-
Previous Article
Existence of radial stationary solutions for a system in combustion theory
- DCDS-B Home
- This Issue
-
Next Article
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.
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.
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.
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.
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.
doi: 10.1006/jtbi.2001.2486. |
[6] |
K. Gopalsamy, Time lags and global stability in two species competition,, Bull Math Biol, 42 (1980), 728.
|
[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. |
[8] |
N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'', Theory and Applications of Hopf Bifurcation, (1981).
|
[9] |
X. He, Stability and delays in a predator-prey system,, J. Math. Anal. Appl., 198 (1996), 355.
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.
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.
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.
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.
doi: 10.1016/S0960-0779(02)00199-6. |
[14] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Delay Differential Equations with Applications in Population Dynamics, (1993).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. D. Murray, "Mathematical Biology,'', Mathematical Biology, (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.
|
[24] |
S. Ruan and J. Wei, Periodic solutions of planar systems with two delays,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017.
|
[25] |
I. Scheuring, Allee effect increases the dynamical stability of populations,, J. Theor. Biol., 199 (1999), 407.
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.
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.
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.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
K. Gopalsamy, Time lags and global stability in two species competition,, Bull Math Biol, 42 (1980), 728.
|
[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. |
[8] |
N. D. Hassard and Y. H. Kazarinoff, "Theory and Applications of Hopf Bifurcation,'', Theory and Applications of Hopf Bifurcation, (1981).
|
[9] |
X. He, Stability and delays in a predator-prey system,, J. Math. Anal. Appl., 198 (1996), 355.
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.
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.
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.
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.
doi: 10.1016/S0960-0779(02)00199-6. |
[14] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Delay Differential Equations with Applications in Population Dynamics, (1993).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. D. Murray, "Mathematical Biology,'', Mathematical Biology, (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.
|
[24] |
S. Ruan and J. Wei, Periodic solutions of planar systems with two delays,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017.
|
[25] |
I. Scheuring, Allee effect increases the dynamical stability of populations,, J. Theor. Biol., 199 (1999), 407.
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.
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.
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.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[1] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[2] |
Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263 |
[3] |
Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020468 |
[4] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[5] |
Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148 |
[6] |
Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021007 |
[7] |
Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328 |
[8] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[9] |
Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 |
[10] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[11] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[12] |
Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020342 |
[13] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[14] |
Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251 |
[15] |
Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 |
[16] |
Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 |
[17] |
Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 |
[18] |
Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020341 |
[19] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[20] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]