| Citation: |
Jinting Lin, Changjin Xu, Yingyan Zhao, Yicheng Pang, Zixin Liu, Jianwei Shen. BIFURCATION AND CONTROL OF A PREDATOR-PREY SYSTEM WITH TWO TIME DELAYS[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 807-859. doi: 10.11948/20250164
|
BIFURCATION AND CONTROL OF A PREDATOR-PREY SYSTEM WITH TWO TIME DELAYS
-
Abstract
To describe the dynamic behavior within ecosystems, we develop a novel predator-prey model that incorporates two distinct time delays in this research. Firstly, by employing fixed-point theorem, inequality technique, and construct an appropriate function, the trait (e.g., existence, uniqueness, non-negativity) of the solution of this delayed predator-prey model is proved. Secondly, the critical value of time delay of this system's stability is sought and the stability and Hopf bifurcation of this model are investigated. A new delay-independent criterion for the stability and bifurcation of this model is presented. Lastly, three delay feedback controllers are designed to adjust the stability region and the onset of bifurcation phenomena of this model. The parameter conditions of this system to maintain stability under the delay feedback controller and the bifurcation phenomenon are discussed. Furthermore, the correctness of the conclusions derived from this study is verified through simulation experiments.
-
Keywords:
- Predator-prey model /
- time delay /
- Hopf bifurcation /
- stability /
- delay feedback controller
-
-
References
[1] N. Bairagi and D. Jana, On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity, Appl. Math. Model., 2011, 35, 3255–3267. [2] Q. Y. Cui, C. J. Xu, W. Ou, Y. C. Pang, Z. X. Liu, P. L. Li and L. Y. Yao, Bifurcation behavior and hybrid controller design of a 2D Lotka-Volterra commensal symbiosis system accompanying delay, Math., 2023, 11, 4808. [3] Y. Q. Gao and N. Li, Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays, J. Comput. Appl. Math., 2023, 83, 1–18. [4] X. He, C. D. Li, T. W. Huang and J. Z. Yu, Bifurcation behaviors of an Euler discretized inertial delayed neuron model, Sci. China Tech. Sci., 2016, 59, 418–427. [5] C. Huang, H. Li and J. Cao, A novel strategy of bifurcation control for a delayed fractional predator–prey model, Appl. Math. Comput., 2019, 347, 808–838. [6] C. Huang, H. Liu, X. P. Chen, M. Zhang, L. Ding, J. Cao and A. Alsaedi, Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator-prey model, Phys. A: Stat. Mech. Appl., 2020, 554, 124136. [7] C. D. Huang, H. Liu, X. P. Chen, J. D. Cao and A. Alsaedie, Extended feedback and simulation strategies for a delayed fractional-order control system, Phys. A Stat. Mech. Appl., 2020, 545, 123127. [8] C. D. Huang, R. X. Wu, H. Li, T. X. Li and S. J. Chen, Stability and bifurcation control in a fractional predator-prey model via extended delay feedback, Int. J. Bifur. Chaos, 2019, 29, 1950150. [9] X. W. Jiang, X. Y. Chen, T. W. Huang and H. C. Yan, Bifurcation and control for a predator-prey system with two delays, IEEE Trans. Circuits Syst., 2021, 68, 376–380. [10] Y. Z. Lan, J. P. Shi and H. Fang, Hopf bifurcation and control of a fractional-order delay stage structure prey-predator model with two fear effects and prey refuge, Symmetry, 2022, 14, 1408. [11] H. L. Li, L. Zhang, C. Hu, Y. L. Jiang and Z. D. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 2017, 54, 435–449. [12] H. R. Li, Y. L. Tian, T. Huang and P. H. Yang, Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model, AIMS Math., 2024, 9, 29608–29632. [13] N. Li and M. T. Yan, Bifurcation control of a delayed fractional-order prey-predator model with cannibalism and disease, Annalen Der Physik, 2022, 600, 127600. [14] P. L. Li, R. Gao, C. J. Xu, Y. Li, A. Akgül and D. Baleanu, Dynamics exploration for a fractional-order delayed zooplankton-phytoplankton system, Chaos Solitons Fractals, 2023, 166, 112975. [15] P. L. Li, Y. J. Lu, C. J. Xu and J. Ren, Bifurcation phenomenon and control technique in fractional BAM neural network models concerning delays, Fract. Fract., 2022, 7, 7. [16] P. L. Li, Y. J. Lu, C. J. Xu and J. Ren, Dynamic exploration and control of bifurcation in a fractional-order Lengyel-Epstein model owing time delay, MATCH Commun. Math. Comput. Chem., 