Geodesic
Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays
Lv, Dingyang1 ; Zhang, Wen2 ; Tang, Yi1
1 Department of Mathematics, Hunan First Normal College, Changsha, 410205 Hunan, P. R. China
2 School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, P. R. China;School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China
Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 6, p. 3479-3490.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we consider a ratio-dependent predator-prey system with multiple delays where the dynamics are logistic with the carrying capacity proportional to prey population. By choosing the sum $\tau$ of two delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.
DOI : 10.22436/jnsa.009.06.03
Classification : 34C25, 37L10
Keywords: Ratio-dependent, delay, Hopf bifurcation, center manifold, periodic solutions.

Lv, Dingyang 1 ; Zhang, Wen 2 ; Tang, Yi 1

1 Department of Mathematics, Hunan First Normal College, Changsha, 410205 Hunan, P. R. China
2 School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, P. R. China;School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China 
@article{JNSA_2016_9_6_a2,
     author = {Lv, Dingyang and Zhang, Wen and Tang, Yi},
     title = {Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {3479-3490},
     publisher = {mathdoc},
     volume = {9},
     number = {6},
     year = {2016},
     doi = {10.22436/jnsa.009.06.03},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.03/}
}
TY  - JOUR
AU  - Lv, Dingyang
AU  - Zhang, Wen
AU  - Tang, Yi
TI  - Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays
JO  - Journal of nonlinear sciences and its applications
PY  - 2016
SP  - 3479
EP  - 3490
VL  - 9
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.03/
DO  - 10.22436/jnsa.009.06.03
LA  - en
ID  - JNSA_2016_9_6_a2
ER  - 
%0 Journal Article
%A Lv, Dingyang
%A Zhang, Wen
%A Tang, Yi
%T Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays
%J Journal of nonlinear sciences and its applications
%D 2016
%P 3479-3490
%V 9
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.03/
%R 10.22436/jnsa.009.06.03
%G en
%F JNSA_2016_9_6_a2
Lv, Dingyang; Zhang, Wen; Tang, Yi. Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays. Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 6, p. 3479-3490. doi : 10.22436/jnsa.009.06.03. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.06.03/

[1] Celik, C. The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos Solitions Fractals, Volume 37 (2008), pp. 87-99

[2] Celik, C. Hopf bifurcation of a ratio-dependent predator-prey system with time delay, Chaos Solitions Fractals, Volume 42 (2009), pp. 1474-1484

[3] Celik, C.; Duman, O. Allee effect in a discrete-time predator-prey system, Chaos Solitions Fractals, Volume 40 (2009), pp. 1956-1962

[4] Cai, S. M.; Hao, J. J.; He, Q. B.; Liu, Z. R. Exponential synchronization of complex delayed dynamical networks via pinning periodically intermittent control, Phys. Lett. A, Volume 375 (2011), pp. 1965-1971

[5] Fang, H.; Li, J. B. On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., Volume 259 (2001), pp. 8-17

[6] Fowler, M.; G. Ruxton Population dynamic consequences of Allee effects, J. Theoret. Biol., Volume 215 (2002), pp. 39-46

[7] Hadjiavgousti, D.; Ichtiaroglou, S. Allee effect in a predator-prey system, Chaos Solitions Fractals, Volume 36 (2008), pp. 334-342

[8] Hassard, B.; Kazarinoff, N.; Wan, Y. Theory and Applications of Hopf bifurcation, Cambridge Univ. Press, Cambridge, 1981

[9] Hu, G.; Li, W. Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonlinear Anal. RWA, Volume 11 (2010), pp. 819-826

[10] Kuang, Y. Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993

[11] Li, Y. K. Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl., Volume 246 (2000), pp. 230-244

[12] Li, Y. K. Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays, Chaos Solitons Fractals, Volume 37 (2008), pp. 288-298

[13] Li, J. B.; He, X. Z.; Liu, Z. R. Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations, Nonlinear Anal., Volume 35 (1999), pp. 457-474

[14] Li, Y. K.; Kuang, Y. Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., Volume 255 (2001), pp. 260-280

[15] Li, Y. K.; Zhang, T. W. On the existence and stability of a unique almost periodic sequence solution in discrete predator-prey models with time delays, Appl. Math. Model., Volume 35 (2011), pp. 5448-5459

[16] Li, Y. K.; Zhao, K. H.; Ye, Y. Multiple positive periodic solutions of n species delay competition systems with harvesting terms, Nonlinear Anal. RWA, Volume 12 (2011), pp. 1013-1022

[17] Li, Y. K.; Zhu, L. F. Positive periodic solutions for a class of higher-demensional state-dependent delay functional differential equations with feedback control , Appl. Math. Comput., Volume 159 (2004), pp. 783-795

[18] Liu, J.; Li, Y. K.; Zhao, L. L. On a periodic predator-prey system with time delays on time scales, Commun. Nonlinear Sci. Numer. Simul., Volume 14 (2009), pp. 3432-3438

[19] Song, Y.; Peng, Y.; Wei, J. Bifurcations for a predator-prey system with two delays, J. Math. Anal. Appl., Volume 337 (2008), pp. 466-479

[20] Sun, H. J.; Cao, H. J. Bifurcations and chaos of a delayed ecological model, Chaos Solitions Fractals, Volume 33 (2007), pp. 1383-1393

[21] Xu, C.; Tang, X.; Liao, M. Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments, Appl. Math. Comput., Volume 216 (2010), pp. 2920-2936

[22] Yan, X.; W. Li Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Appl. Math. Comput., Volume 177 (2006), pp. 427-445

[23] Yin, F. Q.; Li, Y. k. Positive periodic solutions of a single species model with feedback regulation and distributed time delays, Appl. Math. Comput, Volume 153 (2004), pp. 475-484

[24] Zhou, J.; Liu, Z. R.; Chen, G. R. Dynamics of periodic delayed neural networks, Neural Networks, Volume 17 (2004), pp. 87-101

[25] Zhou, S.; Liu, Y.; G. Wang The stability of predator-prey systems subject to the Allee effects, Theor. Population Biol., Volume 67 (2005), pp. 23-31

Cité par Sources :