Advertisement

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

  • Juancho A. ColleraEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 295)

Abstract

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behaviour of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results.

Keywords

Delay differential equations Numerical continuation Numerical bifurcation analysis Time delay systems 

Notes

Acknowledgements

The author acknowledges the support of University of the Philippines Baguio, CIMPA, IMU-CDC, SEAMS, and Universiti Sains Malaysia for his participation to SEAMS School 2018 on Dynamical Systems and Bifurcation Analysis. The author also would like to thank the referees for their valuable reviews that improved the quality of this paper.

References

  1. 1.
    Buono, P.-L., Collera, J.A.: Symmetry-breaking bifurcations in rings of delay-coupled semiconductor lasers. SIAM J. Appl. Dyn. Syst. 14, 1868–1898 (2015).  https://doi.org/10.1137/140986487MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Collera, J.A.: Symmetry-breaking bifurcations in two mutually delay-coupled lasers. Phil. Sci. Tech. 8, 17–21 (2015)CrossRefGoogle Scholar
  3. 3.
    Collera, J.A.: Symmetry-breaking bifurcations in laser systems with all-to-all coupling. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (Eds.) Mathematical and Computational Approaches in Advancing Modern Science and Engineering, pp. 81–88. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-30379-6_8CrossRefGoogle Scholar
  4. 4.
    Collera, J.A., Magpantay, F.M.G: Dynamics of a stage structured intraguild predation model. In: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Wang, X. (Eds.) Recent Advances in Mathematical and Statistical Methods, pp. 327–337. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-99719-3_30Google Scholar
  5. 5.
    Dankowicz, H., Schilder, F.: Recipes for Continuation. SIAM, Philadelphia (2013).  https://doi.org/10.1137/1.9781611972573
  6. 6.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. (TOMS) 29, 141–164 (2003).  https://doi.org/10.1145/779359.779362MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Doedel, E., Oldeman, B.: AUTO-07P Manual—Continuation and bifurcation software for ordinary differential equations. https://sourceforge.net/projects/auto-07p/
  8. 8.
    Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Department of Computer Science, K. U. Leuven, Leuven (2001)Google Scholar
  9. 9.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993).  https://doi.org/10.1007/978-1-4612-4342-7CrossRefGoogle Scholar
  10. 10.
    Pender, J., Rand, R.H., Wesson, E.: Queues with choice via delay differential equations. Int. J. Bifurcat. Chaos 27, 1730016 (2017).  https://doi.org/10.1142/S0218127417300166MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G., Roose D.: DDE-BIFTOOL v.3.1.1 Manual—Bifurcation analysis of delay differential equations. http://arxiv.org/abs/1406.7144
  12. 12.
    Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York (2011).  https://doi.org/10.1007/978-1-4419-7646-8CrossRefGoogle Scholar
  13. 13.
    Szalai, R.: Knut: a continuation and bifurcation software for delay-differential equations (version 8), Department of Engineering Mathematics, University of Bristol (2013). http://rs1909.github.io/knut/
  14. 14.
    Toaha, S., Hassan, M.A.: Stability analysis of predator-prey population model with time delay and constant rate of harvesting. Punjab Univ. J. Math. 40, 37–48 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.University of the Philippines BaguioBaguioPhilippines

Personalised recommendations