2024, 92(2), 437–482. [17] P. L. Li, J. L. Yan, C. J. Xu, R. Gao and Y. Li, Understanding dynamics and bifurcation control mechanism for a fractional-order delayed duopoly game model in insurance market, Fract. Fract., 2022, 6, 270. [18] S. Li, C. D. Huang, S. Guo and X. Y. Song, Fractional modeling and control in a delayed predator-prey system: Extended feedback scheme, Adv. Diff. Equ., 2020, 1, 1–18. [19] J. T. Lin, C. J. Xu, Y. Y. Xu, Y. Y. Zhao, Y. C. Pang, Z. X. Liu and J. W. Shen, Bifurcation and controller design in a 3D delayed predator-prey model, AIMS Math., 2024, 9(12), 33891–33929. [20] J. T. Lin, C. J. Xu, Y. Y. Zhao and Q. W. Deng, Hopf bifurcation and controller design for a predator-prey model with double delays, AIP Adv., 2025, 15(7), 075329. [21] Y. Y. Liu and J. J Wei, Double hopf bifurcation of a diffusive predator-prey system with strong allee effect and two delays, Nonlinear Anal. : Model. Conrol, 2021, 26, 72–92. [22] K. Mokni, M. Ch-Chaoui, B. Mondal and U. Ghosh, Hybrid control of Hopf bifurcation in a Lotka-Volterra predator-prey model with two delays, Adv. Diff. Equ., 2017, 2017, 387. [23] W. Ou, C. J. Xu, Q. Y. Cui, Z. X. Liu, Y. C. Pang and M. Farman, Bifurcation dynamics and control mechanism of a fractional-order delayed brusselator chemical reaction model, MATCH Commun. Math. Comput. Chem., 2023, 89(1), 73–106. [24] W. Ou, C. J. Xu, Q. Y. Cui, Z. X. Liu, Y. C. Pang, M. Farman, S. Ahmad and A. Zeb, Mathematical study on bifurcation dynamics and control mechanism of tri-neuron bidirectional associative memory neural networks including delay, Math. Meth. Appl. Sci., 2025, 48(7), 7820–7844. [25] A. Mahmoud Abd-Rabo, Y. W. Tao, Q. G. Yuan and S. Mohamed, Bifurcation analysis of glucose model with obesity effect, Alexandria Eng. J., 2021, 60, 4919–4930. [26] M. Rakshana and P. Balasubramaniam, Dynamical analysis of non-electric guitar-like instruments with single saddle via Hopf bifurcation, The Eur. Phys. J. Special Top., 2024, 234(8), 1–9. [27] M. Rakshana and P. Balasubramaniam, Hopf bifurcation control of memristor-based fractional delayed tri-diagonal bidirectional associative memory neural networks under various controllers, Commun. Nonlinear Sci. Numer. Simul., 2025, 140, 108440. [28] M. Rakshana and P. Balasubramaniam, Hopf bifurcation of general fractional delayed TdBAM neural networks, Neural Process. Lett., 2023, 55(6), 8095–8113. [29] M. Rakshana and P. Balasubramaniam, Washout filter-based Hopf bifurcation anti-control of fractional TdBAM neural networks, Int. J. Dyn. Control, 2025, 13(4), 1–13. [30] S. Sharma and G. P. Samanta, Dynamical behaviour of a two prey and one predator system, Diff. Equ. Dyn. Syst., 2014, 22, 125–145. [31] Y. L. Tang and F. Li, Multiple stable states for a class of predator-prey systems with two harvesting rates, J. Appl. Anal. Comput., 2024, 14, 506–514. [32] Y. S. Wu and F. Li, Weak centers and local bifurcation of critical periods in a Z2-equivariant vector field of degree 5, Int. J. Bifur. Chaos, 2023, 33(3), 2350029. [33] R. T. Xing, M. Xiao, Y. Z. Zhang and J. L. Qiu, Stability and Hopf bifurcation analysis of an (n plus m)-neuron double-ring neural network model with multiple time delays, J. Syst. Sci. Complex., 2022, 35, 159–178. [34] C. J. Xu, C. Aouiti, Z. X. Liu, P. L. Li and L. Y. Yao, Bifurcation caused by delay in a fractional-order coupled oregonator model in chemistry, MATCH Commun. Math. Comput. Chem., 2022, 88, 371–396. [35] C. J. Xu, D. Mu, Z. X. Liu, Y. C. Pang, M. X. Liao and C. Aouiti, New insight into bifurcation of fractional-order 4D neural networks incorporating two different time delays, Commun. Nonlinear Sci. Numer. Simul., 2023, 118, 107043. [36] C. J. Xu, D. Mu, Y. L. Pan, C. Aouiti and L. Y. Yao, Exploring bifurcation in a fractional-order predator-prey system with mixed delays, App. Math. Lett., 2023, 13, 1119–1136. [37] C. J. Xu, W. Ou, Y. C. Pang, Q. Y. Cui, M. U. Rahman, M. Farman, S. Ahmad and A. Zeb, Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller, MATCH Commun. Math. Comput. Chem., 2024, 91, 367–413. [38] M. R. Xu, S. Liu and Y. Lou, Persistence and extinction in the anti-symmetric Lotka-Volterra systems, J, Diff, Equ., 2024, 387, 