Analysis of the Hopf bifurcation in a Diffusive Gierer-Meinhardt Model
DOI:
https://doi.org/10.12775/TMNA.2022.050Słowa kluczowe
Gierer-Meinhardt model, Hopf bifurcation, reaction-diffusion, center manifold, normal formAbstrakt
In this work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.Bibliografia
S.S. Chen, J.P. Shi and J.J. Wei, Bifurcation analysis of the Gierer–Meinhardt system with saturation in the activator production, Appl. Anal. 93 (2014), 1115–1134.
M. Chen, R. Wu and L. Chen, Pattern dynamics in a diffusive Gierer–Meinhardt model, Int. J. Bifurcation and Chaos 30 (2020), 2030035.
N.T. Fadai, M.J. Ward and J.C. Wei, Delayed reaction kinetic and the stability of spikes in the Gierer–Meinhardt model, SIAM J. Appl. Math. 77 (2017), 664–696.
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30–39.
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer, London, Dordrecht, Heidelberg, New York, 2011.
Y. Li, J. Wang and X. Hou, Stripe and spot patterns for general Gierer–Meinhardt model with common sources, Int. J. Bifurcation and Chaos 27 (2017a), 1750018.
Y. Li, J. Wang and X.Hou, Stripe and spot patterns for the Gierer–Meinhardt model with saturated activator production, J. Math. Anal. Appl. 449 (2017b), 1863–1879.
J.X. Liu, F.Q. Yi and J.J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer–Meinhardt model of morphogenesis, Int. J. Bifurcation and Chaos 20 (2010), 1007–1025.
Y.L. Song, R. Yang and G.Q. Sun, Pattern dynamics in a Gierer-Meinhardt model with a saturating term, Appl. Math. Model. 46 (2017), 476–491.
A. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. B-Biol. Sci. 237 (1952), 37–72.
J.L. Wang, Y. Li, S.H. Zhong and X.J. Hou, Analysis of bifurcation, chaos and pattern formation in a discrete-time and space Gierer–Meinhardt system, Chaos Solittons Fractals 118 (2019), 1–17.
J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Springer–Verlag, London, 2014.
R. Wu, Y. Zhou, Y. Shao and L.Chen, Bifurcation and Turing patterns of reactiondiffusion activator-inhibitor model, Physica A 482 (2017), 597–610.
X.P. Yan, Y.J. Ding and C.H. Zhang, Dynamics analysis in a Gierer–Meinhardt reaction-diffusion model with homogeneous Neumann boundary condition, Int. J. Bifurcation and Chaos 29 (2019), 1930025.
N. Yana, Explicitly solvable eigenvalue problem and bifurcation delay in sub-diffusive Gierer–Meinhardt model, Eur. J. Appl. Math. 27 (2016), 699–725.
R. Yang and Y.L. Song, Spatial resonance and Turing–Hopf bifurcations in the Gierer–Meinhardt model, Nonlinear Anal. 31 (2016), 356–387.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2023 Rasoul Asheghi
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
