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

Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure

Jaume Llibre 1, and  Claudio Vidal 2,

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.
Keywords: predator-prey model, Hopf bifurcation, Ratio--dependence, averaging theory..
Mathematics Subject Classification: Primary: 34D23, 92D2.
Citation: Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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