A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term
This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a d-dimensional box (d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitude...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2013/934745 |
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| _version_ | 1832558895974842368 |
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| author | Shengmao Fu Ji Liu |
| author_facet | Shengmao Fu Ji Liu |
| author_sort | Shengmao Fu |
| collection | DOAJ |
| description | This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a d-dimensional box (d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model. |
| format | Article |
| id | doaj-art-d1d305d4996b4a44a622f89ae8680ff0 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-d1d305d4996b4a44a622f89ae8680ff02025-02-03T01:31:19ZengWileyAdvances in Mathematical Physics1687-91201687-91392013-01-01201310.1155/2013/934745934745A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source TermShengmao Fu0Ji Liu1College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinaCollege of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinaThis paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a d-dimensional box (d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.http://dx.doi.org/10.1155/2013/934745 |
| spellingShingle | Shengmao Fu Ji Liu A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term Advances in Mathematical Physics |
| title | A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term |
| title_full | A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term |
| title_fullStr | A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term |
| title_full_unstemmed | A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term |
| title_short | A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term |
| title_sort | mathematical characterization for patterns of a keller segel model with a cubic source term |
| url | http://dx.doi.org/10.1155/2013/934745 |
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