# American Institute of Mathematical Sciences

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 & 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.  doi: 10.1016/j.chaos.2007.10.045.  Google Scholar [2] C. Çelik, Hopf bifurcation of a ratio-dependent predator-prey system with time delay,, Chaos, Solitons & Fractals, 42 (2009), 1474.  doi: 10.1016/j.chaos.2009.03.071.  Google Scholar [3] X. Chen, Periodicity in a nonlinear disrete predator-prey system with state dependent delays,, Nonlinear Anal. RWA, 8 (2007), 435.  doi: 10.1016/j.nonrwa.2005.12.005.  Google Scholar [4] C. Çelik and O. Duman, Allee effect in a discrete-time predator-prey system,, Chaos, Solitons & Fractals, 40 (2009), 1956.  doi: 10.1016/j.chaos.2007.09.077.  Google Scholar [5] M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects,, J. Theor. Biol., 215 (2002), 39.  doi: 10.1006/jtbi.2001.2486.  Google Scholar [6] K. Gopalsamy, Time lags and global stability in two species competition,, Bull Math Biol, 42 (1980), 728.   Google Scholar [7] D. Hadjiavgousti and S. Ichtiaroglou, Allee effect in a predator-prey system,, Chaos, Solitons & Fractals, 36 (2008), 334.  doi: 10.1016/j.chaos.2006.06.053.  Google Scholar [8] N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'', Theory and Applications of Hopf Bifurcation, (1981).   Google Scholar [9] X. He, Stability and delays in a predator-prey system,, J. Math. Anal. Appl., 198 (1996), 355.  doi: 10.1006/jmaa.1996.0087.  Google Scholar [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.  doi: 10.1016/S0096-3003(03)00635-0.  Google Scholar [11] S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model,, J. Diff. Equ. Appl., 12 (2006), 165.  doi: 10.1080/10236190500539238.  Google Scholar [12] G. Jiang and Q. Lu, Impulsive state feedback of a predator-prey model,, J. Comput. Appl. Math., 200 (2007), 193.  doi: 10.1016/j.cam.2005.12.013.  Google Scholar [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.  doi: 10.1016/S0960-0779(02)00199-6.  Google Scholar [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Delay Differential Equations with Applications in Population Dynamics, (1993).   Google Scholar [15] A. Leung, Periodic solutions for a prey-predator differential delay equation,, J. Differential Equations, 26 (1977), 391.  doi: 10.1016/0022-0396(77)90087-0.  Google Scholar [16] X. Liao and G. Chen, Hopf bifurcation and chaos analysis of Chen's system with idtribted delays,, Chaos, Solitons & Fractals, 25 (2005), 197.  doi: 10.1016/j.chaos.2004.11.007.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.05.014.  Google Scholar [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.  doi: 10.1016/j.cam.2005.06.023.  Google Scholar [19] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system,, Chaos, Solitons & Fractals, 32 (2007), 80.  doi: 10.1016/j.chaos.2005.10.081.  Google Scholar [20] W. Ma and Y. Takeuchi, Stability analysis on a predator-prey system with disributed delays,, J. Comput. Appl. Math., 88 (1998), 79.  doi: 10.1016/S0377-0427(97)00203-3.  Google Scholar [21] M. A. McCarthy, The Allee effect, finding mates and theoretical models,, Ecological Modelling, 103 (1997), 99.  doi: 10.1016/S0304-3800(97)00104-X.  Google Scholar [22] J. D. Murray, "Mathematical Biology,'', Mathematical Biology, (1993).  doi: 10.1007/b98869.  Google Scholar [23] S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,, Quart. Appl. Math., 59 (2001), 159.   Google Scholar [24] S. Ruan and J. Wei, Periodic solutions of planar systems with two delays,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017.   Google Scholar [25] I. Scheuring, Allee effect increases the dynamical stability of populations,, J. Theor. Biol., 199 (1999), 407.  doi: 10.1006/jtbi.1999.0966.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.11.038.  Google Scholar [27] Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delays,, J. Comput. Appl. Math., 197 (2006), 302.  doi: 10.1016/j.cam.2005.11.006.  Google Scholar [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.  doi: 10.1155/S1687183904401058.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.11.102.  Google Scholar [30] X. Wen and Z. Wang, The existence of periodic solutions for some models with delay,, Nonlinear Anal. RWA, 3 (2002), 567.   Google Scholar [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.  doi: 10.1016/j.cam.2005.08.017.  Google Scholar [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.  doi: 10.1016/j.cam.2005.09.001.  Google Scholar [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.  doi: 10.1016/j.amc.2008.08.022.  Google Scholar [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.   Google Scholar [35] L. Zhou and Y. Tang, Stability and Hopf bifurcation for a delay competition diffusion system,, Chaos, Solitons & Fractals, 14 (2002), 1201.  doi: 10.1016/S0960-0779(02)00068-1.  Google Scholar [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.  doi: 10.1016/j.chaos.2007.09.034.  Google Scholar

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.  doi: 10.1016/j.chaos.2007.10.045.  Google Scholar [2] C. Çelik, Hopf bifurcation of a ratio-dependent predator-prey system with time delay,, Chaos, Solitons & Fractals, 42 (2009), 1474.  doi: 10.1016/j.chaos.2009.03.071.  Google Scholar [3] X. Chen, Periodicity in a nonlinear disrete predator-prey system with state dependent delays,, Nonlinear Anal. RWA, 8 (2007), 435.  doi: 10.1016/j.nonrwa.2005.12.005.  Google Scholar [4] C. Çelik and O. Duman, Allee effect in a discrete-time predator-prey system,, Chaos, Solitons & Fractals, 40 (2009), 1956.  doi: 10.1016/j.chaos.2007.09.077.  Google Scholar [5] M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects,, J. Theor. Biol., 215 (2002), 39.  doi: 10.1006/jtbi.2001.2486.  Google Scholar [6] K. Gopalsamy, Time lags and global stability in two species competition,, Bull Math Biol, 42 (1980), 728.   Google Scholar [7] D. Hadjiavgousti and S. Ichtiaroglou, Allee effect in a predator-prey system,, Chaos, Solitons & Fractals, 36 (2008), 334.  doi: 10.1016/j.chaos.2006.06.053.  Google Scholar [8] N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'', Theory and Applications of Hopf Bifurcation, (1981).   Google Scholar [9] X. He, Stability and delays in a predator-prey system,, J. Math. Anal. Appl., 198 (1996), 355.  doi: 10.1006/jmaa.1996.0087.  Google Scholar [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.  doi: 10.1016/S0096-3003(03)00635-0.  Google Scholar [11] S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model,, J. Diff. Equ. Appl., 12 (2006), 165.  doi: 10.1080/10236190500539238.  Google Scholar [12] G. Jiang and Q. Lu, Impulsive state feedback of a predator-prey model,, J. Comput. Appl. Math., 200 (2007), 193.  doi: 10.1016/j.cam.2005.12.013.  Google Scholar [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.  doi: 10.1016/S0960-0779(02)00199-6.  Google Scholar [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Delay Differential Equations with Applications in Population Dynamics, (1993).   Google Scholar [15] A. Leung, Periodic solutions for a prey-predator differential delay equation,, J. Differential Equations, 26 (1977), 391.  doi: 10.1016/0022-0396(77)90087-0.  Google Scholar [16] X. Liao and G. Chen, Hopf bifurcation and chaos analysis of Chen's system with idtribted delays,, Chaos, Solitons & Fractals, 25 (2005), 197.  doi: 10.1016/j.chaos.2004.11.007.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.05.014.  Google Scholar [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.  doi: 10.1016/j.cam.2005.06.023.  Google Scholar [19] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system,, Chaos, Solitons & Fractals, 32 (2007), 80.  doi: 10.1016/j.chaos.2005.10.081.  Google Scholar [20] W. Ma and Y. Takeuchi, Stability analysis on a predator-prey system with disributed delays,, J. Comput. Appl. Math., 88 (1998), 79.  doi: 10.1016/S0377-0427(97)00203-3.  Google Scholar [21] M. A. McCarthy, The Allee effect, finding mates and theoretical models,, Ecological Modelling, 103 (1997), 99.  doi: 10.1016/S0304-3800(97)00104-X.  Google Scholar [22] J. D. Murray, "Mathematical Biology,'', Mathematical Biology, (1993).  doi: 10.1007/b98869.  Google Scholar [23] S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,, Quart. Appl. Math., 59 (2001), 159.   Google Scholar [24] S. Ruan and J. Wei, Periodic solutions of planar systems with two delays,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017.   Google Scholar [25] I. Scheuring, Allee effect increases the dynamical stability of populations,, J. Theor. Biol., 199 (1999), 407.  doi: 10.1006/jtbi.1999.0966.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.11.038.  Google Scholar [27] Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delays,, J. Comput. Appl. Math., 197 (2006), 302.  doi: 10.1016/j.cam.2005.11.006.  Google Scholar [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.  doi: 10.1155/S1687183904401058.  Google Scholar [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.  doi: 10.1016/j.chaos.2005.11.102.  Google Scholar [30] X. Wen and Z. Wang, The existence of periodic solutions for some models with delay,, Nonlinear Anal. RWA, 3 (2002), 567.   Google Scholar [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.  doi: 10.1016/j.cam.2005.08.017.  Google Scholar [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.  doi: 10.1016/j.cam.2005.09.001.  Google Scholar [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.  doi: 10.1016/j.amc.2008.08.022.  Google Scholar [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.   Google Scholar [35] L. Zhou and Y. Tang, Stability and Hopf bifurcation for a delay competition diffusion system,, Chaos, Solitons & Fractals, 14 (2002), 1201.  doi: 10.1016/S0960-0779(02)00068-1.  Google Scholar [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.  doi: 10.1016/j.chaos.2007.09.034.  Google Scholar
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