October  2017, 10(5): 1149-1164. doi: 10.3934/dcdss.2017062

Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model

1. 

School of Science, Heilongjiang University of Science and Technology, Harbin, Heilongjiang Province, 150022, China

2. 

School of Science, Harbin Institute of Technology, Harbin, Heilongjiang Province, 150001, China

3. 

School of Information and Electronics, Beijing Institute of Technology, Beijing 100089, China

* Corresponding author: Hongyan Zhang

Received  December 2016 Revised  January 2017 Published  June 2017

Fund Project: The first author is supported by NSF of China grant 11371108

In this paper, we are mainly considered with a kind of homogeneous diffusive Thomas model arising from biochemical reaction. Firstly, we use the invariant rectangle technique to prove the global existence and uniqueness of the positive solutions of the parabolic system, and then use the maximum principle to show the existence of attraction region which attracts all the solutions of the system regardless of the initial values. Secondly, we consider the long time behaviors of the solutions of the system; Thirdly, we derive precise parameter ranges where the system does not have non-constant steady states by using use some useful inequalities and a priori estimates; Finally, we prove the existence of Turing patterns by using the steady state bifurcation theory.

Citation: Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062
References:
[1]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001. Google Scholar

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall: Englewood Cliffs, NJ, 1964. Google Scholar

[3]

L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17-29. Google Scholar

[4]

M. Mimura and J. Muarry, Spatial structures in a model substrate-inhibition reaction diffusion system, Z. Naturforsh, 33 (1978), 580-586. Google Scholar

[5]

W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9. Google Scholar

[6]

Y. Nishiura, Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037. Google Scholar

[7]

F. Seelig, Chemical oscillations by substrate inhibition: A parametrically universal oscillator type in homogeneous catalysis by metal complex formation, Z. Naturforsh, 31 (1976), 731-738. doi: 10.1515/zna-1976-0710. Google Scholar

[8]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[9]

D. Thomas, Artifical enzyme membranes, transport, memory, and oscillatory phenomena, In: D. Thomas and J. Kernevez (eds) Analysis and Control of Immobilized Enzymes Systems, Berlin Heidelberg New York: Springer 1975,115-150.Google Scholar

[10]

F. YiS. Liu and N. Tuncer, Spatiotemporal patterns of a reaction-diffusion substrate-inhibition Seelig model, J. Dyna. Differential Equations, 29 (2017), 219-241. doi: 10.1007/s10884-015-9444-z. Google Scholar

[11]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogenous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

show all references

References:
[1]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001. Google Scholar

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall: Englewood Cliffs, NJ, 1964. Google Scholar

[3]

L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17-29. Google Scholar

[4]

M. Mimura and J. Muarry, Spatial structures in a model substrate-inhibition reaction diffusion system, Z. Naturforsh, 33 (1978), 580-586. Google Scholar

[5]

W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9. Google Scholar

[6]

Y. Nishiura, Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037. Google Scholar

[7]

F. Seelig, Chemical oscillations by substrate inhibition: A parametrically universal oscillator type in homogeneous catalysis by metal complex formation, Z. Naturforsh, 31 (1976), 731-738. doi: 10.1515/zna-1976-0710. Google Scholar

[8]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[9]

D. Thomas, Artifical enzyme membranes, transport, memory, and oscillatory phenomena, In: D. Thomas and J. Kernevez (eds) Analysis and Control of Immobilized Enzymes Systems, Berlin Heidelberg New York: Springer 1975,115-150.Google Scholar

[10]

F. YiS. Liu and N. Tuncer, Spatiotemporal patterns of a reaction-diffusion substrate-inhibition Seelig model, J. Dyna. Differential Equations, 29 (2017), 219-241. doi: 10.1007/s10884-015-9444-z. Google Scholar

[11]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogenous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[1]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[2]

Julijana Gjorgjieva, Jon Jacobsen. Turing patterns on growing spheres: the exponential case. Conference Publications, 2007, 2007 (Special) : 436-445. doi: 10.3934/proc.2007.2007.436

[3]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183

[4]

Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042

[5]

Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025

[6]

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032

[7]

Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347

[8]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119

[9]

Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265

[10]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551

[11]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[12]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[13]

Tyson Loudon, Stephen Pankavich. Mathematical analysis and dynamic active subspaces for a long term model of HIV. Mathematical Biosciences & Engineering, 2017, 14 (3) : 709-733. doi: 10.3934/mbe.2017040

[14]

Debora Amadori, Stefania Ferrari, Luca Formaggia. Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels. Networks & Heterogeneous Media, 2007, 2 (1) : 99-125. doi: 10.3934/nhm.2007.2.99

[15]

Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034

[16]

Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873

[17]

Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105

[18]

Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357

[19]

Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273

[20]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (17)
  • HTML views (23)
  • Cited by (0)

Other articles
by authors

[Back to Top]