May  2014, 19(3): 747-760. doi: 10.3934/dcdsb.2014.19.747

Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model

1. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada, Canada, Canada

Received  September 2013 Revised  November 2013 Published  February 2014

In recent studies, global Hopf branches were investigated for delayed model of HTLV-I infection with delay-independent parameters. It is shown in [8,9] that when stability switches occur, global Hopf branches tend to be bounded, and different branches can overlap to produce coexistence of stable periodic solutions. In this paper, we investigate global Hopf branches for delayed systems with delay-dependent parameters. Using a delayed predator-prey model as an example, we demonstrate that stability switches caused by varying the time delay are accompanied by bounded global Hopf branches. When multiple Hopf branches exist, they are nested and the overlap produces coexistence of two or possibly more stable limit cycles.
Citation: Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
References:
[1]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[2]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00, A MATLAB Package for Bifurcation Analysis of Delay Differential Equations, Tech. rep., Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001.

[3]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[4]

X. Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198 (1996), 355-370. doi: 10.1006/jmaa.1996.0087.

[5]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.

[6]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York, 1998.

[7]

J. P. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, 1976.

[8]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Bio., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7.

[9]

M. Y. Li, X. Lin and H. Wang, Global Hopf branches of a delayed HTLV-1 infection model: Coexistence of multiple attracting limit cycles, Canadian Appl. Math. Quarterly, 20 (2012), 39-50.

[10]

R. M. May, Time delays versus stability in population models with two or three trophic levels, Ecology, 54 (1973), 315-325. doi: 10.2307/1934339.

[11]

H. Shu, L, Wang and J. Wu, Global dynamics of Nicholson's blow y equation revisited: Onset and termination of nonlinear oscillations, Journal of Differential Equations, 255 (2013), 2565-2586. doi: 10.1016/j.jde.2013.06.020.

[12]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21. doi: 10.1016/j.jmaa.2004.06.056.

[13]

H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Math. Biosci., 221 (2009), 1-10. doi: 10.1016/j.mbs.2009.06.004.

show all references

References:
[1]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[2]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00, A MATLAB Package for Bifurcation Analysis of Delay Differential Equations, Tech. rep., Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001.

[3]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[4]

X. Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198 (1996), 355-370. doi: 10.1006/jmaa.1996.0087.

[5]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.

[6]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York, 1998.

[7]

J. P. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, 1976.

[8]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Bio., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7.

[9]

M. Y. Li, X. Lin and H. Wang, Global Hopf branches of a delayed HTLV-1 infection model: Coexistence of multiple attracting limit cycles, Canadian Appl. Math. Quarterly, 20 (2012), 39-50.

[10]

R. M. May, Time delays versus stability in population models with two or three trophic levels, Ecology, 54 (1973), 315-325. doi: 10.2307/1934339.

[11]

H. Shu, L, Wang and J. Wu, Global dynamics of Nicholson's blow y equation revisited: Onset and termination of nonlinear oscillations, Journal of Differential Equations, 255 (2013), 2565-2586. doi: 10.1016/j.jde.2013.06.020.

[12]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21. doi: 10.1016/j.jmaa.2004.06.056.

[13]

H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Math. Biosci., 221 (2009), 1-10. doi: 10.1016/j.mbs.2009.06.004.

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