October  2011, 16(3): 719-738. doi: 10.3934/dcdsb.2011.16.719

Dynamical behavior of a ratio dependent predator-prey system with distributed delay

1. 

Department of Mathematics and Computer Sciences, Bahçeşehir University, Istanbul, 34353, Turkey

Received  April 2010 Revised  July 2010 Published  June 2011

In this paper, we consider a predator-prey system with distributed time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. In [1] and [2], we studied the impact of the discrete time delay on the stability of the model, however in this paper, we investigate the effect of the distributed delay for the same model. By choosing the delay time $\tau $ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau $ passes some critical values. Using normal form theory and central manifold argument, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations for justifying the theoretical analysis are also presented.
Citation: Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719
References:
[1]

C. Çelik, The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos, Solitons & Fractals, 37 (2008), 87-99. doi: 10.1016/j.chaos.2007.10.045.

[2]

C. Çelik, Hopf bifurcation of a ratio-dependent predator-prey system with time delay, Chaos, Solitons & Fractals, 42 (2009), 1474-1484. doi: 10.1016/j.chaos.2009.03.071.

[3]

X. Chen, Periodicity in a nonlinear disrete predator-prey system with state dependent delays, Nonlinear Anal. RWA, 8 (2007), 435-446. doi: 10.1016/j.nonrwa.2005.12.005.

[4]

C. Çelik and O. Duman, Allee effect in a discrete-time predator-prey system, Chaos, Solitons & Fractals, 40 (2009), 1956-1962. doi: 10.1016/j.chaos.2007.09.077.

[5]

M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects, J. Theor. Biol., 215 (2002), 39-46. doi: 10.1006/jtbi.2001.2486.

[6]

K. Gopalsamy, Time lags and global stability in two species competition, Bull Math Biol, 42 (1980), 728-737.

[7]

D. Hadjiavgousti and S. Ichtiaroglou, Allee effect in a predator-prey system, Chaos, Solitons & Fractals, 36 (2008), 334-342. doi: 10.1016/j.chaos.2006.06.053.

[8]

N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.

[9]

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

[10]

H.-F. Huo and W.-T. Li, Existence and global stability of periodic solutions of a discrete predator-prey system with delays, Appl. Math. Comput., 153 (2004), 337-351. doi: 10.1016/S0096-3003(03)00635-0.

[11]

S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Diff. Equ. Appl. ,12 (2006), 165-181. doi: 10.1080/10236190500539238.

[12]

G. Jiang and Q. Lu, Impulsive state feedback of a predator-prey model, J. Comput. Appl. Math., 200 (2007), 193-207. doi: 10.1016/j.cam.2005.12.013.

[13]

S. Krise and SR. Choudhury, Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons & Fractals, 16 (2003), 59-77. doi: 10.1016/S0960-0779(02)00199-6.

[14]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Boston: Academic Press, 1993.

[15]

A. Leung, Periodic solutions for a prey-predator differential delay equation, J. Differential Equations, 26 (1977), 391-403. doi: 10.1016/0022-0396(77)90087-0.

[16]

X. Liao and G. Chen, Hopf bifurcation and chaos analysis of Chen's system with idtribted delays, Chaos, Solitons & Fractals, 25 (2005), 197-220. doi: 10.1016/j.chaos.2004.11.007.

[17]

Z. Liu and R. Yuan, Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos, Solitons & Fractals, 27 (2006), 1395-1407. doi: 10.1016/j.chaos.2005.05.014.

[18]

B. Liu, Z. Teng and L. Chen, Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. Comput. Appl. Math., 193 (2006), 347-362. doi: 10.1016/j.cam.2005.06.023.

[19]

X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32 (2007), 80-94. doi: 10.1016/j.chaos.2005.10.081.

[20]

W. Ma and Y. Takeuchi, Stability analysis on a predator-prey system with disributed delays, J. Comput. Appl. Math., 88 (1998), 79-94. doi: 10.1016/S0377-0427(97)00203-3.

[21]

M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecological Modelling, 103 (1997), 99-102. doi: 10.1016/S0304-3800(97)00104-X.

[22]

J. D. Murray, "Mathematical Biology,'' Springer-Verlag, New York, 1993. doi: 10.1007/b98869.

[23]

S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.

[24]

S. Ruan and J. Wei, Periodic solutions of planar systems with two delays, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017-1032.

[25]

I. Scheuring, Allee effect increases the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.

[26]

C. Sun, M. Han, Y. Lin and Y. Chen, Global qualitative analysis for a predator-prey system with delay, Chaos, Solitons & Fractals, 32 (2007), 1582-1596. doi: 10.1016/j.chaos.2005.11.038.

[27]

Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delays, J. Comput. Appl. Math., 197 (2006), 302-321. doi: 10.1016/j.cam.2005.11.006.

[28]

L.-L. Wang, W.-T. Li and P.-H. Zhao, Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays, Adv. Difference Equ., 4 (2004), 321-336. doi: 10.1155/S1687183904401058.

[29]

F. Wang and G. Zeng, Chaos in Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays, Chaos, Solitons & Fractals, 32 (2007), 1499-1512. doi: 10.1016/j.chaos.2005.11.102.

[30]

X. Wen and Z. Wang, The existence of periodic solutions for some models with delay, Nonlinear Anal. RWA, 3 (2002), 567-581.

[31]

R. Xu and Z. Wang, Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays, J. Comput. Appl. Math., 196 (2006), 70-86. doi: 10.1016/j.cam.2005.08.017.

[32]

X. P. Yan and Y. D. Chu, Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system, J. Comput. Appl. Math., 196 (2006), 198-210. doi: 10.1016/j.cam.2005.09.001.

[33]

J. Yu, K. Zhang, S. Fei and T. Li, Simplified exponential stability analysis for recurrent neural networks with discrete and distributed time-varying delays, Applied Mathematics and Computation, 205 (2008), 465-474. doi: 10.1016/j.amc.2008.08.022.

[34]

S. R. Zhou, Y. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Population Biol., 67 (2005), 23-31.

[35]

L. Zhou and Y. Tang, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, Solitons & Fractals, 14 (2002), 1201-1225. doi: 10.1016/S0960-0779(02)00068-1.

[36]

X. Zhou, Y. Wu, Y. Li and X. Yau, Stability and Hopf Bifurcation analysis on a two neuron network with discrete and distributed delays, Chaos, Solitons & Fractals, 40 (2009), 1493-1505. doi: 10.1016/j.chaos.2007.09.034.

show all references

References:
[1]

C. Çelik, The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos, Solitons & Fractals, 37 (2008), 87-99. doi: 10.1016/j.chaos.2007.10.045.

[2]

C. Çelik, Hopf bifurcation of a ratio-dependent predator-prey system with time delay, Chaos, Solitons & Fractals, 42 (2009), 1474-1484. doi: 10.1016/j.chaos.2009.03.071.

[3]

X. Chen, Periodicity in a nonlinear disrete predator-prey system with state dependent delays, Nonlinear Anal. RWA, 8 (2007), 435-446. doi: 10.1016/j.nonrwa.2005.12.005.

[4]

C. Çelik and O. Duman, Allee effect in a discrete-time predator-prey system, Chaos, Solitons & Fractals, 40 (2009), 1956-1962. doi: 10.1016/j.chaos.2007.09.077.

[5]

M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects, J. Theor. Biol., 215 (2002), 39-46. doi: 10.1006/jtbi.2001.2486.

[6]

K. Gopalsamy, Time lags and global stability in two species competition, Bull Math Biol, 42 (1980), 728-737.

[7]

D. Hadjiavgousti and S. Ichtiaroglou, Allee effect in a predator-prey system, Chaos, Solitons & Fractals, 36 (2008), 334-342. doi: 10.1016/j.chaos.2006.06.053.

[8]

N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.

[9]

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

[10]

H.-F. Huo and W.-T. Li, Existence and global stability of periodic solutions of a discrete predator-prey system with delays, Appl. Math. Comput., 153 (2004), 337-351. doi: 10.1016/S0096-3003(03)00635-0.

[11]

S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Diff. Equ. Appl. ,12 (2006), 165-181. doi: 10.1080/10236190500539238.

