Communications on Pure and Applied Analysis (CPAA)

Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system

Pages: 307 - 338, Volume 11, Issue 1, January 2012      doi:10.3934/cpaa.2012.11.307

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Yasumasa Nishiura - Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0813, Japan (email)
Takashi Teramoto - Faculty of Photonics Science, Chitose Institute of Science and Technology, Chitose 066-8655, Japan (email)
Xiaohui Yuan - Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences, Harbin 150081, China (email)

Abstract: Spatially localized patterns form a representative class of patterns in dissipative systems. We study how the dynamics of traveling spots in two-dimensional space change when heterogeneities are introduced in the media. The simplest but fundamental one is a line heterogeneity of jump type. When spots encounter the jump, they display various outputs including penetration, rebound, and trapping depending on the incident angle and its height. The system loses translational symmetry by the heterogeneity, but at the same time, it causes the emergence of various types of heterogeneity-induced-ordered-patterns (HIOPs) replacing the homogeneous constant state. We study these issues by using a three-component reaction-diffusion system with one activator and two inhibitors. The above outputs can be obtained through the interaction between the HIOPs and the traveling spots. The global bifurcation and eigenvalue behavior of HISPs are the key to understand the underlying mechanisms for the transitions among those dynamics. A reduction to a finite dimensional system is presented here to extract the model-independent nature of the dynamics. Selected numerical techniques for the bifurcation analysis are also provided.

Keywords:  Reaction-diffusion equations, bifurcation, heterogeneous media, traveling spot.
Mathematics Subject Classification:  Primary: 35K57; Secondary: 35B32.

Received: February 2010;      Revised: January 2011;      Available Online: September 2011.