一类具有时滞的 Gierer-Meinhardt 活化抑制模型的分支分析

Abstract

Abstract:

In this paper, we consider a class of Gierer-Meinhardt activation inhibition model with time delay diffusion under homogeneous Neumann boundary conditions. Firstly, the local asymptotic stability of the positive equilibrium of the model is obtained by using spectral theory; second, choosing the time delay as the bifurcation parameter, the existence of the Hopf bifurcation of the model is analyzed; next, the formulae to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by applying the center manifold theorem and normal form theory for partial differential equation are derived; finally, numerical simulations are also carried out to simulate the Hopf bifurcation that the model go through near the critical point.

Key words: Time delay, Gierer-Meinhardt model, Hopf bifurcation, Stability, Simulation

CLC Number:

O175.12

Ma Yani, Yuan Hailong. Bifurcation Analysis of a Class of Gierer-Meinhardt Activation Inhibition Model with Time Delay[J].Acta mathematica scientia,Series A, 2023, 43(6): 1774-1788.

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References 32

[1]	欧阳颀. 反应扩散系统中的斑图动力学. 上海: 上海科技教育出版社, 2000
	Ou Y X. Turing Dynamics in Reaction Diffusion Systems. Shanghai: Shanghai Science and Technology Education Press, 2000
[2]	Gierer A, Meinhardt H. A theory of biological pattern formation. Kybernetik, 1972, 12: 30-39 pmid: 4663624
[3]	Gierer A, Meinhardt H. Application of a theory of biological pattern formation based on lateral inhibition. J Cell Sci, 1974, 15(2): 321-346 doi: 10.1242/jcs.15.2.321 pmid: 4859215
[4]	Gierer A, Meinhardt H. Generation and regeneration of sequence of structures during morphogenesis. J Theor Biol, 1980, 3: 429-450
[5]	Wei J C, Winter M. Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case. J Differ Equ, 2002, 178(2): 478-518 doi: 10.1006/jdeq.2001.4019
[6]	Ni W M, Suzuki K, Takagi I. The dynamics of a kinetic activator-inhibitor system. J Differ Equ, 2006, 229(2): 426-465 doi: 10.1016/j.jde.2006.03.011
[7]	Takashi M, Maini P K. Speed of pattern appearance in reaction-diffusion models: Implications in the pattern formation of limb bud mesenchyme cells. Bull Math Biol, 2004, 66(4): 627-649 doi: 10.1016/j.bulm.2003.09.009
[8]	Gonpot P, Collet J S A J, Sookia N U H. Gierer-Meinhardt model: Bifurcation analysis and pattern formation. Trends Appl Sci Res, 2008, 3: 115-128 doi: 10.3923/tasr.2008.115.128
[9]	Yang R, Song Y L. Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model. Nonlinear Anal: Real World Appl, 2016, 31: 356-387 doi: 10.1016/j.nonrwa.2016.02.006
[10]	杨文彬, 吴建华. 空间齐次和非齐次下活化-抑制模型动力学分析. 数学物理学报, 2017, 37A(22): 390-400
	Yang W B, Wu J H. Dynamics analysis of activation-inhibition models under spatial homogeneous and heterogeneou. Acta Math Sci, 2017, 37A(22): 390-400
[11]	Liu J X, Yi F Q, Wei J J. Multiple bifurcation analysis and spatiotemporal patterns in a $1$-D Gierer-Meinhardt model of morphogenesis. Int J Bifurcat Chaos, 2010, 20(4): 1007-1025 doi: 10.1142/S0218127410026289
[12]	Wang J L, Hou X J, Jing Z J. Stripe and spot patterns in a Gierer Meinhardt activator-inhibitor model with different sources. Int J Bifurcat Chaos, 2015, 25(8): 108-155
[13]	Wang J F, Wei J J, Shi J P. Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems. J Diff Equ, 2016, 260(4): 3495-3523 doi: 10.1016/j.jde.2015.10.036
[14]	叶其孝, 李正元, 王明新, 吴雅萍. 反应扩散方程引论. 北京: 科学出版社, 2011
	Ye Q X, Li Z Y, Wang M X, Wu Y P. Introduction to Reaction-Diffusion Equations. Beijing: Science Press, 2011
