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Renormalized solutions to a reaction-diffusion system applied to image denoising
Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure
1. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia |
2. | Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Universidad del Bio Bio, Concepción, Avda. Collao 1202, Chile |
  Here we characterize the existence of Hopf bifurcations for the systems. In particular we provide a positive answer to the above question showing for such models the existence of small--amplitude Hopf limit cycles being the equilibrium point $E^*$ unstable.
References:
[1] |
W. G. Aiello and H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.
doi: 10.1016/0025-5564(90)90019-U. |
[2] |
W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.
doi: 10.1137/0152048. |
[3] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[4] |
Z. Li, M. Han and F. Chen, Global stability of a predator-prey system with stage structure and mutual interference, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 173-187.
doi: 10.3934/dcdsb.2014.19.173. |
[5] |
K. G. Magnusson, Destabilizing effect of cannibalism on a structured predator-prey system, Math. Biosci., 155 (1999), 61-75.
doi: 10.1016/S0025-5564(98)10051-2. |
[6] |
W. Wang and L. Chen, A predator-prey system with stage structure for predator, Comput. Math. Appl., 33 (1997), 83-91.
doi: 10.1016/S0898-1221(97)00056-4. |
[7] |
R. Xu, Global convergence of a predator-prey model with stage structure and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273-291.
doi: 10.3934/dcdsb.2011.15.273. |
[8] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Persistence and global stability of a ratio-dependent predator-prey model with stage structure, Appl. Math. Comput., 158 (2004), 729-744.
doi: 10.1016/j.amc.2003.10.012. |
[9] |
X. Zhang and L. Chen, The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168 (2000), 201-210.
doi: 10.1016/S0025-5564(00)00033-X. |
show all references
References:
[1] |
W. G. Aiello and H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.
doi: 10.1016/0025-5564(90)90019-U. |
[2] |
W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.
doi: 10.1137/0152048. |
[3] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[4] |
Z. Li, M. Han and F. Chen, Global stability of a predator-prey system with stage structure and mutual interference, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 173-187.
doi: 10.3934/dcdsb.2014.19.173. |
[5] |
K. G. Magnusson, Destabilizing effect of cannibalism on a structured predator-prey system, Math. Biosci., 155 (1999), 61-75.
doi: 10.1016/S0025-5564(98)10051-2. |
[6] |
W. Wang and L. Chen, A predator-prey system with stage structure for predator, Comput. Math. Appl., 33 (1997), 83-91.
doi: 10.1016/S0898-1221(97)00056-4. |
[7] |
R. Xu, Global convergence of a predator-prey model with stage structure and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273-291.
doi: 10.3934/dcdsb.2011.15.273. |
[8] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Persistence and global stability of a ratio-dependent predator-prey model with stage structure, Appl. Math. Comput., 158 (2004), 729-744.
doi: 10.1016/j.amc.2003.10.012. |
[9] |
X. Zhang and L. Chen, The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168 (2000), 201-210.
doi: 10.1016/S0025-5564(00)00033-X. |
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