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January  2011, 15(1): 217-230. doi: 10.3934/dcdsb.2011.15.217

On spatiotemporal pattern formation in a diffusive bimolecular model

Rui Peng 1, and  Fengqi Yi 2,

1. 

Institute of Nonlinear Complex Systems, College of Science, China Three Gorges University, Yichang, 443002, Hubei, China
2. 

Department of Mathematics, Harbin Engineering University, Harbin, 150001, China

Received  December 2009 Revised  April 2010 Published  October 2010

This paper continues the analysis on a bimolecular autocatalytic reaction-diffusion model with saturation law. An improved result of steady state bifurcation is derived and the effect of various parameters on spatiotemporal patterns is discussed. Our analysis provides a better understanding on the rich spatiotemporal patterns. Some numerical simulations are performed to support the theoretical conclusions.
Keywords: Diffusive bimolecular model, steady state bifurcation., spatiotemporal pattern, autocatalysis and saturation law.
Mathematics Subject Classification: Primary: 3532, 35J55, 35K57, 92C15; Secondary: 92C4.
Citation: Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217
References:
[1]

M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229. doi: doi:10.1016/j.jtbi.2006.09.036.  Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: doi:10.1137/0517094.  Google Scholar

[3]

L. L. Bonilla and M. G. Velarde, Singular perturbations approach to the limit cycle and global patterns in a nonlinear diffusion-reaction problem with autocatalysis and saturation law, J. Math. Phys., 20 (1979), 2692-2703. doi: doi:10.1063/1.524034.  Google Scholar

[4]

Y. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 777-809.  Google Scholar

[5]

Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. doi: doi:10.1006/jdeq.1997.3394.  Google Scholar

[6]

Y. Du and Y. Lou, Qualitative behavior of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349. doi: doi:10.1017/S0308210500000895.  Google Scholar

[7]

Y. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in "Nonlinear Dynamics and Evolution Equations," 95-135, Fields Inst. Commun., 48, Amer. Math. Soc., Providence, RI, 2006.  Google Scholar

[8]

Y. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593. doi: doi:10.1090/S0002-9947-07-04262-6.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," Reprint of the 1998 edition, "Classics in Mathematics," Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

J. L. Ibanez and M. G. Velarde, Multiple steady states in a simple reaction-diffusion model with Michaelis-Menten (first-order Hinshelwood-Langmuir) saturation law: The limit of large separation in the two diffusion constants, J. Math. Phys., 19 (1978), 151-156. doi: doi:10.1063/1.523532.  Google Scholar

[11]

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: doi:10.1007/s10884-004-2782-x.  Google Scholar

[12]

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,, submitted for publication., ().   Google Scholar

[13]

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

[14]

R. Peng, J. P. Shi and M. X. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: doi:10.1088/0951-7715/21/7/006.  Google Scholar

[15]

R. Peng and J. P. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886. doi: doi:10.1016/j.jde.2009.03.008.  Google Scholar

[16]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. doi: doi:10.1016/0022-1236(71)90030-9.  Google Scholar

[17]

W. H. Ruan, Asymptotic behavior and positive steady-state solutions of a reaction-diffusion model with autocatalysis and saturation law, Nonlinear Anal: TMA, 21 (1993), 439-456. doi: doi:10.1016/0362-546X(93)90127-E.  Google Scholar

[18]

J. P. Shi, Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Frontier of Mathematics in China, 4 (2009), 407-424. doi: doi:10.1007/s11464-009-0026-4.  Google Scholar

[19]

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: doi:10.1016/j.jde.2008.09.009.  Google Scholar

[20]

Y. Su, J. Wei and J. P. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. doi: doi:10.1016/j.jde.2009.04.017.  Google Scholar

[21]

M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620. doi: doi:10.1016/S0022-0396(02)00100-6.  Google Scholar

[22]

F. Q. Yi, J. X. Liu and J. J. Wei, Spatiotemporal pattern formation and multiple bifurcations in a diffusibve bimolecular model, Nonl. Anal: RWA, 11 (2010), 3770-3781. doi: doi:10.1016/j.nonrwa.2010.02.007.  Google Scholar

