Diffusion-driven instability of both the equilibrium solution and the periodic solutions for the diffusive Sporns-Seelig model

Nan Xiang; Aying Wan; Hongyan Lin; Nan Xiang; Aying Wan; Hongyan Lin

doi:10.3934/era.2022043

Electronic Research Archive

2022, Volume 30, Issue 3: 813-829. doi: 10.3934/era.2022043

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Diffusion-driven instability of both the equilibrium solution and the periodic solutions for the diffusive Sporns-Seelig model

1.
College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, Heilongjiang Province, 150001, China
2.
College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang Province, 150001, China
3.
School of Mathematics and Statistics, Hulunbuir University, Hailar, Inner Mongolia, 021008, China

Received: 14 December 2021 Revised: 13 January 2022 Accepted: 09 February 2022 Published: 01 March 2022

In this paper, a reaction-diffusion Sporn-Seelig model subject to homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. Of our particular interests, we are concerned with diffusion-driven instability of both the positive constant equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions. To strengthen our analytical results, we also include some numerical simulations. These results allow for the clearer understanding the mechanisms of the spatiotemporal pattern formations of this chemical reaction model.
- the diffusive Sporns-Seelig model,
- Hopf bifurcations,
- spatially homogeneous periodic solutions,
- diffusion-driven instability,
- spatiotemporal patterns
Citation: Nan Xiang, Aying Wan, Hongyan Lin. Diffusion-driven instability of both the equilibrium solution and the periodic solutions for the diffusive Sporns-Seelig model[J]. Electronic Research Archive, 2022, 30(3): 813-829. doi: 10.3934/era.2022043

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Abstract

In this paper, a reaction-diffusion Sporn-Seelig model subject to homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. Of our particular interests, we are concerned with diffusion-driven instability of both the positive constant equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions. To strengthen our analytical results, we also include some numerical simulations. These results allow for the clearer understanding the mechanisms of the spatiotemporal pattern formations of this chemical reaction model.

References

[1]	A. M. Turing, The chemical basis of morphogenesis, Proc. Royal Soc. B, 237 (1952), 37–72. https://doi.org/10.1007/978-3-642-70911-1_16 doi: 10.1007/978-3-642-70911-1_16
[2]	P. De Kepper, V. Castets, E. Dulos, J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic-acid reaction, Physica D, 49 (1991), 161–169. https://doi.org/10.1016/0167-2789(91)90204-M doi: 10.1016/0167-2789(91)90204-M
[3]	K. Maginu, Stability of spatially homogeneous periodic solutions of reaction-diffusion equations, J. Differ. Equ., 31 (1979), 130–138. https://doi.org/10.1016/0022-0396(79)90156-6 doi: 10.1016/0022-0396(79)90156-6
[4]	S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Nat. Resour. Model, 11 (1998), 131–141. https://doi.org/10.1111/j.1939-7445.1998.tb00304.x doi: 10.1111/j.1939-7445.1998.tb00304.x
[5]	M. Kuwamura, H. Izuhara, Diffusion-driven destabilization of spatially homogeneous limit cycles in reaction diffusion systems, Chaos, 27 (2017), 033112. https://doi.org/10.1063/1.4978924 doi: 10.1063/1.4978924
[6]	Y. Morita, Instability of spatially homogeneous periodic solutions to delay-diffusion equations, North-Holland Math. Stud., 98 (1984), 107–124. https://doi.org/10.1016/S0304-0208(08)71495-6 doi: 10.1016/S0304-0208(08)71495-6
[7]	F. Yi, Turing instability of the periodic solutions for reaction-diffusion systems with cross-diffusion and the patch model with cross-diffusion-like coupling, J. Differ. Equ., 281 (2021), 397–410. https://doi.org/10.1016/j.jde.2021.02.006 doi: 10.1016/j.jde.2021.02.006
[8]	O. Sporns, F. Seelig, Oscillations in theoretical models of induction, BioSystems, 19 (1986), 83–89. https://doi.org/10.1016/0303-2647(86)90019-5 doi: 10.1016/0303-2647(86)90019-5
[9]	B. D. Hassard, N. D. Kazarinoff, Y-H Wan, Theory and Application of Hopf Bifurcation, Cambridge: Cambridge University Press, 1981.
[10]	W. Jiang, X. Cao, C. Wang, Turing instability and pattern formation for reaction-diffusion systems on 2D bounded domain, Discrete Contin. Dyn. Syst.-Ser. B, 27 (2022), 1163. https://doi.org/10.3934/dcdsb.2021085 doi: 10.3934/dcdsb.2021085
[11]	W. Ni, M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953–3969. https://doi.org/10.1090/S0002-9947-05-04010-9 doi: 10.1090/S0002-9947-05-04010-9
[12]	F. Yi, S. Liu, N. Tuncer, Spatiotemporal patterns of a reaction-diffusion Seelig model, J. Dyns. Diff. Equ., 29 (2017), 219–247. https://doi.org/10.1007/s10884-015-9444-z doi: 10.1007/s10884-015-9444-z
[13]	F. Yi, J. Wei, J. Shi, Bifurcation and spatiotemporal patterns in a homogenous diffusive predator-prey system, J. Differ. Equ., 246 (2009), 1944–1977. https://doi.org/10.1016/j.jde.2008.10.024 doi: 10.1016/j.jde.2008.10.024
[14]	K. Nadjah, A. M. Salah, Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting, Electron. Res. Archive, 29 (2021), 1641–1660. https://doi:10.3934/era.2020084 doi: 10.3934/era.2020084
[15]	X. Wang, H. Gu, B. Lu, Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron, Electron. Res. Archive, 29 (2021), 2987–3015. https://doi:10.3934/era.2021023 doi: 10.3934/era.2021023
[16]	T. Hou, Y. Wang, X. Xie, Instability and bifurcation of a cooperative system with periodic coefficients, Electron. Res. Archive, 29 (2021), 3069–3079. https://doi:10.3934/era.2021026 doi: 10.3934/era.2021026
[17]	S. Chen, C. A. Santos, M. Yang, J. Zhou, Bifurcation analysis for a modified quasilinear equation with negative exponent, Adv. Nonlinear Anal., 11 (2022), 684–701. https://doi.org/10.1515/anona-2021-0215 doi: 10.1515/anona-2021-0215
[18]	A. Acharya, N. Fonseka, J. Quiroa, R. Shivaji, $\Sigma$-Shaped Bifurcation Curves, Adv. Nonlinear Anal., 10 (2021), 1255–1266. https://doi.org/10.1515/anona-2020-0180 doi: 10.1515/anona-2020-0180

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