-
Previous Article
Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems
- DCDS-B Home
- This Issue
-
Next Article
Approximate tracking of periodic references in a class of bilinear systems via stable inversion
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), 220-229.
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. |
[1] |
Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 |
[2] |
Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182 |
[3] |
Xueying Sun, Renhao Cui. Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4503-4520. doi: 10.3934/dcdss.2021120 |
[4] |
Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062 |
[5] |
Aung Zaw Myint. Positive solutions of a diffusive two competitive species model with saturation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3625-3641. doi: 10.3934/dcdsb.2021199 |
[6] |
Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3747-3785. doi: 10.3934/dcds.2022031 |
[7] |
Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 |
[8] |
Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 2949-2973. doi: 10.3934/dcdss.2020132 |
[9] |
Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 |
[10] |
Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 |
[11] |
Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure and Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347 |
[12] |
Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301 |
[13] |
Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613 |
[14] |
Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941 |
[15] |
Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069 |
[16] |
Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027 |
[17] |
Zhijun Liu, Lianwen Wang, Ronghua Tan. Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2767-2790. doi: 10.3934/dcdsb.2021159 |
[18] |
Daifeng Duan, Ben Niu, Junjie Wei. Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3683-3706. doi: 10.3934/dcdsb.2021202 |
[19] |
Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026 |
[20] |
Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]