Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

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2016, 13(4): 857-885. doi: 10.3934/mbe.2016021

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

1.	School of Mathematics and Statistics, Southwest University, Chongqing, 400715

Received August 2015 Revised February 2016 Published May 2016

Abstract
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This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

Keywords: steady-state bifurcation, Plant-wrack model, stability, hopf bifurcation, nonconstant positive solutions., turing instability.

Mathematics Subject Classification: Primary: 35J55, 35K57; Secondary: 92C15, 92C4.

Citation: Jun Zhou. Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences & Engineering, 2016, 13 (4) : 857-885. doi: 10.3934/mbe.2016021

References:

[1]	Q. Y. Bie, Pattern formation in a general two-cell Brusselator model, J. Math. Anal. Appl., 376 (2011), 551-564. doi: 10.1016/j.jmaa.2010.10.066.
[2]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.
[3]	A. J. Catllá, A. McNamara and C. M. Topaz, Instabilities and patterns in coupled reaction-diffusion layers, Phy. Rev. E Stat. Nonlinear & Soft Matter Physics, 85 (2012), 489-500.
[4]	W. Chen, Localized Patterns in the Gray-scott Model: An Asymptotic and Numerical Study of Dynamics and Stability, PhD thesis, University of British Columbia,, 2009., ().
[5]	F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507-516. doi: 10.1017/S0308210500000275.
[6]	A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563. doi: 10.1088/0951-7715/10/2/013.
[7]	L. L. Du and M. X. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485. doi: 10.1016/j.jmaa.2010.02.002.
[8]	J. E. Furter and J. C. Eilbeck, Analysis of bifurcations in reaction-diffusion systems with no-flux boundary conditions: The Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413-438. doi: 10.1017/S0308210500028109.
[9]	M. Ghergu, Steady-state solutions for a general Brusselator system, In Modern Aspects of the Theory of Partial Differential Equations, volume 216 of Oper. Theory Adv. Appl., pages 153-166. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0069-3_9.
[10]	M. Ghergu and V. Rădulescu, Turing patterns in general reaction-diffusion systems of Brusselator type, Commun. Contemp. Math., 12 (2010), 661-679. doi: 10.1142/S0219199710003968.
[11]	M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345. doi: 10.1088/0951-7715/21/10/007.
[12]	M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Verlag, 2012. doi: 10.1007/978-3-642-22664-9.
[13]	A. A. Golovin, B. J. Matkowsky and V. A. Volpert, Turing pattern formation in the Brusselator model with superdiffusion, SIAM J. Appl. Math., 69 (2008), 251-272. doi: 10.1137/070703454.
[14]	J. K. Hale, L. A. Peletier and W. C. Troy, Stability and instability in the Gray-Scott model: The case of equal diffusivities, Appl. Math. Lett., 12 (1999), 59-65. doi: 10.1016/S0893-9659(99)00035-X.
[15]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41. CUP Archive, 1981.
[16]	D. Iron, J. C. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390. doi: 10.1007/s00285-003-0258-y.
[17]	J. Jang, W. M. Ni and M. X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320. doi: 10.1007/s10884-004-2782-x.
[18]	J. Y. Jin, J. P. Shi, J. J. Wei and F. Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of cima chemical reaction, Roc. Mount.J. Math., 43 (2013), 1637-1674. doi: 10.1216/RMJ-2013-43-5-1637.
[19]	T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Phys. D, 214 (2006), 63-77. doi: 10.1016/j.physd.2005.12.005.
[20]	J. van de Koppel and C. M. Crain, Scale-dependent inhibition drives regular tussock spacing in a freshwater marsh, Amer. Natu., 168 (2006), 36-47. doi: 10.1086/508671.
[21]	J. López-Gómez, J. C. Eilbeck, M. Molina and K. N. Duncan, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal., 12 (1992), 405-428. doi: 10.1093/imanum/12.3.405.
[22]	Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.
[23]	W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable gray-scott model, Math. Compu. in Simulation, 40 (1996), 371-396. doi: 10.1016/0378-4754(95)00044-5.
[24]	J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal. Real World Appl., 5 (2004), 105-121. doi: 10.1016/S1468-1218(03)00020-8.