[15]	Yi F Q, Wei J J, Shi J P. Diffusion-Driven instability and bifurcation in the Lengyel-Epstein system. Nonlinear Anal: Real World Appl, 2008, 9(3): 1038-1051 doi: 10.1016/j.nonrwa.2007.02.005
[16]	Galkhar S, Negi K, Sahani S K. Effects of seasonal growth on ratio dependent delayed prey predator system. Commun Nonlinear Sci Numer Simul, 2009, 14(3): 850-862 doi: 10.1016/j.cnsns.2007.10.013
[17]	Guan X N, Wang W M, Cai Y L. Spatiotemporal dynamics of a leslie-gower predator-prey model incorporating a prey refuge. Nonlinear Anal: Real World Appl, 2011, 12(4): 2385-2395 doi: 10.1016/j.nonrwa.2011.02.011
[18]	Lee S S, Gaffney E A, Monk N A M. The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems. Bull Math Biol, 2010, 72(8): 2139-2160 doi: 10.1007/s11538-010-9532-5
[19]	Dutta S, Ray D S. Effects of delay in a reaction-diffusion system under the influence of an electric field. Phys Rev E, 2008, 77(3): 036202 doi: 10.1103/PhysRevE.77.036202
[20]	Ghosh P. Control of the Hopf-turing transition by time-delayed global feedback in a reaction-diffusion System. Phys Rev E, 2011, 84: 016222 doi: 10.1103/PhysRevE.84.016222
[21]	Ghosh P, Sen S, Ray D S. Reaction-Cattaneo systems with fluctuating relaxation time. Phys Rev E, 2010, 81(2): 026205 doi: 10.1103/PhysRevE.81.026205
[22]	Kyrychko Y, Blyuss K B, Hogan S J, et al. Control of spatiotemporal patterns in the Gray-Scott model. Chaos, 2009, 19(4): 043126 doi: 10.1063/1.3270048
[23]	Sen S, Ghosh P, Syed S, et al. Time-Delay-Induced instabilities in reaction diffusion systems. Phys Rev E, 2008, 80(4): 046212 doi: 10.1103/PhysRevE.80.046212
[24]	Suleiman A L. Stability analysis of the Gierer-Meinhardt system with activator degradation. Dutse J Pure Appl Sci, 2016, 2(1): 48-54
[25]	Li C C, Guo S J. Stability and bifurcation of a delayed reaction-diffusion model with robin boundary condition in heterogeneous environment. Int J Bifurcat Chaos, 2023, 33(2): 2350018 doi: 10.1142/S0218127423500189
[26]	Wen T T, Wang X L, Zhang G H. Hopf bifurcation in a two-species reaction-diffusion-advection competitive model with nonlocal delay. Commun Pur Appl Anal, 2023, 22(5): 1517-1544
[27]	Ma L, Wei D. Hopf bifurcation of a delayed reaction-diffusion model with advection term. Nonlinear Anal: An Inter Mult J, 2021, 212(2): 112455
[28]	王雅迪, 袁海龙. 一类具有时滞的营养-微生物扩散模型的 Hopf 分支研究. 西南师范大学学报(自然科学版), 2023, 48(5): 1-13
	Wang Y D, Yuan H L. Hopf bifurcation of a nutritional microbial diffusion model with time delay. Journal of Southwest Normal University (Natural Science Edition), 2023, 48(5): 1-13
[29]	Dong Y Y, Li S B, Zhang S L. Hopf bifurcation in a reaction-diffusion model with Degn-Harrison reaction scheme. Nonlinear Anal: Real World Appl, 2017, 33: 284-297 doi: 10.1016/j.nonrwa.2016.07.002
[30]	Lin X D, So J W H, Wu J H. Center manifolds for partial differential equations with delay. P Roy Soc Edinb A: Math, 1992, 122(3/4): 237-254
[31]	Wu J H. Theory and Applications of Partial Functional Differential Equations. New York: Springer-Verlag, 1996
[32]	万阿英, 衣风岐, 郑立飞. 一类扩散的 Gierer-Meinhardt 的模型的振动模式和 Hopf 分支分析. 数学物理学报, 2015, 35A(2): 381-394
	Wan A Y, Yi F Q, Zeng L F. Oscillating patterns and Hopf bifurcation analysis of a class of diffusion Gierer-Meinhardt models. Acta Math Sci, 2015, 35A(2): 381-394

Bifurcation Analysis of a Class of Gierer-Meinhardt Activation Inhibition Model with Time Delay

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