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Dynamic analysis of a fractional order delayed predator–prey system with harvesting

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Abstract

In the study, we consider a fractional order delayed predator–prey system with harvesting terms. Our discussion is divided into two cases. Without harvesting, we investigate the stability of the model, as well as deriving some criteria by analyzing the associated characteristic equation. With harvesting, we investigate the dynamics of the system from the aspect of local stability and analyze the influence of harvesting to prey and predator. Finally, numerical simulations are presented to verify our theoretical results. In addition, using numerical simulations, we investigate the effects of fractional order and harvesting terms on dynamic behavior. Our numerical results show that fractional order can affect not only the stability of the system without harvesting terms, but also the switching times from stability to instability and to stability. The harvesting can convert the equilibrium point, the stability and the stability switching times.

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References

  • Abbas S, Banerjee M, Momani S (2011) Dynamical analysis of fractional-order modified logistic model. Comput Math Appl 62(3):1098–1104

    Article  Google Scholar 

  • Aguirre P, Flores JD, González-Olivares E (2014) Bifurcations and global dynamics in a predator-prey model with a strong allee effect on the prey, and a ratio-dependent functional response. Nonlinear Anal Real World Appl 16:235–249

    Article  Google Scholar 

  • Ahmed E, El-Sayed A, El-Saka H (2007) Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J Math Anal Appl 325(1):542–553

    Article  Google Scholar 

  • Bagley RL, Calico R (1991) Fractional order state equations for the control of viscoelasticallydamped structures. J Guid Control Dyn 14(2):304–311

    Article  Google Scholar 

  • Celik C (2008) The stability and hopf bifurcation for a predator-prey system with time delay. Chaos Solitons Fractals 37(1):87–99

    Article  Google Scholar 

  • Chakraborty K, Chakraborty M, Kar T (2011) Optimal control of harvest and bifurcation of a prey-predator model with stage structure. Appl Math Comput 217(21):8778–8792

    Google Scholar 

  • Chang X, Wei J (2012) Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting. Nonlinear Anal Model Control 17(4):379–409

    Google Scholar 

  • Cheng Y, Jiang W (2012) Stability of single-species model with fractional order delay. J Jiamusi Univ (Natural Science Edition) 30(3):468–469

    Google Scholar 

  • Cui J, Song X (2004) Permanence of predator-prey system with stage structure. Discrete Contin Dyn Syst Ser B 4:547–554

    Article  Google Scholar 

  • Deng W, Li C, Lü J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416

    Article  Google Scholar 

  • Duarte Ortigueira M, Tenreiro Machado J (2003) Fractional signal processing and applications. Signal Process 83(11):2285–2286

    Article  Google Scholar 

  • El-Sayed A, El-Mesiry A, El-Saka H (2007) On the fractional-order logistic equation. Appl Math Lett 20(7):817–823

    Article  Google Scholar 

  • Friedrich C (1991) Relaxation and retardation functions of the maxwell model with fractional derivatives. Rheol Acta 30(2):151–158

    Article  CAS  Google Scholar 

  • Guo Y (2014) The stability of solutions for a fractional predator-prey system. Abstr Appl Anal. doi:10.1155/2014/124145

  • Haque M, Sarwardi S, Preston S, Venturino E (2011) Effect of delay in a lotka-volterra type predator-prey model with a transmissible disease in the predator species. Math Biosci 234(1):47–57

    Article  PubMed  Google Scholar 

  • Hu Z, Teng Z, Zhang L (2011) Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Anal Real World Appl 12(4):2356–2377

    Article  Google Scholar 

  • Huang Z, Cheng Y, Jiang W (2013) The asymptotic stability of a class of fractional order predator-prey model. J Hefei Univ (Natural Sciences) 23(1):8–11

  • Javidi M, Nyamoradi N (2013) Dynamic analysis of a fractional order prey-predator interaction with harvesting. Appl Math Model 37(20):8946–8956

    Article  Google Scholar 

  • Li C, Zhao Z (2009) Asymptotical stability analysis of linear fractional differential systems. J Shanghai Univ (English Edition) 13:197–206

    Article  Google Scholar 

  • Li T, Wang Y (2014) Stability of a class of fractional-order nonlinear systems. doi:10.1155/2014/724270

  • Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic Press

  • Rihan FA (2013) Numerical modeling of fractional-order biological systems. doi:10.1155/2013/816803

  • Rihan FA, Lakshmanan S, Hashish AH, Rakkiyappan R, Ahmed E (2015) Fractional-order delayed predatorcprey systems with holling type-ii functional response. Nonlinear Dyn 80(1–2):777–789

    Article  Google Scholar 

  • Rivero M, Trujillo JJ, Vázquez L, Pilar Velasco M (2011) Fractional dynamics of populations. Appl Math Comput 218(3):1089–1095

    Google Scholar 

  • Tian J, Yu Y, Wang H (2014) Stability and bifurcation of two kinds of three-dimensional fractional lotka-volterra systems. Math Probl Eng. doi:10.1155/2014/695871

  • Tong D, Wang R, Yang H (2005) Exact solutions for the flow of non-newtonian fluid with fractional derivative in an annular pipe. Sci China Ser G Phys Mech Astron 48(4):485–495

    Article  Google Scholar 

  • Wei F, Wu L, Fang Y (2014) Stability and hopf bifurcation of delayed predator-prey system incorporating harvesting. Abstr Appl Anal. doi:10.1155/2014/624162

  • Xu C, Li P (2013) Stability analysis in a fractional order delayed predator-prey model. Int J Math Comput Phys Quantum Eng 7(5):556–559

    Google Scholar 

  • Zhou S, Liu Y, Wang G (2005) The stability of predator-prey systems subject to the allee effects. Theor Popul Biol 67(1):23–31

    Article  PubMed  Google Scholar 

Download references

Acknowledgments

We would like to express our gratitude to the editor and anonymous referees for their valuable comments and suggestions, which helped to improve this manuscript. The work was supported by National Natural Science Foundation of China (Grant Nos. 61174155 and 11571170), as well as being sponsored by the Qing Lan Project of Jiangsu and Jiangsu Province Science Foundation (BK20150420).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Ping Song, Hongyong Zhao & Xuebing Zhang

  2. Department of Basic Course, Huaian College of Information Technology, Huaian, 223003, People’s Republic of China

    Xuebing Zhang

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  1. Ping Song
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  2. Hongyong Zhao
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Correspondence to Hongyong Zhao.

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Song, P., Zhao, H. & Zhang, X. Dynamic analysis of a fractional order delayed predator–prey system with harvesting. Theory Biosci. 135, 59–72 (2016). https://doi.org/10.1007/s12064-016-0223-0

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