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 |
|---|---|
| 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
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Cite this
Karali, G., Suzuki, T., & Yamada, Y. (2013). Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems- Series A, 33(7), 2885-2900. https://doi.org/10.3934/dcds.2013.33.2885