Abstract : One way to understand the complex dynamics in dissipative systems is to decompose the object into subsystems with different spatiotemporal scales. The resulting subsystems could be unified by singular perturbation, fast-slow method, unfolding of singularities, bifurcation, and data-driven approaches. We collect 12 talks to present the state-of-the-art multi-scale pattern formation arising in biology, chemical reaction, fluid dynamics, and materials science in homogeneous and heterogeneous media. One of our goals is to find a promising direction in the newly emerging field of multiscale pattern formation problems.
Abstract : One way to understand the complex dynamics in dissipative systems is to decompose the object into subsystems with different spatiotemporal scales. The resulting subsystems could be unified by singular perturbation, fast-slow method, unfolding of singularities, bifurcation, averaging, and data-driven approaches. I will try to present the state-of-the-art multi-scale pattern formation arising in biology, chemical reaction, fluid dynamics, and materials science in homogeneous and heterogeneous media.
[01983] Localized spot dynamics: curvature and instability
Format : Talk at Waseda University
Author(s) :
Justin Tzou (Macquarie University)
Abstract : For localized spots solutions of singularly perturbed reaction-diffusion systems, we discuss two aspects of slow drift dynamics: the oscillatory instabilities, and effects of domain curvature. The problem with curvature is analyzed within the context of a model of vegetation patterns on a curved terrain, which incorporates advection effects due to flow of water downhill. The stability analysis on flat domains centers on understanding how domain geometry selects dominant modes of oscillation.
[01901] Spiky patterns in a three-component consumer chain model
Format : Talk at Waseda University
Author(s) :
Shuangquan Xie (Hunan University)
Abstract : We study a cooperative consumer chain model with one producer and two consumers, which is a three-component extension of the Schankenberg model. We show that consumers can survive and have a profile of a spike for sufficiently high consumption. We then study the stability of spike solutions. When the consumption constant is further increased to a certain threshold, the system undergoes a Hopf bifurcation and the spiky pattern begins to oscillate.
[01941] Emergence of locomotion by autonomous parameter tuning
Format : Talk at Waseda University
Author(s) :
Keiichi Ueda (University of Toyama)
Takumi Horita (University of Toyama)
Abstract : A peristaltic locomotion model with parameter tuning is presented. The parameter tuning system is described by the dynamical system with selection algorithm. The model autonomously generates stable elongation-contraction wave and finds appropriate anchor timing. We model the parameter tuning system as distributed system in order that the system achieves adaptability to various environmental changes.
[01917] Symmetry-Breaking for a Compartmental-Reaction Diffusion System
Format : Talk at Waseda University
Author(s) :
Michael Ward (University of British Columbia)
Abstract : We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where two bulk
diffusing species are coupled to nonlinear intracellular reactions that are confined within a
disjoint collection of small circular compartments within the domain. The bulk
species are coupled to the spatially segregated intracellular reactions
through Robin conditions across the cell boundaries. For this compartmental-reaction
diffusion system, symmetry-breaking bifurcations,
regulated by a membrane binding rate ratio, occur even when the
bulk species have equal diffusivities.
[01765] Bayesian Model Selection of PDEs for Pattern Formation
Format : Talk at Waseda University
Author(s) :
Natsuhiko Yoshinaga (Tohoku University)
Satoru Tokuda (Kyushu University)
Abstract : Partial differential equations (PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying their formation. PDE models often rely on the pre-request knowledge of physical laws and developing a model to reproduce a desired pattern remains difficult. We propose a method to estimate the best PDE from one snapshot of an objective pattern under the stationary state. We apply our method to complex patterns, such as quasi-crystals.
