2023 Volume 13 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(3): 1119-1136

Next Article Previous Article

Changjin Xu, Dan Mu, Yuanlu Pan, Chaouki Aouiti, Lingyun Yao. EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1119-1136. doi: 10.11948/20210313

Citation:

Changjin Xu, Dan Mu, Yuanlu Pan, Chaouki Aouiti, Lingyun Yao. EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1119-1136. doi: 10.11948/20210313

EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS

1.
Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
3.
Library, Guizhou University of Finance and Economics, Guiyang 550025, China
4.
Faculty of Sciences of Bizerta, UR13ES47 Research Units of Mathematics and Applications, University of Carthage, Bizerta 7021, Tunisia

Author Bio: Email address: md980929@126.com(D. Mu); Email address: 137415148@qq.com(Y. Pan); Email address: chaouki.aouiti@fsb.rnu.tn(C. Aouiti); Email address: lingyunyao2015@126.com((L. Yao)
Corresponding author: Email address: xcj403@126.com(C. Xu)
Fund Project: The authors were supported by National Natural Science Foundation of China(12261015, 62062018), Project of High-level Innovative Talents of Guizhou Province([2016]5651), Guizhou Key Laboratory of Big Data Statistical Analysis([2019]5103), Basic Research Program of Guizhou Province(ZK[2022]025), Natural Science Project of the Education Department of Guizhou Province(KY[2021]031), Key Project of Hunan Education Department(17A181), University Science and Technology Top Talents Project of Guizhou Province(KY[2018]047), Foundation of Science and Technology of Guizhou Province([2019]1051), Guizhou University of Finance and Economics(2018XZD01)

Abstract

Abstract

This work chiefly develops and discusses a fractional-order predator-prey model with distributed delay and discrete delay. Applying skilly an appropriate variable substitution, a novel equivalent form of the fractional-order predator-prey model with distributed delay and discrete delay is derived. By virtue of the stability theorem and bifurcation principle of fractional-order dynamical system, we establish a delay-independent stability and bifurcation criterion ensuring the stability and the onset of Hopf bifurcation for the involved predator-prey system. The role of the time delay in stabilizing system and controlling Hopf bifurcation of the considered fractional-order predator-prey model is displayed. Software simulation results are presented to support the key theoretical fruits.
- Fractional-order predator-prey model /
- stability /
- Hopf bifurcation /
- discrete delay /
- distributed delay
MSC: 34A08, 92B20, 93D05, 34C23, 26A33, 34K18, 37GK15, 39A11

FullText(HTML)

