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

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Dynamical Systems, Bifurcation Analysis and Applications (DySBA 2018)

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Analysis and Geometry, Faculty of Mathematics and Natural Sciences, Institut Teknologi, Bandung, Indonesia

    Johan M. Tuwankotta, Eric Harjanto & Livia Owen

  2. Department of Mathematics, University of Parahyangan, Bandung, Indonesia

    Livia Owen

Authors
  1. Johan M. Tuwankotta
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  2. Eric Harjanto
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  3. Livia Owen
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Corresponding author

Correspondence to Johan M. Tuwankotta .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Mohd Hafiz Mohd

  2. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Norazrizal Aswad Abdul Rahman

  3. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Nur Nadiah Abd Hamid

  4. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Yazariah Mohd Yatim

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Tuwankotta, J.M., Harjanto, E., Owen, L. (2019). Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation. In: Mohd, M., Abdul Rahman, N., Abd Hamid, N., Mohd Yatim, Y. (eds) Dynamical Systems, Bifurcation Analysis and Applications. DySBA 2018. Springer Proceedings in Mathematics & Statistics, vol 295. Springer, Singapore. https://doi.org/10.1007/978-981-32-9832-3_3

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