# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062
##### References:
 [1] E. Conway, D. 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. [2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall: Englewood Cliffs, NJ, 1964. [3] L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17-29. [4] M. Mimura and J. Muarry, Spatial structures in a model substrate-inhibition reaction diffusion system, Z. Naturforsh, 33 (1978), 580-586. [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. [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. [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. [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. [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. [10] F. Yi, S. 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. [11] F. Yi, J. 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.

show all references

##### References:
 [1] E. Conway, D. 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. [2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall: Englewood Cliffs, NJ, 1964. [3] L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17-29. [4] M. Mimura and J. Muarry, Spatial structures in a model substrate-inhibition reaction diffusion system, Z. Naturforsh, 33 (1978), 580-586. [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. [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. [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. [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. [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. [10] F. Yi, S. 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. [11] F. Yi, J. 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.
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