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Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems
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 |
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.
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.
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.
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.
|
[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.
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.
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, 48 (2006), 95.
|
[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.
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, (1998).
|
[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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
|
[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.
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.
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, 48 (2006), 95.
|
[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.
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, (1998).
|
[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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
doi: doi:10.1016/j.jde.2008.10.024. |
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