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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

 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.
Citation: Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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