Biological pattern formation on two-dimensional spatial domains: A nonlinear bifurcation analysis

被引:35
作者
Cruywagen, GC
Maini, PK
Murray, JD
机构
[1] Sea Fisheries Res Inst, ZA-8012 Cape Town, South Africa
[2] Math Inst, Ctr Math Biol, Oxford OX1 3LB, England
[3] Univ Washington, Dept Appl Math FS20, Seattle, WA 98195 USA
关键词
pattern formation; nonlinear bifurcation analyses; tissue interaction;
D O I
10.1137/S0036139996297900
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
A tissue interaction model for skin organ pattern formation is presented. Possible spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional perturbation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation.
引用
收藏
页码:1485 / 1509
页数:25
相关论文
共 19 条
[1]   ON A TISSUE INTERACTION-MODEL FOR SKIN PATTERN-FORMATION [J].
CRUYWAGEN, GC ;
MURRAY, JD .
JOURNAL OF NONLINEAR SCIENCE, 1992, 2 (02) :217-240
[2]  
CRUYWAGEN GC, 1992, IMA J MATH APPL MED, V9, P227
[3]   TRAVELING WAVES IN A TISSUE INTERACTION-MODEL FOR SKIN PATTERN-FORMATION [J].
CRUYWAGEN, GC ;
MAINI, PK ;
MURRAY, JD .
JOURNAL OF MATHEMATICAL BIOLOGY, 1994, 33 (02) :193-210
[4]   BIFURCATING SPATIAL PATTERNS ARISING FROM TRAVELING WAVES IN A TISSUE INTERACTION-MODEL [J].
CRUYWAGEN, GC ;
MAINI, PK ;
MURRAY, JD .
APPLIED MATHEMATICS LETTERS, 1994, 7 (03) :63-66
[5]   CELL-ADHESION MOLECULES IN THE REGULATION OF ANIMAL FORM AND TISSUE PATTERN [J].
EDELMAN, GM .
ANNUAL REVIEW OF CELL BIOLOGY, 1986, 2 :81-116
[6]   NEURON-GLIA CELL-ADHESION MOLECULE INTERACTS WITH NEURONS AND ASTROGLIA VIA DIFFERENT BINDING MECHANISMS [J].
GRUMET, M ;
EDELMAN, GM .
JOURNAL OF CELL BIOLOGY, 1988, 106 (02) :487-503
[7]   A NONLINEAR-ANALYSIS OF A MECHANICAL MODEL FOR BIOLOGICAL PATTERN-FORMATION [J].
MAINI, PK ;
MURRAY, JD .
SIAM JOURNAL ON APPLIED MATHEMATICS, 1988, 48 (05) :1064-1072
[8]   NONLINEAR DYNAMIC STABILITY - A FORMAL THEORY [J].
MATKOWSKY, BJ .
SIAM JOURNAL ON APPLIED MATHEMATICS, 1970, 18 (04) :872-+
[9]   CELL TRACTION MODELS FOR GENERATING PATTERN AND FORM IN MORPHOGENESIS [J].
MURRAY, JD ;
OSTER, GF .
JOURNAL OF MATHEMATICAL BIOLOGY, 1984, 19 (03) :265-279
[10]   THRESHOLD BIFURCATION IN TISSUE INTERACTION MODELS FOR SPATIAL PATTERN GENERATION [J].
MURRAY, JD ;
CRUYWAGEN, GC .
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1994, 347 (1685) :661-676