The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds
Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two- or three-dimensional spatial patterns, either stationary or subject to certain motion. Such struct...
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| Published in | Journal of mathematical biology Vol. 25; no. 6; pp. 597 - 610 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Germany
01.12.1987
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0303-6812 1432-1416 |
| DOI | 10.1007/BF00275496 |
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| Abstract | Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two- or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. |
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| AbstractList | Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two- or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two- or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus.Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two- or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. |
| Author | Grindrod, Peter Gomatam, Jagannathan |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/3437227$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1007/978-1-4684-0152-3 10.1109/JRPROC.1962.288235 10.1137/0146062 10.1007/978-3-662-22492-2 10.1016/0022-0396(77)90116-4 10.1007/BFb0089647 10.1007/978-3-642-93046-1 |
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| References | CR2 D. Henry (CR4) 1981 J. J. Tyson (CR11) 1976 J. D. Murray (CR7) 1977 J. Smoller (CR10) 1983 A. T. Winfree (CR12) 1980 P. S. Gradsteyn (CR3) 1980 CR14 J. Norbury (CR9) 1985 J. P. Keener (CR5) 1986; 46 D. G. Aronson (CR1) 1975 J. Nagumo (CR8) 1962; 50 J. P. Keener (CR6) 1986; 21D A. T. Winfree (CR13) 1984; 13D |
| References_xml | – volume-title: Shock waves and reaction-difiusion equations year: 1983 ident: CR10 doi: 10.1007/978-1-4684-0152-3 – volume-title: Free boundary problems: application and theory, vol. IV year: 1985 ident: CR9 – volume-title: Partial differential equations and related topics. Lect. Notes Math., vol. 446 year: 1975 ident: CR1 – volume: 50 start-page: 2061 year: 1962 ident: CR8 publication-title: Proc. IRE doi: 10.1109/JRPROC.1962.288235 – ident: CR14 – volume: 46 start-page: 1039 year: 1986 ident: CR5 publication-title: SIAM J. Appl. Math. doi: 10.1137/0146062 – volume-title: Tables of integrals, series and products year: 1980 ident: CR3 – volume: 21D start-page: 307 year: 1986 ident: CR6 publication-title: Physica – volume-title: The geometry of biological time year: 1980 ident: CR12 doi: 10.1007/978-3-662-22492-2 – volume: 13D start-page: 221 year: 1984 ident: CR13 publication-title: Physica – ident: CR2 doi: 10.1016/0022-0396(77)90116-4 – volume-title: Geometric theory of semilinear parabolic equations year: 1981 ident: CR4 doi: 10.1007/BFb0089647 – volume-title: Lecture notes on nonlinear differential equation models in biology year: 1977 ident: CR7 – volume-title: The Belousov-Zhabotinsky reaction year: 1976 ident: CR11 doi: 10.1007/978-3-642-93046-1 |
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| Title | The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds |
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