References
- J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations. I. Evolution equations and the steady state solutions, Bull. Math. Biology 37 (1975), no. 4, 323-365. https://doi.org/10.1007/bf02459519
- N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol. 136 (1989), no. 1, 57-66. https://doi.org/10.1016/S0022-5193(89)80189-4
- N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990), no. 6, 1663-1688. https://doi.org/10.1137/0150099
- K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal. 24 (1995), no. 12, 1713-1725. https://doi.org/10.1016/0362-546X(94)00218-7
- A. J. Catlla, A. McNamara, and C. M. Topaz, Instabilities and patterns in coupled reaction-diffusion layers, Phy. Rev. E 85 (2012), no. 2, 026215-1. https://doi.org/10.1103/PhysRevE.85.026215
- W. Chen, Localized patterns in the gray-scott model: an asymptotic and numerical study of dynamics and stability, A thesis submitted in partial fulfillment of the requirments for the degree of doctor of philosophy in the faculty of graduate studies (Mathematics), The Uinversity of British Columbia(Vancouver) July, 2009.
- C. Chris and C. R. Stephen, Spatial Ecology via Reaction-Diusion Equations, John Wiley & Sons, Ltd., Chichester, 2003.
- 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), no. 3, 507-516. https://doi.org/10.1017/S0308210500000275
- A. Doelman, T. J. Kaper, and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity 10 (1997), no. 2, 523-563. https://doi.org/10.1088/0951-7715/10/2/013
- L. Du and M. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl. 366 (2010), no. 2, 473-485. https://doi.org/10.1016/j.jmaa.2010.02.002
- 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), no. 2, 413-438. https://doi.org/10.1017/S0308210500028109
- M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity 21 (2008), no. 10, 2331-2345. https://doi.org/10.1088/0951-7715/21/10/007
- M. Ghergu and V. Radulescu, Turing patterns in general reaction-diffusion systems of Brusselator type, Commun. Contemp. Math. 12 (2010), no. 4, 661-679. https://doi.org/10.1142/S0219199710003968
- M. Ghergu and V. Radulescu, Nonlinear PDEs, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-22664-9
- J. K. Hale, L. A. Peletier, and W. C. Troy, Stability and instability in the Gray-Scott model: the case of equal diusivities, Appl. Math. Lett. 12 (1999), no. 4, 59-65. https://doi.org/10.1016/S0893-9659(99)00035-X
- B. D. Hassard, N. D. Kazarino, and Y. H. Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge, 1981.
- M. Herschkowitz-Kaufman, Bifurcation analysis of nonlinear reaction-diffusion equations. II. Steady state solutions and comparison with numerical simulations, Bull. Math. Biology 37 (1975), no. 6, 589-636. https://doi.org/10.1007/BF02459527
- D. Iron, J. Wei, and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol. 49 (2004), no. 4, 358-390. https://doi.org/10.1007/s00285-003-0258-y
- J. Jang, W.-M. Ni, and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations 16 (2004), no. 2, 297-320. https://doi.org/10.1007/s10884-004-2782-x
- J. Y. Jin, J. P. Shi, J. J. Wei, and F. Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math. 43 (2013), no. 5, 1637-1674. https://doi.org/10.1216/RMJ-2013-43-5-1637
- T. Kolokolnikov, T. Erneux, and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Phys. D 214 (2006), no. 1, 63-77. https://doi.org/10. 1016/j.physd.2005.12.005 https://doi.org/10.1016/j.physd.2005.12.005
- C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1-27. https://doi.org/10.1016/0022-0396(88)90147-7
- J. Lopez-Gomez, 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), no. 3, 405-428. https://doi.org/10.1093/imanum/12.3.405
- Y. Lou and W.-M. Ni, Diusion, self-diffusion and cross-diffusion, J. Differential Equa- tions 131 (1996), no. 1, 79-131. https://doi.org/10.1006/jdeq.1996.0157
- W. Mazin, K. Rasmussen, E. Mosekilde, P. Borckmans, and G. Dewel, Pattern formation in the bistable gray-scott model, Math. Comput. Simu. 40 (1996), no. 3, 371-396. https://doi.org/10.1016/0378-4754(95)00044-5
- J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal. Real World Appl. 5 (2004), no. 1, 105-121. https://doi.org/10.1016/S1468-1218(03)00020-8
- W.-M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume 1, Chapter 3, 157-233, 2004.
