometric singular perturbation theory (Fenichel) together with invariant manifold theory (Hirsch-Pugh-Shub). The questions of well-posedness of the model and stability of traveling waves are open, and I will be happy for any advice in this direction. The talk is based on joint work with S. Tikhomirov and Ya. Efendiev (SIMA, arXiv:2401.05981).

Threshold phenomena in local and nonlocal bistable reaction-diffusion equations

Lionel Roques INRAE France Co-Author(s):

Abstract: Threshold phenomena are a basic feature of bistable reaction-diffusion equations. They refer to the existence of critical initial data separating extinction from invasion, with convergence to a ground state at the threshold. I will first discuss the classical local case $\partial_t u = \partial_{xx} u + f(u)$, with emphasis on the role of the amplitude and the spatial arrangement of the initial condition, in particular through fragmentation effects. I will then turn to a nonlocal equation of the form $\partial_t u = \partial_{xx} u + r(x) u - u \int_{\mathbb{R}} u - h(u)$, where the threshold structure appears to be richer and may involve two distinct thresholds.

Freidlin-Gartner formula and asymptotic profile in reaction-diffusion equations

Luca Rossi Sapienza University of Rome Italy Co-Author(s):

Abstract: We address the question of the large-time behavior of solutions of reaction-diffusion equations in periodic media. We will start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin-Gartner formula. We will then present some recent results for the bistable equation, obtained in collaboration with H. Guo and F. Hamel, about a regular version of the Freidlin-Gartner formula, as well as the convergence of the profile of the solution towards pulsating traveling fronts.

Reaction-Diffusion Model for Pattern Formation in Tape Peeling

Keisuke K Taga Tokyo University of Science Japan Co-Author(s):    Keisuke Taga

Abstract: Depending on the peeling velocity, the traces formed on the surface of adhesive tape exhibit a variety of patterns. At an intermediate peeling velocity, a random fractal pattern resembling a Sierpi\`{n}ski gasket is observed. Several models have been proposed to explain this pattern formation. In this talk, we introduce our model, formulated as a reaction-diffusion system. The model is constructed by describing the tape-peeling dynamics as an excitable oscillatory system and incorporating spatial interactions arising from viscoelasticity together with a stochastic term. It has the form of a Li\`{e}nard system and, under an appropriate transformation, can be related to a Bonhoeffer--van der Pol type reaction-diffusion system, which is known to exhibit Sierpi\`{n}ski-gasket patterns. We will also present results on the dynamical scaling properties of this model.

Reaction-diffusion approximation for nonlocal interactions

Yoshitaro Tanaka Future University Hakodate Japan Co-Author(s):    Hiroshi Ishii

Abstract: In this talk, we study the mathematical relationship between nonlocal interactions of convolution type and systems of multiple diffusive substances in high-dimensional space. Motivated by the observation that nonlocal evolution equations can reproduce similar patterns to those arising in reaction-diffusion systems, we approximate nonlocal interactions in evolution equations by solutions to appropriate reaction-diffusion systems with multiple components in Euclidean space of arbitrary dimension. It is shown that any absolutely integrable radial kernel can be approximated by a linear combination of specific Green functions to elliptic partial differential equations. This enables us to demonstrate that a linear sum of auxiliary diffusive substances can approximate a broad class of nonlocal interactions of convolution type. Our results establish a connection between a broad class of nonlocal interactions and diffusive chemical reactions in dynamical systems.

Global existence for mass controlled reaction-diffusion systems

Bao Q Tang University of Graz, Department of Mathematics and Scientific Computing Austria Co-Author(s):

Abstract: In this talk I discuss some recent advances on global existence of reaction-diffusion systems, where only natural assumptions such as quasi-positivity and mass controlled are imposed. It is shown that, in the case of smooth diffusion, the system with quadratic growing nonlinearities has a unique global classical solution, which is also bounded uniformly in time provided that the mass is dissipated. For non-smooth diffusion coefficients, the global existence of bounded weak solutions is shown by a generalised $L^p$-energy method and intermediate sum condition with sub-critical growth of nonlinearities.