[23]

F. Q. Yi, J. J. Wei and J. P. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonl. Anal: RWA, 9 (2008), 1038-1051. doi: doi:10.1016/j.nonrwa.2010.02.007.  Google Scholar

[24]

F. Q. Yi, J. J. Wei, J. P. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55. doi: doi:10.1016/j.aml.2008.02.003.  Google Scholar

[25]

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: doi:10.1016/j.jde.2008.10.024.  Google Scholar

show all references

References:
[1]

M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229. doi: doi:10.1016/j.jtbi.2006.09.036.  Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: doi:10.1137/0517094.  Google Scholar

[3]

L. L. Bonilla and M. G. Velarde, Singular perturbations approach to the limit cycle and global patterns in a nonlinear diffusion-reaction problem with autocatalysis and saturation law, J. Math. Phys., 20 (1979), 2692-2703. doi: doi:10.1063/1.524034.  Google Scholar

[4]

Y. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 777-809.  Google Scholar

[5]

Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. doi: doi:10.1006/jdeq.1997.3394.  Google Scholar

[6]

Y. Du and Y. Lou, Qualitative behavior of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349. doi: doi:10.1017/S0308210500000895.  Google Scholar

[7]

Y. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in "Nonlinear Dynamics and Evolution Equations," 95-135, Fields Inst. Commun., 48, Amer. Math. Soc., Providence, RI, 2006.  Google Scholar

[8]

Y. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593. doi: doi:10.1090/S0002-9947-07-04262-6.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," Reprint of the 1998 edition, "Classics in Mathematics," Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

J. L. Ibanez and M. G. Velarde, Multiple steady states in a simple reaction-diffusion model with Michaelis-Menten (first-order Hinshelwood-Langmuir) saturation law: The limit of large separation in the two diffusion constants, J. Math. Phys., 19 (1978), 151-156. doi: doi:10.1063/1.523532.  Google Scholar

[11]

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: doi:10.1007/s10884-004-2782-x.  Google Scholar

[12]

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,, submitted for publication., ().   Google Scholar

[13]

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

[14]

R. Peng, J. P. Shi and M. X. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: doi:10.1088/0951-7715/21/7/006.  Google Scholar

[15]

R. Peng and J. P. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886. doi: doi:10.1016/j.jde.2009.03.008.  Google Scholar

[16]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. doi: doi:10.1016/0022-1236(71)90030-9.  Google Scholar

[17]

W. H. Ruan, Asymptotic behavior and positive steady-state solutions of a reaction-diffusion model with autocatalysis and saturation law, Nonlinear Anal: TMA, 21 (1993), 439-456. doi: doi:10.1016/0362-546X(93)90127-E.  Google Scholar

[18]

J. P. Shi, Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Frontier of Mathematics in China, 4 (2009), 407-424. doi: doi:10.1007/s11464-009-0026-4.  Google Scholar

[19]

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: doi:10.1016/j.jde.2008.09.009.  Google Scholar

[20]

Y. Su, J. Wei and J. P. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. doi: doi:10.1016/j.jde.2009.04.017.  Google Scholar

[21]

M. X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620. doi: doi:10.1016/S0022-0396(02)00100-6.  Google Scholar

[22]

F. Q. Yi, J. X. Liu and J. J. Wei, Spatiotemporal pattern formation and multiple bifurcations in a diffusibve bimolecular model, Nonl. Anal: RWA, 11 (2010), 3770-3781. doi: doi:10.1016/j.nonrwa.2010.02.007.  Google Scholar

[23]

F. Q. Yi, J. J. Wei and J. P. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonl. Anal: RWA, 9 (2008), 1038-1051. doi: doi:10.1016/j.nonrwa.2010.02.007.  Google Scholar

[24]

F. Q. Yi, J. J. Wei, J. P. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett., 22 (2009), 52-55. doi: doi:10.1016/j.aml.2008.02.003.  Google Scholar

[25]

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: doi:10.1016/j.jde.2008.10.024.  Google Scholar

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