Abstract
Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, τ = s+1/p-1 . Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
Original language | English |
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Pages (from-to) | 2885-2900 |
Number of pages | 16 |
Journal | Discrete and Continuous Dynamical Systems- Series A |
Volume | 33 |
Issue number | 7 |
DOIs | |
Publication status | Published - 2013 Jul |
Keywords
- Asymptotic behavior of the solution
- Gierer-Meinhardt system
- Hamilton structure
- Reaction-diffusion equation
- Turing pattern
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis