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Applicable Analysis
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Articles

Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

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Pages 1115-1134 | Received 15 Jan 2013, Accepted 17 Jun 2013, Published online: 09 Jul 2013

References

  • Gierer A, Meinhardt H. A theory of biological pattern formation. Kybernetik. 1972;12:30–39.
  • Gierer A. Generation of biological patterns and form: some physical, mathematical, and logical aspects. Prog. Biophys. Mol. Biol. 1981;37:1–47.
  • Meinhardt H, Gierer A. Applications of a theory of biological pattern formation based on lateral inhibition. J. Cell. Sci. 1974;15:321–346.
  • Edelstein-Keshet L. Mathematical models in biology. Classics in Applied Mathematics. Vol. 46. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); 2005. Reprint of the 1988 original.
  • Murray JD. Mathematical biology. Interdisciplinary Applied Mathematics. 3rd ed. Vol. 17. New York: Springer-Verlag; 2002.
  • Iron D, Ward MJ, Wei JC. The stability of spike solutions to the one-dimensional Gierer–Meinhardt model. Phys. D. 2001;150(1–2):25–62.
  • Kolokolnikov T, Sun W, Ward MJ, Wei JC. The stability of a stripe for the Gierer–Meinhardt model and the effect of saturation. SIAM J. Appl. Dyn. Syst. 2006;5(2):313–363.
  • Takagi I. Stability of bifurcating solutions of the Gierer–Meinhardt systems. Tôhoku Math. J. (2). 1979;31(2):221–246.
  • Takagi I. Point-condensation for a reaction–diffusion system. J. Differential Equations. 1986;61(2):208–249.
  • Wei JC, Winter M. On the two-dimensional Gierer–Meinhardt system with strong coupling. SIAM J. Math. Anal. 1999;30(6):1241–1263.
  • Wei JC, Winter M. Spikes for the two-dimensional Gierer–Meinhardt system: the weak coupling case. J. Nonlinear Sci. 2001;11(6):415–458.
  • Wei JC, Winter M. Spikes for the Gierer–Meinhardt system in two dimensions: the strong coupling case. J. Differential Equations. 2002;178(2):478–518.
  • Wei JC, Winter M. Existence, classification and stability analysis of multiple-peaked solutions for the Gierer–Meinhardt system in R1. Methods Appl. Anal. 2007;14(2):119–163.
  • Kurata K, Morimoto K. Construction and asymptotic behavior of multi-peak solutions to the Gierer–Meinhardt system with saturation. Commun. Pure Appl. Anal. 2008;7(6):1443–1482.
  • Morimoto K. Construction of multi-peak solutions to the Gierer–Meinhardt system with saturation and source term. Nonlinear Anal. 2009;71(7–8):2532–2557.
  • Morimoto K. Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2010;27(4):973–995.
  • Wei JC, Winter M. On the Gierer–Meinhardt system with saturation. Commun. Contemp. Math. 2004;6(2):259–277.
  • Liu J, Yi F, Wei J. Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer–Meinhardt model of morphogenesis. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2010;20(4):1007–1025.
  • Ni WM, Tang M. Turing patterns in the lengyel-epstein system for the cima reaction. Transactions of the American Mathematical Society. 2005;357(19):3953–3970.
  • Peng R, Wang M. On multiplicity and stability of positive solutions of a diffusive prey–predator model. J. Math. Anal. Appl. 2006;316(1):256–268.
  • Peng R, Wang M. Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment. Proc. Amer. Math. Soc. 2008;136(3):859–865.
  • Yi F, Liu J, Wei J. Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model. Nonlinear Anal. Real World Appl. 2010;11(5):3770–3781.
  • Zuo W, Wei J. Multiple bifurcations and spatiotemporal patterns for a coupled two-cell brusselator model. Dynam. Part. Differ. Eq. 2011;8:363–384.
  • Chen S, Shi J. Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays. Nonlinear Anal. Real World Appl. 2013;14(4):1871–1886.
  • Jin J, Shi J, Wei J, Yi F. Bifurcations of patterned solutions in diffusive Lengyel–Epstein dystem of CIMA chemical reaction. Rocky Moun. J. Math. in press.
  • Wang J, Shi J, Wei J. Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. J. Differential Equations. 2011;251(4–5):1276–1304.
  • Yi F, Wei J, Shi J. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differential Equations. 2009;246(5):1944–1977.
  • Turing AM. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences,. 1952;237(641):37–72.
  • Yi F, Wei J, Shi J. Diffusion-driven instability and bifurcation in the Lengyel–Epstein system. Nonlinear Anal. Real World Appl. 2008;9(3):1038–1051.
  • Yi F, Wei J, Shi J. Global asymptotical behavior of the Lengyel–Epstein reaction–diffusion system. Appl. Math. Lett. 2009;22(1):52–55.
  • Dancer EN, Du Y. Effects of certain degeneracies in the predator–prey model. SIAM J. Math. Anal. 2002;34(2):292–314.
  • Du Y. Effects of a degeneracy in the competition model. I. Classical and generalized steady-state solutions. J. Differential Equations. 2002;181(1):92–132.
  • Du Y. Effects of a degeneracy in the competition model. II. Perturbation and dynamical behaviour. J. Differential Equations. 2002;181(1):133–164.
  • Du Y. Spatial patterns for population models in a heterogeneous environment. Taiwanese J. Math. 2004;8(2):155–182.
  • Du Y, Hsu SB. A diffusive predator–prey model in heterogeneous environment. J. Differential Equations. 2004;203(2):331–364.
  • Du Y, Shi J. Some recent results on diffusive predator–prey models in spatially heterogeneous environment. Nonlinear dynamics and evolution equations. Fields Inst. Commun. Vol. 48. Providence, RI: Amer. Math. Soc.; 2006.
  • Du Y, Shi. J. Allee effect and bistability in a spatially heterogeneous predator–prey model. Trans. Amer. Math. Soc. 2007;359(9):4557–4593.
  • Lou Y. On the effects of migration and spatial heterogeneity on single and multiple species. J. Differential Equations. 2006;223(2):400–426.
  • Wiggins S. Introduction to applied nonlinear dynamical system and chaos. New York: Springer-Verlag; 1990.

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