# American Institute of Mathematical Sciences

December  2011, 4(6): 1413-1428. doi: 10.3934/dcdss.2011.4.1413

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

 1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States, United States, United States 2 Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556 3 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618

Received  May 2009 Revised  September 2009 Published  December 2010

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.
Citation: Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese, Yong-Tao Zhang. Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1413-1428. doi: 10.3934/dcdss.2011.4.1413
##### References:

show all references

##### References:
 [1] Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1 [2] Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053 [3] Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163 [4] Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181 [5] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [6] Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515 [7] Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304 [8] A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65 [9] Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49 [10] Piermarco Cannarsa, Alexander Khapalov. Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1293-1311. doi: 10.3934/dcdsb.2010.14.1293 [11] Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373 [12] Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105 [13] Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 [14] C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872 [15] Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191 [16] Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21 [17] Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815 [18] Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415 [19] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [20] Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

2018 Impact Factor: 0.545