Abstract : Reaction-diffusion systems reveal rich phenomena related to mathematical biology and evolutionary dynamics, like pattern formation, propagation of traveling waves. These results are also related to the free boundary problems coming from the fast reaction limits. In order for the communication with other researchers, we organize a minisymposium consisting of the topics on patterns, traveling waves and entire solutions for reaction-diffusion systems. The speakers introduce their works from the approach by center manifold reduction, comparison principles and variational methods. We expect to have new and positive contributions to these fields.
[03601] Cross-diffusion derived from predator-prey models with two behavioral states in predators
Format : Talk at Waseda University
Author(s) :
Hirofumi Izuhara (University of Miyazaki)
Masato Iida (University of Miyazaki)
Ryusuke Kon (University of Miyazaki)
Abstract : Cross-diffusion may be an important driving force of pattern formation in population models. Recently, a relation between cross-diffusion and reaction-diffusion systems has been revealed from the mathematical modeling point of view. In this talk, we derive a predator-prey model with cross-diffusion from a simple reaction-diffusion system with two behavioral states in the predator population and examine whether cross-diffusion can induce spatial patterns in predator-prey models.
[00453] Weak entire solutions of reaction–interface systems
Format : Talk at Waseda University
Author(s) :
YANYU CHEN (National Taiwan University)
Abstract : In this talk, the singular limit problems arising from FitzHugh–Nagumo–type reaction–diffusion systems are studied, which are called reaction–interface systems. All weak entire solutions originating from finitely many excited intervals are completely characterized. For weak entire solutions originating from infinitely many excited intervals, periodic wave trains and time periodic solutions are discussed.
[03042] Pulse bifurcations in a three-component FitzHugh-Nagumo system
Format : Talk at Waseda University
Author(s) :
Kei Nishi (Kyoto Sangyo University)
Abstract : Pulse dynamics in a three-component FitzHugh-Nagumo system in one dimensional space is considered. The system admits a pulse solution of bistable type, which exhibits a variety of interface dynamics, not observed for the two-component FitzHugh-Nagumo system. In order to analytically investigate the mechanism for the pulse behavior, we apply the multiple scales method to the original reaction-diffusion system, and derive finite-dimensional ordinary differential equations which describe the motions of the pulse interfaces. The reduced ODEs enable us to reveal the global bifurcation structures of the pulse solutions, and to clarify the mechanism behind the variety of the pulse dynamics from a view point of bifurcation theory.
[05009] The Motion of Weakly Interacting Waves for Reaction-Diffusion Equations in a Cylinder
Format : Talk at Waseda University
Author(s) :
Chih-Chiang Huang (National Chung Cheng University )
Shin-Ichiro Ei (Hokkaido University)
Abstract : In the talk, I am going to introduce the well-known results of the weakly interaction of two waves in a real line, for the Allen-Cahn equation, the FitzHugh-Nagumo system and competition-diffusion systems. Next, I would like to discuss such an interaction for reaction-diffusion equations with a triple-well potential in a cylinder. In this case, we can construct a stable traveling wave which is made up by two repulsive fronts. Based on a perturbation theory, the wave profile and wave speed can be characterized by a small parameter. This work is joint with Prof. Shin-Ichiro Ei.
[02140] Some Progress on the spreading properties of two-species Lotka-Volterra competition-diffusion systems
Format : Talk at Waseda University
Author(s) :
Chang-Hong Wu (National Yang Ming Chiao Tung University)
Abstract : The Lotka-Volterra competition-diffusion system is a well-established model for understanding the interactions between competing species. In particular, the two-species case has been extensively studied, revealing the existence of traveling waves that can provide insight into the spreading behaviors of the species. In this presentation, we will present some recent progress on the spreading properties of this system.
[02366] Weak interaction between traveling wave solutions in the three-species competition-diffusion systems
Format : Talk at Waseda University
Author(s) :
Chueh-Hsin Chang (National Chung Cheng University, Department of Mathematics)
Abstract : In this talk we consider the weak interaction between two trivial three-species traveling wave solutions (one component is trivial) of the threes-species Lotka-Volterra competition-diffusion systems. By the asymptotic behavior of the trivial threes-species waves and the existence results of the three-species waves from gluing bifurcation approaches in our previous results, we can observe the dynamics of the distance between the trivial three-species waves. The interaction between the two trivial three-species waves are attractive or repulsive due to different conditions of parameters. We let the growth rate of the third species as the bifurcation parameter.
[04983] Defects in the segmented pattern for oscillated reaction-diffusion systems
Format : Talk at Waseda University
Author(s) :
Ayuki Sekisaka (Meiji university)
Abstract : In this talk, the existence and stability of the modulated waves of a certain three-component reaction-diffusion system will be discussed. The modulated waves appearing in this equation are called defects, and their formulation and basic properties have been investigated by Sandstede and Scheel. In this talk, we will discuss the basic properties of the deffects appearing in the equations and their stability using the infinite dimensional Evans function.
[04560] Linearized eigenvalue problems in a mass-conserved reaction-diffusion compartment model
Format : Talk at Waseda University
Author(s) :
Tsubasa Sukekawa (Institute for the Advanced Study of Human Biology (ASHBi), Kyoto University Institute for Advanced Study, Kyoto University)
Abstract : In a mass-conserved reaction-diffusion system, we can observe by numerical simulations that a transient pattern
such as a stripe one converges to a spatially monotone pattern.
To understand the dynamics theoretically, we introduce a reaction-diffusion compartment model. This model equation is defined on multiple regions (compartments), and each compartment is connected by diffusive coupling.
In this talk, we analyze linearized eigenvalue problems of spatially non-monotone stationary solutions
in mass-conserved reaction-diffusion compartment model.