References

[1]	J. Alidousti, Stability and bifurcation analysis for a fractional prey-predator scavenger model, Appl. Math. Model., 2020, 81, 342–355. doi: 10.1016/j.apm.2019.11.025 CrossRef Google Scholar
[2]	B. Bandyopadhyay and S. Kamal, Stabliization and Control of Fractional Order Systems: A Sliding Mode Approach, Springer, Heidelberg, 2015, 317. Google Scholar
[3]	D. Barman, J. Roy, H. Alrabaiah, P. Panja, S. P. Mondal and S. Alam, Impact of predator incited fear and prey refuge in a fractional order prey predator model, Chaos Solitons Fractals, 2021, 142, 110420. doi: 10.1016/j.chaos.2020.110420 CrossRef Google Scholar
[4]	W. Deng, C. Li and J. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 2007, 48(4), 409–416. doi: 10.1007/s11071-006-9094-0 CrossRef Google Scholar
[5]	A. S. Deshpande, V. Daftardar-Gejji and Y. V. Sukale, On Hopf bifurcation in fractional dynamical systems, Chaos Solitons Fractals, 2017, 98, 189–198. doi: 10.1016/j.chaos.2017.03.034 CrossRef Google Scholar
[6]	B. Ghanbari and S. Djilali, Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population, Chaos Solitons Fractals, 2020, 138, 109960. doi: 10.1016/j.chaos.2020.109960 CrossRef Google Scholar
[7]	H. Guo and L. Chen, The effects of impulsive harvest on a predator-prey system with distributed time delay, Commun. Nonlinear Sci. Numerical Simul., 2009, 14(5), 2301–2309. doi: 10.1016/j.cnsns.2008.05.010 CrossRef Google Scholar
[8]	C. Huang and J. Cao, Comparative study on bifurcation control methods in a fractional-order delayed predator-prey system, Sci. China Tech. Sci., 2019, 62(2), 298–307. doi: 10.1007/s11431-017-9196-4 CrossRef Google Scholar
[9]	C. Huang, H. Liu, Y. Chen, X. Chen and F. Song, Dynamics of a fractional-order BAM neural network with leakage delay and communication delay, Fractals, 2021, 29(03), 2150073. doi: 10.1142/S0218348X21500730 CrossRef Google Scholar
[10]	C. Huang, H. Liu, X. 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. doi: 10.1016/j.physa.2020.124136 CrossRef Google Scholar
[11]	C. Huang, J. Wang, X. Chen and J. Cao, Bifurcations in a fractional-order BAM neural network with four different delays, Neural Netw., 2021, 141, 344–354. doi: 10.1016/j.neunet.2021.04.005 CrossRef Google Scholar
[12]	H. Li, C. Huang and T. Li, Dynamic complexity of a fractional-order predator-prey system with double delays, Phys. A: Stat. Mech. Appl., 2019, 526, 120852. doi: 10.1016/j.physa.2019.04.088 CrossRef Google Scholar
[13]	G. Lin and R. Yuan, Periodic solution for a predator-prey system with distributed delay, Math. Comput. Model., 2005, 42(9–10), 959–966. doi: 10.1016/j.mcm.2005.05.015 CrossRef Google Scholar
[14]	M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal.: Hybrid Syst., 2018, 28, 87–104. doi: 10.1016/j.nahs.2017.10.004 CrossRef Google Scholar
[15]	Q. Liu, D. Jiang, X. He, T. Hayat and A. Alsaedi, Stationary distribution of a stochastic predator-prey model with distributed delay and general functional response, Phys. A: Stat. Mech. Appl., 2019, 513, 273–287. doi: 10.1016/j.physa.2018.09.033 CrossRef Google Scholar
[16]	D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl., 1996, 2, 963–968. Google Scholar
[17]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. Google Scholar
[18]	Y. Saito, Permanence and global stability for general Lotka-Volterra predator-prey systems with distributed delays, Nonlinear Anal.: TMA, 2001, 47(9), 6157–6168. doi: 10.1016/S0362-546X(01)00680-0 CrossRef Google Scholar
[19]	S. Wang, G. Hu, T. Wei and L. Wang, Stability in distribution of a stochastic predator-prey system with S-type distributed time delays, Phys. A: Stat. Mech. Appl., 2018, 505, 919–930. doi: 10.1016/j.physa.2018.03.078 CrossRef Google Scholar
[20]	X. Wang, Z. Wang and J. Xia, Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders, Jouranl of the Franklin Institute 2019, 356(15), 8278–8295. doi: 10.1016/j.jfranklin.2019.07.028 CrossRef Google Scholar
