Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 19, Issue 3-4


Analysis of a Delayed Predator–Prey System with Harvesting

Wei Liu
  • School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
  • School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yaolin Jiang
Published Online: 2018-05-11 | DOI: https://doi.org/10.1515/ijnsns-2017-0094


This article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

Keywords: Leslie-Gower; gestation delay; bifurcation; asymptotically stable; harvest

PACS: 87.10.Ed; 05.45.-a; 87.23.Cc


  • [1]

    Berryman A. A., The origins and evolution of predator-prey theory, Ecology 73 (1992), 1530-1535.CrossrefGoogle Scholar

  • [2]

    Leslie P. H., Gower J. C., The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika 47 (1960), 219-234.CrossrefGoogle Scholar

  • [3]

    Chen L. S., Mathematical models and methods in ecology, Press Science, Beijing, 1988. (in Chinese).Google Scholar

  • [4]

    Kot M., Elements of Mathematical Biology, Cambridge University Press, Cambridge, 2001.Google Scholar

  • [5]

    Vasilova M., Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay, Math. Comput. Modell. 57 (2013), 764-781.CrossrefGoogle Scholar

  • [6]

    Teng Z. D., Chen L., Global asymptotic stability of periodic Lotka-Volterra systems with delays, Nonlinear Analysis: Theory, Method. Appl. 45 (2001), 1081-1095.CrossrefGoogle Scholar

  • [7]

    Al Noufaey K. S., Marchant T. R., Edwards M. P., The diffusive Lotka-Volterra predator-prey system with delay, Math. Biosci. 270 (2015), 30-40.Google Scholar

  • [8]

    Tripathi J. P., Abbas S., Thakur M., A density dependent delayed predator-prey model with Beddington-DeAngelis type function response incorporating a prey refuge, Commun. Nonl. Sci. Numer. Simu. 22 (2015), 427-450.CrossrefGoogle Scholar

  • [9]

    Karaoglu E., Merdan H., Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays, Chaos, Solitons & Fractals 68 (2014), 159-168.Web of ScienceCrossrefGoogle Scholar

  • [10]

    Adak D., Bairagi N., Complexity in a predator-prey-parasite model with nonlinear incidence rate and incubation delay, Chaos, Solitons & Fractals 81 (2015), 271-289.Web of ScienceCrossrefGoogle Scholar

  • [11]

    Martin A., Ruan S., Predator-prey models with delay and prey harvesting, J. Math. Biol. 43 (2001), 247-267.CrossrefGoogle Scholar

  • [12]

    Al-Omari J. F. M., The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput. 271 (2015), 142-153.Web of ScienceGoogle Scholar

  • [13]

    Li Y., Wang M. X., Hopf bifurcation and global stability of a delayed predator-prey model with prey harvesting, Comput. Math. Appl. 69 (2015), 398-410.CrossrefWeb of ScienceGoogle Scholar

  • [14]

    Zhang X. B., Zhao H. Y., Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters, J. Theor. Biol. 363 (2014), 390-403.Web of ScienceCrossrefGoogle Scholar

  • [15]

    Kar T. K., Pahari U. K., Non-selective harvesting in prey-predator models with delay, Commun. Nonl. Sci. Numer. Simu. 11 (2006), 499-509.CrossrefGoogle Scholar

  • [16]

    Zhang G. D., Shen Y., Chen B. S., Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn. 73 (2013), 2119-2131.CrossrefWeb of ScienceGoogle Scholar

  • [17]

    Chen B. S., Chen J. J., Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comput. 219 (2012), 2371-2386.Google Scholar

  • [18]

    Wu X. Y., Chen B. S., Bifurcations and stability of a discrete singular bioeconomic system, Nonlinear Dyn. 73 (2013), 1813-1828.Web of ScienceCrossrefGoogle Scholar

  • [19]

    Zhang G. D., Shen Y., Chen B. S., Bifurcation analysis in a discrete differential-algebraic predator-prey system, Appl. Mathe. Modell. 38 (2014), 4835-4848.Google Scholar

  • [20]

    Liu W. Y., C. Fu J., B. Chen S., Stability and Hopf bifurcation of a predator-prey biological economic system with nonlinear harvesting rate, Int. J. Nonlinear Sci. Numer. Simul. 16 (2015), 249-258.Web of ScienceGoogle Scholar

  • [21]

    Ruan S., Wei J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls, Syst. Ser. A Math. Anal. 10 (2003), 863-874.Google Scholar

  • [22]

    Hale J. K., Theory of functional differential equations, Springer, New York, 1997.Google Scholar

  • [23]

    Cooke K. L., Grossman Z., Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982), 592-627.CrossrefGoogle Scholar

  • [24]

    Hassard B., Kazarinoff D., Wan Y., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.Google Scholar

  • [25]

    Gordon H. S., Economic theory of a common property resource: the fishery, J. Polit. Econ. 62 (1954), 124-142.CrossrefGoogle Scholar

  • [26]

    Mankiw N. G., Principles of Economics, Peking University Press, Beijing, 2015.Google Scholar

  • [27]

    Faria T., Maglhalães L. T., Normal form for retarded functional differential equations with parameters and applications to Hopf Bifurcation, J. Differ. Equ. 122 (1995), 181-200.CrossrefGoogle Scholar

  • [28]

    Faria T., Maglhalães L. T., Normal form for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differ. Equ. 122 (1995), 201-224.CrossrefGoogle Scholar

  • [29]

    Guckenheimer J., Holmes P., Oscillations Nonlinear, Systems Dynamical, and Bifurcations of Vector Fields, Springer, New York, USA, 1983.Google Scholar

  • [30]

    Chen B. S., X. Liao X., Y. Liu Q., Normal forms and bifurcations for the differential-algebraic systems, Acta Math. Appl. Sin. 23 (2000), 429-443. (in Chinese).Google Scholar

  • [31]

    Nussbaum R. D., Periodic solutions of some nonlinear autonomous functional equations, Ann. Mat. Pura Appl. 10 (1974), 263-306.Google Scholar

  • [32]

    Erbe L. H., Geba K., Krawcewicz W., Wu J., S1-degree and global Hopf bifurcations, J. Differ. Equ. 98 (1992), 277-298.Google Scholar

  • [33]

    Reich S., On the local qualitative behavior of differential-algebraic equations, Circ. Syst. Sig. Process 14 (1995), 427-443.CrossrefGoogle Scholar

  • [34]

    Venkatasubramanian V., Schättler H., Zaborszky J., Local bifurcation and feasibility regions in differential-algebraic systems, Trans IEEE. Automat. Contr. 40 (1995), 1992-2013.CrossrefGoogle Scholar

About the article

Received: 2017-04-24

Accepted: 2018-01-30

Published Online: 2018-05-11

Published in Print: 2018-06-26

This work is supported by the National Natural Science Foundations of China (Grants No. 11371287 and 61663043), Science and Technology Projects Founded by the Education Department of Jiangxi Province (Grants no. GJJ14774 and GJJ14775).

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 335–349, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0094.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in