Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation

  • Johan M. TuwankottaEmail author
  • Eric Harjanto
  • Livia Owen
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 295)


We study a two dimensional system of ordinary differential equations of a predator-prey type. We use the Holling type IV functional response which models the group defence mechanism. For this system we discuss the number of equilibria in the system and prove it using a geometrical approach. Using the classical Lagrange Multiplier method, we compute fold and cusp bifurcations for equilibrium in the system. As we turn on to numerics, we compute the other bifurcations for equilibrium, namely Hopf bifurcations, and homoclinic bifurcations. As for bifurcation of periodic solution we compute the Fold of Limit Cycle bifurcation. We also include time-periodic variation in the system which translates most of the bifurcation sets for equilibria into bifurcation sets for periodic solutions. Furthermore, we found the swallowtail bifurcation for periodic solution in the system.


Predator-prey Bogdanov-Takens bifurcation Bautin bifurcation Cusp bifurcation Swallowtail bifurcation 



J.M. Tuwankotta research is supported by Riset KK B, Institut Teknologi Bandung (2019).


  1. 1.
    Balagaddé, F.K., Song, H., Ozaki, J., Collins, C.H., Barnet, M., Arnold, F.H., Quake, S.R., You, L.: A synthetic escherichia coli predator-prey ecosystem. Mol. Syst. Biol. 4, 187 (2008)CrossRefGoogle Scholar
  2. 2.
    Berryman, A.A.: The orgins and evolution of predator-prey theory. Ecology 73, 1530–1535 (1992)CrossRefGoogle Scholar
  3. 3.
    Briggs, G.E., Haldane, J.B.S.: A note on the kinetics of enzyme action. Biochem. J. 19, 338 (1925)CrossRefGoogle Scholar
  4. 4.
    Broer, H., Naudot, V., Roussarie, R., Saleh, K.: Bifurcations of a predator-prey model with non-monotonic response function. Comptes Rendus Mathématique 341, 601–604 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Broer, H., Naudot, V., Roussarie, R., Saleh, K.: A predator-prey model with non-monotonic response function. Regul. Chaotic Dyn. 11, 155–165 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Broer, H., Saleh, K., Naudot, V., Roussarie, R.: Dynamics of a predator-prey model with non-monotonic response function. Discret. Contin. Dyn. Syst. A 18, 221–251 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cai, Z., Wang, Q., Liu, G.: Modeling the natural capital investment on tourism industry using a predator-prey model. In: Advances in Computer Science and its Applications, Springer, pp. 751–756 (2014)Google Scholar
  8. 8.
    Casagrandi, R., Rinaldi, S.: A theoretical approach to tourism sustainability. Conserv. Ecol. 6 (2002)Google Scholar
  9. 9.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X., et al.: Continuation and Bifurcation Software for Ordinary Differential Equations (with Homcont), AUTO97. Concordia University, Canada (1997)Google Scholar
  10. 10.
    Feinstein, C.H., Dobb, M.: Socialism, capitalism & economic growth, CUP Archive (1967)Google Scholar
  11. 11.
    Fenton, A., Perkins, S.E.: Applying predator-prey theory to modelling immune-mediated, within-host interspecific parasite interactions. Parasitology 137, 1027–1038 (2010)CrossRefGoogle Scholar
  12. 12.
    Grimme, C., Lepping, J.: Integrating niching into the predator-prey model using epsilon-constraints. In: Proceedings of the 13th Annual Conference Companion on Genetic and Evolutionary Computation, ACM, pp. 109–110 (2011)Google Scholar
  13. 13.
    Harjanto, E., Tuwankotta, J.: Bifurcation of periodic solution in a predator-prey type of systems with non-monotonic response function and periodic perturbation. Int. J. Non Linear Mech. 85, 188–196 (2016)CrossRefGoogle Scholar
  14. 14.