- W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc. 357 (2005), no. 10, 3953-3969. https://doi.org/10.1090/S0002-9947-05-04010-9
- R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations 241 (2007), no. 2, 386-398. https://doi.org/10.1016/j.jde.2007.06.005
- R. Peng, J. Shi, and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity 21 (2008), no. 7, 1471-1488. https://doi.org/10.1088/0951-7715/21/7/006
- R. Peng and F. Sun, Turing pattern of the Oregonator model, Nonlinear Anal. 72 (2010), no. 5, 2337-2345. https://doi.org/10.1016/j.na.2009.10.034
- R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl. 309 (2005), no. 1, 151-166. https://doi.org/10.1016/j.jmaa.2004.12.026
- 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), no. 4, 569-573. https://doi.org/10.1016/j.aml.2008.06.032
- R. Peng, M. Wang, and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling 44 (2006), no. 9-10, 945-951. https://doi.org/10.1016/j.mcm.2006.03.001
- J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol. 81 (1979), no. 3, 389-400. https://doi.org/10.1016/0022-5193(79)90042-0
- E. Sel'Kov, Self-oscillations in glycolysis, Eur. J. Bioch. 4 (1968), no. 1, 79-86. https://doi.org/10.1111/j.1432-1033.1968.tb00175.x
- J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations 246 (2009), no. 7, 2788-2812. https://doi.org/10.1016/j.jde.2008.09.009
- A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), no. 641, 37-72. https://doi.org/10.1098/rstb.1952.0012
- J. Wang, J. Shi, and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations 251 (2011), no. 4-5, 1276-1304. https://doi.org/10.1016/j.jde.2011.03.004
- M. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equa- tions 190 (2003), no. 2, 600-620. https://doi.org/10.1016/S0022-0396(02)00100-6
- M. Wang and P. Y. H. Pang, Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model, Appl. Math. Lett. 21 (2008), no. 11, 1215-1220. https://doi.org/10.1016/j.aml.2007.10.026
- M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math. 109 (2002), no. 3, 229-264. https://doi.org/10.1111/1467-9590.00223
- J. Wei, Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability, Phys. D 148 (2001), no. 1-2, 20-48. https://doi.org/10.1016/S0167-2789(00)00183-4
- J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol. 57 (2008), no. 1, 53-89. https://doi.org/10.1007/s00285-007-0146-y
- J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol. 64 (2012), no. 1-2, 211-254. https://doi.org/10.1007/s00285-011-0412-x
- C. Xu and J.Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl. 13 (2012), no. 4, 1961-1977. https://doi.org/10.1016/j.nonrwa.2012.01.001
- L. Xu, G. Zhang, and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model, WSEAS Trans. Math. 10 (2011), no. 6, 201-209.
- G. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl. 22 (2015), 54-65. https://doi.org/10.1016/j.nonrwa.2014.07.013
- F. Yi, J. Wei, and J. Shi, Diusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl. 9 (2008), no. 3, 1038-1051. https://doi.org/10.1016/j.nonrwa.2007.02.005
- F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predatorprey system, J. Differential Equations 246 (2009), no. 5, 1944-1977. https://doi.org/10.1016/j.jde.2008.10.024
- F. Yi, J. Wei, and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett. 22 (2009), no. 1, 52-55. https://doi.org/10.1016/j.aml.2008.02.003
- Y. You, Global dynamics of the Brusselator equations, Dyn. Partial Differ. Equ. 4 (2007), no. 2, 167-196. https://doi.org/10.4310/DPDE.2007.v4.n2.a4
- Y. You, Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems, Commun. Pure Appl. Anal. 10 (2011), no. 5, 1415-1445. https://doi.org/10.3934/cpaa.2011.10.1415
- Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Anal. 74 (2011), no. 5, 1969-1986. https://doi.org/10.1016/j.na.2010.11.004
- Y. You, Global dynamics of the Oregonator system, Math. Methods Appl. Sci. 35 (2012), no. 4, 398-416. https://doi.org/10.1002/mma.1591
- Y. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations 24 (2012), no. 3, 495-520. https://doi.org/10.1007/s10884-012-9252-7
- J. Zhou and C. Mu, Pattern formation of a coupled two-cell Brusselator model, J. Math. Anal. Appl. 366 (2010), no. 2, 679-693. https://doi.org/10.1016/j.jmaa.2009.12.021