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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning

Pages: 1413 - 1428, Volume 4, Issue 6, December 2011      doi:10.3934/dcdss.2011.4.1413

 
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Wenrui Hao - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Jonathan D. Hauenstein - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Bei Hu - Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, United States (email)
Yuan Liu - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Andrew J. Sommese - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Yong-Tao Zhang - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, United States (email)

Abstract: The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579--589] has multiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size.

Keywords:  Steady states, reaction-diffusion, polynomial systems, dorsal-ventral patterning.
Mathematics Subject Classification:  Primary: 65H04, 65H20, 65M06, 65N06, 92B05.

Received: May 2009;      Revised: September 2009;      Available Online: December 2010.

 References