A generalized predator–prey system with multiple discrete delays and habitat complexity

  • Zhihui MaEmail author
  • Shufan Wang
Original Paper


A generalized predator–prey model with multiple discrete delays and the effect of habitat complexity is proposed in this paper. Firstly, the stability of the considered model system and existence of Hopf bifurcations are investigated by the differential equation theory. Secondly, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Most importantly and interestingly, this paper gives a class of the corresponding model systems, and some published works become special cases of ours.


Predator–prey system Discrete delay Habitat complexity Stability Hopf bifurcation 

Mathematics Subject Classification

37C75 34K18 92B05 92D25 93D20 



This work was supported by the National Natural Science Foundation of China (no. 11301238) and the Fundamental Research Funds for the Central Universities (no. lzujbky-2017-166).


  1. 1.
    Ma, Z., Wang, S., Wang, T., Tang, H., Li, Z.: A generalized predator-prey system with habitat complexity. J. Biol. Syst. 25, 495–520 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    August, P.V.: The role of habitat complexity and heterogeneity in structuring tropical mammal communities. Ecology 64, 1495–1507 (1983)CrossRefGoogle Scholar
  3. 3.
    Auger, P., Charles, S., Viala, M., Poggiale, J.C.: Aggregation and emergence in ecological modelling: interaction of ecological levels. Ecol. Model. 127, 11–20 (2000)CrossRefGoogle Scholar
  4. 4.
    Jana, D., Bairagi, N.: Habitat complexity, dispersal and metapopulations: macroscopic study of a predator-prey system. Ecol. Complex. 17, 131–139 (2014)CrossRefGoogle Scholar
  5. 5.
    Bairagi, N., Jana, D.: Age-structured predator-prey model with habitat complexity: oscillations and control. Dyn. Syst. 27, 475–499 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bell, S.S.: Habitat complexity of polychaete tube caps: influence of architecture on dynamics of a meioepibenthic assemblage. J. Mar. Res. 43, 647–657 (1985)CrossRefGoogle Scholar
  7. 7.
    Canion, C.R., Heck, K.L.: Effect of habitat complexity on predation success: re-evaluating the current paradigm in seagrass beds. Mar. Ecol. Prog. Ser. 393, 37–46 (2009)CrossRefGoogle Scholar
  8. 8.
    Bell, S., McCoy, E., Mushinsky, H.: Habitat structure: the physical arrangement of objects in space. Chapman and Hall, London (1991)CrossRefGoogle Scholar
  9. 9.
    Ellner, S.P.: Habitat structure and population persistence in an experimental community. Nature 412, 538–543 (2001)CrossRefGoogle Scholar
  10. 10.
    Ylikarjula, J., Alaja, S., Laakso, J., Tesar, D.: Effects of patch number and dispersal patterns on population dynamics and synchrony. J. Theor. Biol. 207, 377–387 (2000)CrossRefGoogle Scholar
  11. 11.
    Deka, B.D., Patra, A., Tushar, J., Dubey, B.: Stability and Hopf-bifurcation in a general Gauss type two-prey and one-predator system. Appl. Math. Model. 40, 5793–5818 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kar, T.K., Jana, S.: Stability and bifurcation analysis of a stage structured predator prey model with time delay. Appl. Math. Comput. 219, 3779–3792 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chen, Y., Song, C.: Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay. Chaos Solitons Fractals 38, 1104–1114 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gan, Q., Xu, R., Yang, P.: Bifurcation and chaos in a ratio-dependent predator-prey system with time delay. Chaos Solitons Fractals 39, 1883–1895 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hu, G.P., Li, W.T., Yan, X.P.: Hopf bifurcations in a predator-prey system with multiple delays. Chaos Solitons Fractals 42, 1273–1285 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Song, X., Li, Y.: Dynamic behaviors of the periodic predator-prey model with modified Leslie–Gower Holling-type II schemes and impulsive effect. Nonlinear Anal. RWA. 9, 64–79 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Song, Y., Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. J. Math. Anal. Appl. 301, 1–21 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ma, Z.P., Li, W.T., Yan, X.P.: Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion. Appl. Maht. Comput. 219, 2713–2731 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yan, X.P., Li, W.T.: Stability of bifurcating periodic solutions in a delayed reaction–diffusion population model. Nonlinearity 23, 1413–1431 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yan, X.P., Zhang, C.H.: Hopf bifurcation in a delayed Lokta–Volterra predator–prey system. Nonlinear Anal-RWA 9, 114–127 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting. J. Math. Anal. Appl. 398, 278–295 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jia, Y., Xue, P.: Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system. Nonlinear Anal. RWA. 32, 229–241 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kar, T.K., Pahari, U.K.: Modelling and analysis of a prey–predator system with stage-structure and harvesting. Nonlinear Anal. RWA. 8, 601–609 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, X., Wei, J.: Dynamics in a diffusive predator–prey system with strong Allee effect and Ivlev-type functional response. J. Math. Anal. Appl. 422, 1447–1462 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bairagi, N., Jana, D.: On the stability and Hopf bifurcation of a delay-induced predator–prey system with habitat complexity. Appl. Math. Model. 35, 3255–3267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Huang, T., Zhang, H.: Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system. Chaos Solitons Fractals 91, 92–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yang, H., Tian, Y.: Hopf bifurcation in REM algorithm with communication delay. Chaos Solitons Fractals 25, 1093–1105 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Qu, Y., Wei, J.: Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure. Nonlinear Dyn. 49, 285–294 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Celik, C.: The stability and Hopf bifurcation of a predator–prey system with time delay. Chaos Solitons Fractals 37, 87–99 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Celik, C.: Hopf bifurcation of a ratio-dependent predator–prey system with time delay. Chaos Solitons Fractals 42, 1474–1484 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sun, C., Lin, Y., Han, M.: Stability and Hopf bifurcation for an epidemic disease model with delay. Chaos Solitons Fractals 30, 204–216 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of CD4+ T-cells. Math. Biosci. 165, 27–39 (2000)CrossRefzbMATHGoogle Scholar
  33. 33.
    Tripathia, J., Tyagia, S., Abbas, S.: Global analysis of a delayed density dependent predator-prey model with Crowley–Martin functional response. Commun. Nonlinear. Sci. Numer. Simul. 30, 45–69 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and application of Hopf bifurcation. Cambridge University, Cambridge (1981)zbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.School of Mathematics and Computer ScienceNorthwest University for NationalitiesLanzhouPeople’s Republic of China

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