Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type
The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we s...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2020/5293748 |
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| author | Lianchao Gu Peiliang Gong Hongqing Wang |
| author_facet | Lianchao Gu Peiliang Gong Hongqing Wang |
| author_sort | Lianchao Gu |
| collection | DOAJ |
| description | The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns. |
| format | Article |
| id | doaj-art-efc1a01645e84867b01ede7d4422b0b6 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-efc1a01645e84867b01ede7d4422b0b62025-02-03T06:43:41ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2020-01-01202010.1155/2020/52937485293748Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion TypeLianchao Gu0Peiliang Gong1Hongqing Wang2College of Science, Beijing Forestry University, No. 35, Tsinghua East Road, Beijing 100083, ChinaSchool of Computer and Information Technology, Beijing Jiaotong University, No. 3, Shangyuancun, Beijing 100044, ChinaCollege of Science, Beijing Forestry University, No. 35, Tsinghua East Road, Beijing 100083, ChinaThe reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns.http://dx.doi.org/10.1155/2020/5293748 |
| spellingShingle | Lianchao Gu Peiliang Gong Hongqing Wang Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type Discrete Dynamics in Nature and Society |
| title | Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type |
| title_full | Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type |
| title_fullStr | Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type |
| title_full_unstemmed | Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type |
| title_short | Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type |
| title_sort | hopf bifurcation and turing instability analysis for the gierer meinhardt model of the depletion type |
| url | http://dx.doi.org/10.1155/2020/5293748 |
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