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Bifurcations of a class of singular biological economic models

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  • Zhang, Xue
  • Zhang, Qing-ling
  • Zhang, Yue

Abstract

This paper studies systematically a prey–predator singular biological economic model with time delay. It shows that this model exhibits two bifurcation phenomena when the economic profit is zero. One is transcritical bifurcation which changes the stability of the system, and the other is singular induced bifurcation which indicates that zero economic profit brings impulse, i.e., rapid expansion of the population in biological explanation. On the other hand, if the economic profit is positive, at a critical value of bifurcation parameter, the system undergoes a Hopf bifurcation, i.e., the increase of delay destabilizes the system and bifurcates into small amplitude periodic solution. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here. In addition, we study numerically that the system undergoes a saddle-node bifurcation when the bifurcation parameter goes through critical value of positive economic profit.

Suggested Citation

  • Zhang, Xue & Zhang, Qing-ling & Zhang, Yue, 2009. "Bifurcations of a class of singular biological economic models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1309-1318.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:3:p:1309-1318
    DOI: 10.1016/j.chaos.2007.09.010
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    References listed on IDEAS

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    4. Sun, Chengjun & Lin, Yiping & Han, Maoan, 2006. "Stability and Hopf bifurcation for an epidemic disease model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 204-216.
    5. Huang, Chuangxia & Huang, Lihong & Feng, Jianfeng & Nai, Mingyong & He, Yigang, 2007. "Hopf bifurcation analysis for a two-neuron network with four delays," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 795-812.
    6. Zhang, Yue & Zhang, Qingling & Zhao, Lichun & Yang, Chunyu, 2007. "Dynamical behaviors and chaos control in a discrete functional response model," Chaos, Solitons & Fractals, Elsevier, vol. 34(4), pages 1318-1327.
    7. H. Scott Gordon, 1954. "The Economic Theory of a Common-Property Resource: The Fishery," Journal of Political Economy, University of Chicago Press, vol. 62(2), pages 124-124.
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    Cited by:

    1. Grigorii A. Vasilev & Aleksandra A. Filkova & Anastasia N. Sveshnikova, 2021. "Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics," Mathematics, MDPI, vol. 9(7), pages 1-12, April.
    2. Zhang, Xue & Zhang, Qing-Ling & Liu, Chao & Xiang, Zhong-Yi, 2009. "Bifurcations of a singular prey–predator economic model with time delay and stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1485-1494.
    3. Zhao, Hongyong & Zhang, Xuebing & Huang, Xuanxuan, 2015. "Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 462-480.

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