[21]	M. Xiao, W. Zheng, J. Lin, G. Jiang, L. Zhao and J. Cao, Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks, J. Franklin Inst., 2017, 354(17), 7643–7667. doi: 10.1016/j.jfranklin.2017.09.009 CrossRef Google Scholar
[22]	C. Xu and C. Aouiti, Comparative analysis on Hopf bifurcation of integer order and fractional order two-neuron neural networks with delay, Int. J. Circuit Theory Appl., 2020, 48(9), 1459–1475. doi: 10.1002/cta.2847 CrossRef Google Scholar
[23]	C. Xu, L. Chen, P. Li and Y. Guo, Oscillatory dynamics in a discrete predator-prey model with distributed delays, PLOS ONE, 2018, 13(12), e0108322. Google Scholar
[24]	C. Xu, M. Liao, P. Li, Y. Guo and Z. Liu, Bifurcation properties for fractional order delayed BAM neural networks, Cogn. Comput., 2021, 13(2), 322–356. doi: 10.1007/s12559-020-09782-w CrossRef Google Scholar
[25]	C. Xu, Z. Liu, C. Aouiti, P. Li, L. Yao and J. Yan, New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays, Cogn. Neurodyn., 2022, doi: https://doi.org/10.1007/s11571-021-09763-1. CrossRef Google Scholar
[26]	C. Xu, Z. Liu, M. Liao, P. Li, Q. Xiao and S. Yuan, Fractional-order bidirectional associate memory (BAM) neural networks with multiple delays: The case of Hopf bifurcation, Math. Comput. Simul., 2021, 182, 471–494. doi: 10.1016/j.matcom.2020.11.023 CrossRef Google Scholar
[27]	C. Xu, Z. Liu, M. Liao and L. Yao, Theoretical analysis and computer simulations of a fractional order bank data model incorporating two unequal time delays, Expert Syst. Appl., 2022, 199, 116859. doi: 10.1016/j.eswa.2022.116859 CrossRef Google Scholar
[28]	C. Xu, D. Mu, Z. Liu, Y. Pang, M. Liao, P. Li, L. Yao and Q. Qin, Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks, Nonlinear Anal. : Model. Control, 2022, doi: https://doi.org/10.15388/namc.2022.27.28491. CrossRef Google Scholar
[29]	C. Xu and Y. Shao, Bifurcations in a predator-prey model with discrete and distributed time delay, Nonlinear Dyn., 2012, 67(3), 2207–2223. doi: 10.1007/s11071-011-0140-1 CrossRef Google Scholar
[30]	C. Xu, W. Zhang, C. Aouiti, Z. Liu, M. Liao and P. Li, Further investigation on bifurcation and their control of fractional-order BAM neural networks involving four neurons and multiple delays, Mathematical Methods in the Applied Sciences, 2021. doi: https://doi.org/10.1002/mma.7581. CrossRef Google Scholar
[31]	C. Xu, W. Zhang, C. Aouiti, Z. Liu and L. Yao, Further analysis on dynamical properties of fractional-order bi-directional associative memory neural networks involving double delays, Math. Meth. Appl. Sci., 2022, doi: 10.1002/mma.8477. CrossRef Google Scholar
[32]	C. Xu, W. Zhang, Z. Liu, P. Li and L. Yao, Bifurcation study for fractional-order three-layer neural networks involving four time delays, Cogn. Comput., 2021, doi: https://doi.org/10.1007/s12559-021-09939-1. CrossRef Google Scholar
[33]	C. Xu, W. Zhang, Z. Liu and L. Yao, Delay-induced periodic oscillation for fractional-order neural networks with mixed delays, Neurocomputing, 2022, 488, 681–693. doi: 10.1016/j.neucom.2021.11.079 CrossRef Google Scholar
[34]	Y. Yang and J. Ye, Hopf bifurcation in a predator-prey system with discrete and distributed delays, Chaos Solitons Fractals, 2009, 42(1), 554–559. doi: 10.1016/j.chaos.2009.01.026 CrossRef Google Scholar
[35]	F. B. Yousef, A. Yousef and C. Maji, Effects of fear in a fractional-order predator-prey system with predator density-dependent prey mortality, Chaos Solitons Fractals, 2021, 145, 110711. doi: 10.1016/j.chaos.2021.110711 CrossRef Google Scholar
[36]	J. Yuan and C. Huang, Quantitative analysis in delayed fractional-order neural networks, Neural Process. Lett., 2020, 51, 1631–1651. doi: 10.1007/s11063-019-10161-2 CrossRef Google Scholar
[37]	J. Yuan, L. Zhao, C. Huang and M. Xiao, Stability and bifurcation analysis of a fractional predator-prey model involving two nonidentical delays, Mathe. Compu. Simul., 2021, 181, 562–580. doi: 10.1016/j.matcom.2020.10.013 CrossRef Google Scholar