Moving and jumping spot in a two-dimensional reaction-diffusion model

Shuangquan Xie, Theodore Kolokolnikov

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

We consider a single spot solution for the Schnakenberg model in a two-dimensional unit disk in the singularly perturbed limit of a small diffusivity ratio. For large values of the reaction-time constant, this spot can undergo two different types of instabilities, both due to a Hopf bifurcation. The first type induces oscillatory instability in the height of the spot. The second type induces a periodic motion of the spot center. We use formal asymptotics to investigate when these instabilities are triggered, and which one dominates. In the parameter regime where spot motion occurs, we construct a periodic solution consisting of a rotating spot, and compute its radius of rotation and angular velocity. Detailed numerical simulations are performed to validate the asymptotic theory, including rotating spots. More complex, non-circular spot trajectories are also explored numerically.

Original languageEnglish
Pages (from-to)1536-1563
Number of pages28
JournalNonlinearity
Volume30
Issue number4
DOIs
Publication statusPublished - 2017 Mar 7
Externally publishedYes

Keywords

  • 35B40
  • 92B99
  • pattern formation
  • reaction-diffusion systems
  • spike patterns Mathematics Subject Classification numbers: 35K57

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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