Advanced Search
Article Contents
Article Contents

On spatiotemporal pattern formation in a diffusive bimolecular model

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 3532, 35J55, 35K57, 92C15; Secondary: 92C40.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    J. Y. Jin, J. P. Shi, J. J. Wei and F. Q. YiBifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, submitted for publication.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(104) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint