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Almost periodic and asymptotically almost periodic solutions of Liénard equations
Dynamical behavior of a ratio dependent predator-prey system with distributed delay
1. | Department of Mathematics and Computer Sciences, Bahçeşehir University, Istanbul, 34353, Turkey |
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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