[25]	W. M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations Stationary Partial Differential Equations, 1 (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.
[26]	W. M. Ni and M. X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9.
[27]	R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166. doi: 10.1016/j.jmaa.2004.12.026.
[28]	R. Peng, M. X. Wang and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling, 44 (2006), 945-951. doi: 10.1016/j.mcm.2006.03.001.
[29]	R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations, 241 (2007), 386-398. doi: 10.1016/j.jde.2007.06.005.
[30]	R. Peng and M. X. Wang, Some nonexistence results for nonconstant stationary solutions to the Gray-Scott model in a bounded domain, Appl. Math. Lett., 22 (2009), 569-573. doi: 10.1016/j.aml.2008.06.032.
[31]	R. Peng and F. Q. Sun, Turing pattern of the Oregonator model, Nonlinear Anal., 72 (2010), 2337-2345. doi: 10.1016/j.na.2009.10.034.
[32]	Y. W. Qi, The development of travelling waves in cubic auto-catalysis with different rates of diffusion, Phys. D, 226 (2007), 129-135. doi: 10.1016/j.physd.2006.11.010.
[33]	E. E. Sel'Kov, Self-oscillations in glycolysis, European Journal of Biochemistry, 4 (1968), 79-86.
[34]	J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.
[35]	J. P. Shi and X. F. 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.
[36]	I. Takagi C. S. Lin and W. M. Ni, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[37]	J. J. Tyson, K. Chen and B. Novak, Network dynamics and cell physiology, Nature Rev. Molecular Cell Bio., 2 (2001), 908-916.
[38]	A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72.
[39]	M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620. doi: 10.1016/S0022-0396(02)00100-6.
[40]	M. X. Wang, R. Peng and J. P. Shi, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: 10.1088/0951-7715/21/7/006.
[41]	M. X. Wang and P. Y. H. Pang, Global asymptotic stability of positive steady states of a diffusive ratio-dependent pre-predator model, Applied Mathematics Letters, 21 (2008), 1215-1220. doi: 10.1016/j.aml.2007.10.026.
[42]	J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004.
[43]	M. J. Ward and J. C. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223.
[44]	J. M. Wei, Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability, Phys. D, 148 (2001), 20-48. doi: 10.1016/S0167-2789(00)00183-4.
[45]	J. C. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.
[46]	J. C. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol., 64 (2012), 211-254. doi: 10.1007/s00285-011-0412-x.
[47]	S. Wiggins and M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2. Springer, 1990. doi: 10.1007/978-1-4757-4067-7.
[48]	L. Xu, G. Zhang and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model, WSEAS Transac. Math, 10 (2011), 201-209.
[49]	C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977. doi: 10.1016/j.nonrwa.2012.01.001.
[50]	F. Q. Yi, J. J. Wei and J. J. Shi, Diffusion-driven instability and bifurcation in the lengyel-epstein system, Nonlinear Anal.: Real World Applications, 9 (2008), 1038-1051. doi: 10.1016/j.nonrwa.2007.02.005.
[51]	F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.
[52]	F. Q. Yi, J. J. Wei and J. P. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55. doi: 10.1016/j.aml.2008.02.003.
[53]	Y. C. You, Global dynamics of the Brusselator equations, Dyn. Partial Differ. Equ., 4 (2007), 167-196. doi: 10.4310/DPDE.2007.v4.n2.a4.
[54]	Y. C. You, Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems, Commun. Pure Appl. Anal., 10 (2011), 1415-1445. doi: 10.3934/cpaa.2011.10.1415.
[55]	Y. C. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Anal., 74 (2011), 1969-1986. doi: 10.1016/j.na.2010.11.004.
[56]	Y. C. You, Global dynamics of the Oregonator system, Math. Methods Appl. Sci., 35 (2012), 398-416. doi: 10.1002/mma.1591.
[57]	Y. C. You, Robustness of Global Attractors for Reversible Gray-Scott Systems, J. Dynam. Differential Equations, 24 (2012), 495-520. doi: 10.1007/s10884-012-9252-7.
[58]	J. Zhou and C. L. Mu, Pattern formation of a coupled two-cell Brusselator model, J. Math. Anal. Appl., 366 (2010), 679-693. doi: 10.1016/j.jmaa.2009.12.021.
[59]	W. J. Zuo and J. J. Wei, Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model, Dyn. Partial Differ. Equ., 8 (2011), 363-384.

