Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources
Abstract We study the pattern generating mechanism of a generalized Gierer–Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic sol...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-07-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-018-1697-5 |
| _version_ | 1818939907707502592 |
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| author | Jinliang Wang You Li Xiaojie Hou |
| author_facet | Jinliang Wang You Li Xiaojie Hou |
| author_sort | Jinliang Wang |
| collection | DOAJ |
| description | Abstract We study the pattern generating mechanism of a generalized Gierer–Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer–Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature. |
| first_indexed | 2024-12-20T06:31:13Z |
| format | Article |
| id | doaj.art-0801677e5b4e46d69976737751aa9613 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-20T06:31:13Z |
| publishDate | 2018-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-0801677e5b4e46d69976737751aa96132022-12-21T19:50:07ZengSpringerOpenAdvances in Difference Equations1687-18472018-07-012018112310.1186/s13662-018-1697-5Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sourcesJinliang Wang0You Li1Xiaojie Hou2LMIB & School of Mathematics and System Science, Beihang UniversityLMIB & School of Mathematics and System Science, Beihang UniversityDepartment of Mathematics and Statistics, University of North Carolina WilmingtonAbstract We study the pattern generating mechanism of a generalized Gierer–Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer–Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature.http://link.springer.com/article/10.1186/s13662-018-1697-5SupercriticalHopf bifurcationTuring instabilitySpatial pattern |
| spellingShingle | Jinliang Wang You Li Xiaojie Hou Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources Advances in Difference Equations Supercritical Hopf bifurcation Turing instability Spatial pattern |
| title | Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources |
| title_full | Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources |
| title_fullStr | Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources |
| title_full_unstemmed | Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources |
| title_short | Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources |
| title_sort | supercritical hopf bifurcation and turing patterns for an activator and inhibitor model with different sources |
| topic | Supercritical Hopf bifurcation Turing instability Spatial pattern |
| url | http://link.springer.com/article/10.1186/s13662-018-1697-5 |
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