# American Institute of Mathematical Sciences

August  2016, 21(6): 1859-1867. doi: 10.3934/dcdsb.2016026

## 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

Received  October 2014 Revised  March 2016 Published  June 2016

A ratio--dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point $E^*$ at the positive octant is unstable.
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.
Citation: Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1859-1867. doi: 10.3934/dcdsb.2016026
##### 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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##### 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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