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Modeling eating behaviors: The role of environment and positive food association learning via a Ratatouille effect
Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack
1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400715 |
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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