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

Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese, Yong Tao Zhang

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)1413-1428
Number of pages16
JournalDiscrete and Continuous Dynamical Systems - Series S
Volume4
Issue number6
DOIs
StatePublished - Dec 2011

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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