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 |
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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
Cite this
Global-in-time behavior of the solution to a Gierer-Meinhardt system. / Karali, Georgia; Suzuki, Takashi; Yamada, Yoshio.
In: Discrete and Continuous Dynamical Systems- Series A, Vol. 33, No. 7, 07.2013, p. 2885-2900.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Global-in-time behavior of the solution to a Gierer-Meinhardt system
AU - Karali, Georgia
AU - Suzuki, Takashi
AU - Yamada, Yoshio
PY - 2013/7
Y1 - 2013/7
N2 - 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.
AB - 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.
KW - Asymptotic behavior of the solution
KW - Gierer-Meinhardt system
KW - Hamilton structure
KW - Reaction-diffusion equation
KW - Turing pattern
UR - http://www.scopus.com/inward/record.url?scp=84872108073&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84872108073&partnerID=8YFLogxK
U2 - 10.3934/dcds.2013.33.2885
DO - 10.3934/dcds.2013.33.2885
M3 - Article
AN - SCOPUS:84872108073
VL - 33
SP - 2885
EP - 2900
JO - Discrete and Continuous Dynamical Systems- Series A
JF - Discrete and Continuous Dynamical Systems- Series A
SN - 1078-0947
IS - 7
ER -