show all references

References:

[1]	Q. Y. Bie, Pattern formation in a general two-cell Brusselator model, J. Math. Anal. Appl., 376 (2011), 551-564. doi: 10.1016/j.jmaa.2010.10.066.
[2]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.
[3]	A. J. Catllá, A. McNamara and C. M. Topaz, Instabilities and patterns in coupled reaction-diffusion layers, Phy. Rev. E Stat. Nonlinear & Soft Matter Physics, 85 (2012), 489-500.
[4]	W. Chen, Localized Patterns in the Gray-scott Model: An Asymptotic and Numerical Study of Dynamics and Stability, PhD thesis, University of British Columbia,, 2009., ().
[5]	F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507-516. doi: 10.1017/S0308210500000275.
[6]	A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563. doi: 10.1088/0951-7715/10/2/013.
[7]	L. L. Du and M. X. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485. doi: 10.1016/j.jmaa.2010.02.002.
[8]	J. E. Furter and J. C. Eilbeck, Analysis of bifurcations in reaction-diffusion systems with no-flux boundary conditions: The Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413-438. doi: 10.1017/S0308210500028109.
[9]	M. Ghergu, Steady-state solutions for a general Brusselator system, In Modern Aspects of the Theory of Partial Differential Equations, volume 216 of Oper. Theory Adv. Appl., pages 153-166. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0069-3_9.
[10]	M. Ghergu and V. Rădulescu, Turing patterns in general reaction-diffusion systems of Brusselator type, Commun. Contemp. Math., 12 (2010), 661-679. doi: 10.1142/S0219199710003968.
[11]	M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345. doi: 10.1088/0951-7715/21/10/007.
[12]	M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Verlag, 2012. doi: 10.1007/978-3-642-22664-9.
[13]	A. A. Golovin, B. J. Matkowsky and V. A. Volpert, Turing pattern formation in the Brusselator model with superdiffusion, SIAM J. Appl. Math., 69 (2008), 251-272. doi: 10.1137/070703454.
[14]	J. K. Hale, L. A. Peletier and W. C. Troy, Stability and instability in the Gray-Scott model: The case of equal diffusivities, Appl. Math. Lett., 12 (1999), 59-65. doi: 10.1016/S0893-9659(99)00035-X.
[15]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41. CUP Archive, 1981.
[16]	D. Iron, J. C. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390. doi: 10.1007/s00285-003-0258-y.
[17]	J. Jang, W. M. Ni and M. X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320. doi: 10.1007/s10884-004-2782-x.
[18]	J. Y. Jin, J. P. Shi, J. J. Wei and F. Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of cima chemical reaction, Roc. Mount.J. Math., 43 (2013), 1637-1674. doi: 10.1216/RMJ-2013-43-5-1637.
[19]	T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Phys. D, 214 (2006), 63-77. doi: 10.1016/j.physd.2005.12.005.
[20]	J. van de Koppel and C. M. Crain, Scale-dependent inhibition drives regular tussock spacing in a freshwater marsh, Amer. Natu., 168 (2006), 36-47. doi: 10.1086/508671.
[21]	J. López-Gómez, J. C. Eilbeck, M. Molina and K. N. Duncan, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal., 12 (1992), 405-428. doi: 10.1093/imanum/12.3.405.
[22]	Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.
[23]	W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable gray-scott model, Math. Compu. in Simulation, 40 (1996), 371-396. doi: 10.1016/0378-4754(95)00044-5.
[24]	J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal. Real World Appl., 5 (2004), 105-121. doi: 10.1016/S1468-1218(03)00020-8.
[25]	W. M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations Stationary Partial Differential Equations, 1 (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.
[26]	W. M. Ni and M. X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9.
[27]	R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166. doi: 10.1016/j.jmaa.2004.12.026.
[28]	R. Peng, M. X. Wang and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling, 44 (2006), 945-951. doi: 10.1016/j.mcm.2006.03.001.
