Abstract : This minisymposium concerns theoretical and numerical studies about partial differential equations on metric graphs. Partial differential equations on metric graphs have been applied to a variety of areas as mathematical models related to phenomena in network structures. In particular, this minisymposium focuses on pattern formation and wave propagation in many reaction-diffusion systems on metric graphs, such as Lotka-Volterra systems, Gierer-Meinhardt systems, Allen-Cahn equations, and so on. This minisymposium will invite researchers to report their recent results on these subjects.
Organizer(s) : Satoru Iwasaki, Yuta Ishii, Shin-Ichiro Ei, Ken-Ichi Nakamura
[05005] Invasion analysis for population dynamics models on simple metric graphs
Format : Talk at Waseda University
Author(s) :
Satoru Iwasaki (Osaka University)
Harunori Monobe (Osaka Metropolitan University)
Abstract : We are concerned with invasion processes of biological species in network shaped domains, that is metric graphs. In our model, invasions of biological species are restricted by some traps. In this presentation, we report the difference between results of one-dimensional domains case and those of metric graphs case. Particularly, in metric graphs case, we know that vanishing occurs with less traps than one-dimensional case due to an effect of junctions in metric graphs.
[02969] Propagation phenomena of Fisher-KPP equation in a shifting environment
Format : Talk at Waseda University
Author(s) :
Jong-Shenq Guo (Tamkang University)
Abstract : In this talk, we shall discuss the propagation phenomena of the Fisher-KPP equation with a shifting intrinsic growth
rate. We divide the heterogeneous shifting term into two different classes, one is the devastating case and the other is the advantageous case. We shall present some results on the existence of forced waves and the spreading dynamics for solutions with compactly supported initial data.
[03115] Pulse dynamics for reaction-diffusion systems on various metric graphs
Format : Talk at Waseda University
Author(s) :
Shin-Ichiro Ei (Hokkaido University)
Haruki Shimatani (Hokkaido University)
Abstract : In this talk, we give pulse dynamics for reaction-diffusion systems on various metric graphs. In particular, we analyze the behavior of the pulse solutions by deriving an equation of the motion. Note an equation of the motion is an expression for the time evolution of the positions of pulse solutions. In the same way, we will also discuss front dynamics for reaction-diffusion systems on various metric graphs.
[03288] Reaction-advection-diffusion equations over simple graphs
Format : Talk at Waseda University
Author(s) :
Bendong Lou (Shanghai Normal University)
Abstract : I will talk about the dynamical behavior of a species spreading in a river network, which is modeled by advective diffusion equations over simple graphs with Fisher-KPP or bistable reactions. Denote by $c_*$ the (minimal) speed of the traveling waves of the corresponding equations without advections, I will present the long time behavior for the solutions in the special case where the water flow speed in upstream is larger than $c_*$ and that in downstream is smaller than $c_*$, which includes washing out, persistence at carrying capacity or persistence below carrying capacity. (joint works with Y. Du, R. Peng, M. Zhou, Y. Morita).
[04149] The effect of advection on spike solutions for the Schnakenberg model on Y-shaped metric graph
Format : Talk at Waseda University
Author(s) :
Yuta Ishii (National Institute of Technology, Ibaraki College)
Abstract : In this talk, we consider one-peak solutions for the Schnakenberg model with advection term on Y-shaped metric graph. The location and amplitude of the spike are decided by the interaction of the advection with the geometry of Y-shaped graph. In particular, the effect of the advection on the location of the spike depends on the choice of boundary conditions strongly.
[03837] Turing instability and bifurcation in reaction-diffusion systems on metric graphs
Format : Talk at Waseda University
Author(s) :
Junping Shi (College of William & Mary)
Abstract : We show that under a general framework, a constant equilibrium in a time-evolution system could lose its stability with
the addition of distinct dispersal rates for different species. Spontaneous symmetry breaking bifurcations occur so non-constant stationary patterns emerge. This is the classical Turing instability and bifurcations. We will show the application of this general scenario to the case of (i) an ODE system on a weighted directed graph, and (ii) a reaction-diffusion system on a metric graph.
[04417] Front propagation for Lotka-Volterra competition-diffusion system on unbounded star graphs
Format : Talk at Waseda University
Author(s) :
Ken-Ichi Nakamura (Meiji University)
Abstract : We consider the 2-component Lotka-Volterra competition-diffusion system in an infinite star graph with a single junction. Under strong competition conditions, we give sufficient conditions for the success/failure of the invasion of superior species beyond the junction. The method is based on a standard argument by constructing super-subsolutions with the help of a new result on the speed of traveling waves for the Lotka-Volterra competition-diffusion system on the whole line.