[12]

G. Jiang and Q. Lu, Impulsive state feedback of a predator-prey model, J. Comput. Appl. Math., 200 (2007), 193-207. doi: 10.1016/j.cam.2005.12.013.

[13]

S. Krise and SR. Choudhury, Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons & Fractals, 16 (2003), 59-77. doi: 10.1016/S0960-0779(02)00199-6.

[14]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Boston: Academic Press, 1993.

[15]

A. Leung, Periodic solutions for a prey-predator differential delay equation, J. Differential Equations, 26 (1977), 391-403. doi: 10.1016/0022-0396(77)90087-0.

[16]

X. Liao and G. Chen, Hopf bifurcation and chaos analysis of Chen's system with idtribted delays, Chaos, Solitons & Fractals, 25 (2005), 197-220. doi: 10.1016/j.chaos.2004.11.007.

[17]

Z. Liu and R. Yuan, Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos, Solitons & Fractals, 27 (2006), 1395-1407. doi: 10.1016/j.chaos.2005.05.014.

[18]

B. Liu, Z. Teng and L. Chen, Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. Comput. Appl. Math., 193 (2006), 347-362. doi: 10.1016/j.cam.2005.06.023.

[19]

X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32 (2007), 80-94. doi: 10.1016/j.chaos.2005.10.081.

[20]

W. Ma and Y. Takeuchi, Stability analysis on a predator-prey system with disributed delays, J. Comput. Appl. Math., 88 (1998), 79-94. doi: 10.1016/S0377-0427(97)00203-3.

[21]

M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecological Modelling, 103 (1997), 99-102. doi: 10.1016/S0304-3800(97)00104-X.

[22]

J. D. Murray, "Mathematical Biology,'' Springer-Verlag, New York, 1993. doi: 10.1007/b98869.

[23]

S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.

[24]

S. Ruan and J. Wei, Periodic solutions of planar systems with two delays, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017-1032.

[25]

I. Scheuring, Allee effect increases the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.

[26]

C. Sun, M. Han, Y. Lin and Y. Chen, Global qualitative analysis for a predator-prey system with delay, Chaos, Solitons & Fractals, 32 (2007), 1582-1596. doi: 10.1016/j.chaos.2005.11.038.

[27]

Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delays, J. Comput. Appl. Math., 197 (2006), 302-321. doi: 10.1016/j.cam.2005.11.006.

[28]

L.-L. Wang, W.-T. Li and P.-H. Zhao, Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays, Adv. Difference Equ., 4 (2004), 321-336. doi: 10.1155/S1687183904401058.

[29]

F. Wang and G. Zeng, Chaos in Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays, Chaos, Solitons & Fractals, 32 (2007), 1499-1512. doi: 10.1016/j.chaos.2005.11.102.

[30]

X. Wen and Z. Wang, The existence of periodic solutions for some models with delay, Nonlinear Anal. RWA, 3 (2002), 567-581.

[31]

R. Xu and Z. Wang, Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays, J. Comput. Appl. Math., 196 (2006), 70-86. doi: 10.1016/j.cam.2005.08.017.

[32]

X. P. Yan and Y. D. Chu, Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system, J. Comput. Appl. Math., 196 (2006), 198-210. doi: 10.1016/j.cam.2005.09.001.

[33]

J. Yu, K. Zhang, S. Fei and T. Li, Simplified exponential stability analysis for recurrent neural networks with discrete and distributed time-varying delays, Applied Mathematics and Computation, 205 (2008), 465-474. doi: 10.1016/j.amc.2008.08.022.

[34]

S. R. Zhou, Y. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Population Biol., 67 (2005), 23-31.

[35]

L. Zhou and Y. Tang, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, Solitons & Fractals, 14 (2002), 1201-1225. doi: 10.1016/S0960-0779(02)00068-1.

[36]

X. Zhou, Y. Wu, Y. Li and X. Yau, Stability and Hopf Bifurcation analysis on a two neuron network with discrete and distributed delays, Chaos, Solitons & Fractals, 40 (2009), 1493-1505. doi: 10.1016/j.chaos.2007.09.034.

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