Abstract
Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.
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Acknowledgments
We acknowledge fruitful discussions with Dirk De Vos and Przemyslaw Klosiewicz. DD acknowledges financial support from the Department of Mathematics and Computer Science of the University of Antwerp.
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This work is part of the Geconcerteerde Onderzoeksactie (G.O.A.) research grant “A System Biology Approach of Leaf Morphogenesis” granted by the research council of the University of Antwerp.
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Draelants, D., Broeckhove, J., Beemster, G.T.S. et al. Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model. J. Math. Biol. 67, 1279–1305 (2013). https://doi.org/10.1007/s00285-012-0588-8
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DOI: https://doi.org/10.1007/s00285-012-0588-8
Keywords
- Bifurcation analysis
- Pattern formation
- Parameter dependence
- Auxin transport model
- Stability
- Periodic solution pattern