299–323. [39] L. L. Zhang and L. H. Huang, Hopf bifurcation analysis for a maglev system with two time delays, Mech. Syst. Sign. Process., 2025, 224, 112006. [40] L. P. Zhang, H. N. Wang and M. Xu, Hopf bifurcation and control for the delayed predator-prey model with nonlinear prey harvesting, J. Appl. Anal. Comput., 2024, 14, 2954–2976. [41] L. P. Zhang, H. N. Wang and M. Xu, Hybrid control of bifurcation in a predator-prey system with three delays, Acta Phys. Sin., 2011, 60, 010506. [42] R. Y. Zhang, Bifurcation analysis for T system with delayed feedback and its application to control of chaos, Nonlinear Dyn., 2013, 72, 629–641. [43] Z. Z. Zhang and H. Z. Yang, Hybrid control of Hopf bifurcation in a two prey one predator system with time delay, in Proceeding of the 33rd Chinese Control Conference, Nanjing, China, 2014, 6895–6900. [44] L. H. Zhu, X. W. Wang, H. H. Zhang, S. L. Shen, Y. M. Li and Y. D. Zhou, Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting, Physica Scripta, 2020, 95, 210–215. [45] L. H. Zhu, X. Zhou, Y. M. Li, Y. X. Zhu and M. X. Liao, Stability and bifurcation analysis on a delayed epidemic model with information-dependent vaccination, Phys. Scr., 2019, 94, 125202. -
-
Figures(16)
Export File
Citation
Jinting Lin, Changjin Xu, Yingyan Zhao, Yicheng Pang, Zixin Liu, Jianwei Shen. BIFURCATION AND CONTROL OF A PREDATOR-PREY SYSTEM WITH TWO TIME DELAYS[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 807-859. doi: 10.11948/20250164
Format
Content
DownLoad:
-
Figure 1.
Matlab simulation graphs of system (7.1) under the delay
$ \tau_1>0, \tau_2=0 $ $ \tau_1=2.0<\tau_{1}^{\ast}=2.4 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 2.
Matlab simulation figures of system (7.1) under the delay
$ \tau_1>0, \tau_2=0 $ $ \tau_1=2.7>\tau_{1}^{\ast}=2.4 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 3.
Matlab simulation graphs of system (7.2) under the delay
$ \tau_1=0, \tau_2>0 $ $ \tau_2=0.1<\tau_{2}^{\ast}=0.4 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 4.
Matlab simulation figures of system (7.2) under the delay
$ \tau_1=0, \tau_2>0 $ $ \tau_2=0.8>\tau_{2}^{\ast}=0.4 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 5.
Matlab simulation graphs of system (7.3) under the delay
$ \tau_1=\tau_2=\tau $ $ \tau=0.1<\tau^{\ast}=0.5 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 6.
Matlab simulation plots of system (7.3) under the delay
$ \tau_1=\tau_2=\tau $ $ \tau=0.92>\tau^{\ast}=0.5 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 7.
Matlab simulation graphs of system (7.4) under the delay
$ \tau_1=2.0, \tau_2=0.2 $ $ \tau_1=2.0<\tau_{1}^{\star}=2.2. $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 8.
Matlab simulation figures of system (7.4) under the delay
$ \tau_1=2.5, \tau_2=0.2 $ $ \tau_1=2.5>\tau_{1}^{\star}=2.2 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 9.
Matlab simulation graphs of system (7.5) under the delay
$ \tau_1=1.8, \tau_2=0.3 $ $ \tau_2=0.3<\tau_{2}\star=0.6. $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 10.
Matlab simulation figures of system (7.5) under the delay
$ \tau_1=1.8, \tau_2=0.2 $ $ \tau_2=0.8>\tau_{2}^{\star}=0.6 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 11.
Matlab simulation graphs of system (7.6) under the delay
$ \tau_1=\tau_2=\tau $ $ \tau=0.55<\tau_{(0)}=0.8. $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 12.
Matlab simulation figures of system (7.6) under the delay
$ \tau_1=\tau_2=\tau $ $ \tau=2.0>\tau_{(0)}=0.8 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 13.
Matlab simulation graphs of system (7.7) under the delay
$ \tau_1=2.0, \tau_2=0 $ $ \tau_1=2.0<\tau_1^{\iota}=2.5. $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 14.
Matlab simulation figures of system (7.7) under the delay
$ \tau_1=2.0, \tau_2=0 $ $ \tau_1=2.7>\tau_{1}^{\iota}=2.5 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 15.
Matlab simulation graphs of system (7.8) under the delay
$ \tau_1=0, \tau_2=0.12 $ $ \tau_2=0.12<\tau_2^{\iota}=0.4 $ $ E(0.1283, 1.3227, 0.4333) $ -
Figure 16.
Matlab simulation figures of system (7.8) under the delay
$ \tau_1=0, \tau_2=1.1 $ $ \tau_2=1.1>\tau_{2}^{\iota}=0.4 $ $ E(0.1283, 1.3227, 0.4333) $