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A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains
Jorge A Castillo1, Faustino Sánchez-Garduño2, Pablo Padilla3
1Universidad Autónoma de Guerrero, C. Carlos E. Adame #54, Col. Garita, C.P. 39650, Acapulco, Gro., Mexico.
Bulletin of Mathematical Biology
|July 15, 2016
Summary
This study explores pattern formation in biological systems using the FitzHugh-Nagumo model. Domain growth significantly influences pattern selection, revealing key insights into morphogenetic mechanisms.
Area of Science:
- Mathematical Biology
- Developmental Biology
- Pattern Formation
Background:
- Morphogenesis involves the emergence of complex patterns during development.
- Turing-Hopf mechanisms are crucial for understanding pattern generation.
- FitzHugh-Nagumo models describe reaction-diffusion systems relevant to biological patterns.
Purpose of the Study:
- To investigate pattern formation on static and growing domains.
- To analyze the bifurcation structure of these patterns.
- To understand the influence of domain growth on pattern selection.
Main Methods:
- Analysis of reaction-diffusion equations on a flat square domain.
- Consideration of both fixed and growing domain scenarios.
- Numerical solutions of initial and boundary value problems.
Main Results:
- Sufficient conditions for specific space-time pattern formation were identified.
- A comparison of patterns on fixed versus growing domains was established.
- The role of domain growth in selecting specific patterns was highlighted.
Conclusions:
- Domain growth is a critical factor in pattern selection during morphogenesis.
- The study provides a foundation for further research into dynamic pattern formation.
- Open problems in pattern formation and morphogenetic mechanisms were identified.
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