[29]	R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations, 241 (2007), 386-398. doi: 10.1016/j.jde.2007.06.005.
[30]	R. Peng and M. X. Wang, Some nonexistence results for nonconstant stationary solutions to the Gray-Scott model in a bounded domain, Appl. Math. Lett., 22 (2009), 569-573. doi: 10.1016/j.aml.2008.06.032.
[31]	R. Peng and F. Q. Sun, Turing pattern of the Oregonator model, Nonlinear Anal., 72 (2010), 2337-2345. doi: 10.1016/j.na.2009.10.034.
[32]	Y. W. Qi, The development of travelling waves in cubic auto-catalysis with different rates of diffusion, Phys. D, 226 (2007), 129-135. doi: 10.1016/j.physd.2006.11.010.
[33]	E. E. Sel'Kov, Self-oscillations in glycolysis, European Journal of Biochemistry, 4 (1968), 79-86.
[34]	J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.
[35]	J. P. Shi and X. F. 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.
[36]	I. Takagi C. S. Lin and W. M. Ni, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[37]	J. J. Tyson, K. Chen and B. Novak, Network dynamics and cell physiology, Nature Rev. Molecular Cell Bio., 2 (2001), 908-916.
[38]	A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72.
[39]	M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620. doi: 10.1016/S0022-0396(02)00100-6.
[40]	M. X. Wang, R. Peng and J. P. Shi, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: 10.1088/0951-7715/21/7/006.
[41]	M. X. Wang and P. Y. H. Pang, Global asymptotic stability of positive steady states of a diffusive ratio-dependent pre-predator model, Applied Mathematics Letters, 21 (2008), 1215-1220. doi: 10.1016/j.aml.2007.10.026.
[42]	J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004.
[43]	M. J. Ward and J. C. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223.
[44]	J. M. Wei, Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability, Phys. D, 148 (2001), 20-48. doi: 10.1016/S0167-2789(00)00183-4.
[45]	J. C. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.
[46]	J. C. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol., 64 (2012), 211-254. doi: 10.1007/s00285-011-0412-x.
[47]	S. Wiggins and M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2. Springer, 1990. doi: 10.1007/978-1-4757-4067-7.
[48]	L. Xu, G. Zhang and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model, WSEAS Transac. Math, 10 (2011), 201-209.
[49]	C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977. doi: 10.1016/j.nonrwa.2012.01.001.
[50]	F. Q. Yi, J. J. Wei and J. J. Shi, Diffusion-driven instability and bifurcation in the lengyel-epstein system, Nonlinear Anal.: Real World Applications, 9 (2008), 1038-1051. doi: 10.1016/j.nonrwa.2007.02.005.
[51]	F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.
[52]	F. Q. Yi, J. J. Wei and J. P. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55. doi: 10.1016/j.aml.2008.02.003.
[53]	Y. C. You, Global dynamics of the Brusselator equations, Dyn. Partial Differ. Equ., 4 (2007), 167-196. doi: 10.4310/DPDE.2007.v4.n2.a4.
[54]	Y. C. You, Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems, Commun. Pure Appl. Anal., 10 (2011), 1415-1445. doi: 10.3934/cpaa.2011.10.1415.
[55]	Y. C. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Anal., 74 (2011), 1969-1986. doi: 10.1016/j.na.2010.11.004.
[56]	Y. C. You, Global dynamics of the Oregonator system, Math. Methods Appl. Sci., 35 (2012), 398-416. doi: 10.1002/mma.1591.
[57]	Y. C. You, Robustness of Global Attractors for Reversible Gray-Scott Systems, J. Dynam. Differential Equations, 24 (2012), 495-520. doi: 10.1007/s10884-012-9252-7.
[58]	J. Zhou and C. L. Mu, Pattern formation of a coupled two-cell Brusselator model, J. Math. Anal. Appl., 366 (2010), 679-693. doi: 10.1016/j.jmaa.2009.12.021.
[59]	W. J. Zuo and J. J. Wei, Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model, Dyn. Partial Differ. Equ., 8 (2011), 363-384.

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