    Hirsch, M.W.: Systems of differential equations which are competitive or cooperative: I. limit sets. SIAM J. Math. Anal. 13, 167–179 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hirsch, M.W.: Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423–439 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hirsch, M.W.: Systems of differential equations which are competitive or cooperative: III. competing species. Nonlinearity 1, 51 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the european pine sawfly. Can. Entomol. 91, 293–320 (1959)CrossRefGoogle Scholar
  18. 18.
    Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)CrossRefGoogle Scholar
  19. 19.
    Huang, Y., Diekmann, O.: Predator migration in response to prey density: what are the consequences? J. Math. Biol. 43, 561–581 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. II—the problem of endemicity. Proc. R. Soc. Lond. A, 138, 55–83 (1932)Google Scholar
  21. 21.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. III—further studies of the problem of endemicity. Proc. R. Soc. Lond. A 141, 94–122 (1933)Google Scholar
  22. 22.
    Kermark, M., Mckendrick, A.: Contributions to the mathematical theory of epidemics. part I. Proc. R. Soc. A 115, 700–721 (1927)Google Scholar
  23. 23.
    Koren, I., Feingold, G.: Aerosol-cloud-precipitation system as a predator-prey problem. In: Proceedings of the National Academy of Sciences (2011)Google Scholar
  24. 24.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer Science & Business Media (2013)Google Scholar
  25. 25.
    Marwan, M., Tuwankotta, J.M., Harjanto, E.: Application of lagrange multiplier method for computing fold bifurcation point in a two-prey one predator dynamical system. J. Indones. Math. Soc. 24, 7–19 (2018)MathSciNetGoogle Scholar
  26. 26.
    Nagano, S., Maeda, Y.: Phase transitions in predator-prey systems. Phys. Rev. E 85, 011915 (2012)CrossRefGoogle Scholar
  27. 27.
    Owen, L., Tuwankotta, J.: Bogdanov-takens bifurcations in three coupled oscillators system with energy preserving nonlinearity. J. Indones. Math. Soc. 18, 73–83 (2012)CrossRefGoogle Scholar
  28. 28.
    Owen, L., Tuwankotta, J.: Computation of cusp bifurcation point in a two-prey one predator model using lagrange multiplier method, in Proceedings of the International Conference on Applied Physics and Mathematics 2019. Chulalongkorn University, Bangkok, Thailand (2019)Google Scholar
  29. 29.
    Rinaldi, S., Muratori, S., Kuznetsov, Y.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull. Math. Biol. 55, 15–35 (1993)CrossRefGoogle Scholar
  30. 30.
    Semenov, N.F.: II. Chemical Kinetics and Chain Reactions (1935)Google Scholar
  31. 31.
    Sharma, A., Singh, N.: Object detection in image using predator-prey optimization. Signal Image Proc. 2 (2011)Google Scholar
  32. 32.
    Tuwankotta, J., Harjanto, E.: Strange attractors in a predator-prey type of systems with nonmonotonic response function and periodic perturbation. J. Comput. Dyn. (2019)Google Scholar
  33. 33.
    Verhulst, F.: The hunt for canards in population dynamics: a predator-prey system. Int. J. Non Linear Mech. 67, 371–377 (2014)CrossRefGoogle Scholar
  34. 34.
    Zhang, T., Xing, Y., Zang, H., Han, M.: Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality. Nonlinear Dyn. 78, 265–277 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhang, T., Zang, H.: Delay-induced turing instability in reaction-diffusion equations. Phys. Rev. E 90, 052908 (2014)CrossRefGoogle Scholar
  36. 36.
    Zhu, H., Campbell, S.A., Wolkowicz, G.S.: Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 63, 636–682 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Johan M. Tuwankotta
    • 1
    Email author
  • Eric Harjanto
    • 1
  • Livia Owen
    • 1
    • 2
  1. 1.Analysis and Geometry, Faculty of Mathematics and Natural SciencesInstitut TeknologiBandungIndonesia
  2. 2.Department of MathematicsUniversity of ParahyanganBandungIndonesia

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