[02979] Spectral Stability of Far-from-Equilibrium Planar Periodic Patterns
Format : Online Talk on Zoom
Author(s) :
Björn de Rijk (Karlsruhe Institute of Technology)
Abstract : We consider the existence and spectral stability of far-from-equilibrium planar periodic patterns in reaction-diffusion-advection systems. The planar periodic traveling waves are constructed by bifurcating from one-dimensional wave trains undergoing a transverse short-wave destabilization. The selected wavenumber matrix and velocity vector at bifurcation are fully determined by the wavenumber, velocity and critical Bloch modes of the underlying wave train. Our spectral analysis of the planar periodic pattern yields an expansion of the critical spectral surface touching the origin due to translational invariance in both spatial directions. In particular, such an expansion allows for an explicit verification of the spectral stability conditions implying nonlinear stability of the planar periodic pattern against spatially localized perturbations. This is joint work with Miguel Rodrigues (Université de Rennes 1, France).
[01867] Fronts in the wake of a slow parameter ramp
Format : Talk at Waseda University
Author(s) :
Ryan Goh (Boston University)
Tasso Kaper (Boston University)
Arnd Scheel (University of Minnesota)
Theodore Vo (Monash University)
Abstract : We discuss front solutions in the presence of a parameter ramp which slowly varies in space, rigidly propagates in time, and moderates the (in)stability of a spatially-homogeneous equilibrium, nucleating a traveling wave in its wake. For moving ramps, the front location is governed by a slow passage between convective and absolute instability; a projectivized fold. For stationary ramps, fronts are governed by slow-passage through a pitchfork and a connecting solution of the Painléve-II equation.
[02507] Front propagation in a multi-variable morphogenetic model of branching
Format : Talk at Waseda University
Author(s) :
Edgar Knobloch (University of California at Berkeley)
Arik Yochelis (Ben-Gurion University of the Negev)
Abstract : We study the existence and stability of propagating fronts in the Meinhardt model of branching in 1D. We identify a sniper bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. We show that propagation failure is a consequence of a T-point in a spatial dynamics description and identify additional T-points responsible for a large multiplicity of different traveling front-peak states.
[02105] CONTROL OF ENGULFMENT FOR BINARY POLYMER PARTICLES
Format : Talk at Waseda University
Author(s) :
Takashi Teramoto (Asahikawa Medical University)
Abstract : Engulfment configurations with separated phases occur in the nanoparticles of a binary polymer mixture. Numerical investigations of the Cahn-Hilliard model with the boundary contact energy show the relationships between the free energies and two types of configurations within confined spheres in three-dimensions. These results are consistent with experimental observations. A Janus-type configuration forms a spherical-cap-shaped interface inside a particle. In the core-shell configuration, one of polymer phases completely engulfs another phase to form concentric interfaces with inner and outer phases. We consider the sharp interface limit of equilibrium configurations and derive the stability condition that each configuration becomes the only minimizer when the contact angle changes between the three phases.
[03494] Instability of Planar Interfaces in Reaction-Diffusion-Advection Equations
Format : Talk at Waseda University
Author(s) :
Paul Carter (University of California, Irvine)
Abstract : We consider planar interfaces between stable homogeneous rest states in singularly perturbed 2-component reaction diffusion advection equations, motivated by the appearance of fronts between bare soil and vegetation in dryland ecosystems, as well as multi-interface solutions, such as vegetation stripes. On sloped terrain, one can find stable traveling interfaces, while on flat ground, one finds that sideband instabilities along the interface can lead to labyrinthine Turing-like patterns. To explore this behavior, using geometric singular perturbation methods, we analyze instability criteria for planar interfaces in reaction diffusion advection systems, focussing on a specific Klausmeier-type model, and examine the effect of terrain slope on the stability of the interfaces.
[04510] Patterns on patterns
Format : Online Talk on Zoom
Author(s) :
Martina Chirilus-Bruckner (Leiden University)
Jolien Kamphuis (Leiden University)
Abstract : The formation of patterns on top of spatially varying background states in the context of reaction-diffusion systems with spatially varying coefficients (such as the extended Klausmeier model) is motivated by the study of vegetation patterns on changing topographies. We present two regimes: (i) At onset when the background state loses stability and small amplitude modulations occur and (ii) in the long wavelength limit where the patterned state is composed of highly localized individual pulses.
