-
Previous Article
Traveling bands for the Keller-Segel model with population growth
- MBE Home
- This Issue
-
Next Article
Mathematical probit and logistic mortality models of the Khapra beetle fumigated with plant essential oils
Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions
1. | Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3 |
2. | Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada |
References:
[1] |
J. Blat and K. J. Brown, Global bifurcation of positive solutions in some system of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353.
doi: 10.1137/0517094. |
[2] |
E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math., 3 (1982), 288-334.
doi: 10.1016/S0196-8858(82)80009-2. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[5] |
E. N. Dancer, On positive solutions of some pairs of differential equations II, J. Diff. Equat., 60 (1985), 236-258.
doi: 10.1016/0022-0396(85)90115-9. |
[6] |
L. R. Fox, Defense and dynamics in plant-herbivore systems, Amer. Zool., 21 (1981), 853-864.
doi: 10.1093/icb/21.4.853. |
[7] |
B. D. Hassard, N. D. KazavinoJ and Y. H. Wan, Theory and Applications of the Hopf Bifurcation, Cambridge University Press, Cambridge, MA, 1981. |
[8] |
L. R. Ginzburg, Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models, J. of Animal Ecology, 67 (1998), 325-327.
doi: 10.1046/j.1365-2656.1998.00226.x. |
[9] |
L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.
doi: 10.1090/S0002-9947-1988-0920151-1. |
[10] |
R. E. Ricklefs, The Economy of Nature, Freeman and Company, New York, 2010. |
[11] |
Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104.
doi: 10.1088/0951-7715/27/1/87. |
[12] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345.
doi: 10.1137/0521018. |
show all references
References:
[1] |
J. Blat and K. J. Brown, Global bifurcation of positive solutions in some system of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353.
doi: 10.1137/0517094. |
[2] |
E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math., 3 (1982), 288-334.
doi: 10.1016/S0196-8858(82)80009-2. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[5] |
E. N. Dancer, On positive solutions of some pairs of differential equations II, J. Diff. Equat., 60 (1985), 236-258.
doi: 10.1016/0022-0396(85)90115-9. |
[6] |
L. R. Fox, Defense and dynamics in plant-herbivore systems, Amer. Zool., 21 (1981), 853-864.
doi: 10.1093/icb/21.4.853. |
[7] |
B. D. Hassard, N. D. KazavinoJ and Y. H. Wan, Theory and Applications of the Hopf Bifurcation, Cambridge University Press, Cambridge, MA, 1981. |
[8] |
L. R. Ginzburg, Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models, J. of Animal Ecology, 67 (1998), 325-327.
doi: 10.1046/j.1365-2656.1998.00226.x. |
[9] |
L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.
doi: 10.1090/S0002-9947-1988-0920151-1. |
[10] |
R. E. Ricklefs, The Economy of Nature, Freeman and Company, New York, 2010. |
[11] |
Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104.
doi: 10.1088/0951-7715/27/1/87. |
[12] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345.
doi: 10.1137/0521018. |
[1] |
Guangyu Sui, Meng Fan, Irakli Loladze, Yang Kuang. The dynamics of a stoichiometric plant-herbivore model and its discrete analog. Mathematical Biosciences & Engineering, 2007, 4 (1) : 29-46. doi: 10.3934/mbe.2007.4.29 |
[2] |
Ya Li, Z. Feng. Dynamics of a plant-herbivore model with toxin-induced functional response. Mathematical Biosciences & Engineering, 2010, 7 (1) : 149-169. doi: 10.3934/mbe.2010.7.149 |
[3] |
Yang Kuang, Jef Huisman, James J. Elser. Stoichiometric Plant-Herbivore Models and Their Interpretation. Mathematical Biosciences & Engineering, 2004, 1 (2) : 215-222. doi: 10.3934/mbe.2004.1.215 |
[4] |
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 |
[5] |
Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805 |
[6] |
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 |
[7] |
Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 |
[8] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
[9] |
John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 |
[10] |
Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 |
[11] |
Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 |
[12] |
Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 |
[13] |
Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 |
[14] |
Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 |
[15] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[16] |
Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 |
[17] |
Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042 |
[18] |
Shu Li, Zhenzhen Li, Binxiang Dai. Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022025 |
[19] |
Jun Zhou. Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences & Engineering, 2016, 13 (4) : 857-885. doi: 10.3934/mbe.2016021 |
[20] |
Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]