H. W. Broer, R. Roussarie, V. Naudot, K. Saleh, “A predator-prey model with non-monotonic response function”, Regul. Chaotic Dyn., 11:2 (2006), 155

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Regular and Chaotic Dynamics, 2006, Volume 11, Issue 2, Pages 155–165
DOI: https://doi.org/10.1070/RD2006v011n02ABEH000342 (Mi rcd665)

This article is cited in 19 scientific papers (total in 19 papers)

On the 70th birthday of L.P. Shilnikov

A predator-prey model with non-monotonic response function

H. W. Broer^a, R. Roussarie^b, V. Naudot^a, K. Saleh^a

^a Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
^b Institut Mathématiques de Bourgogne, CNRS, 9, avenue Alain Savary, B.P. 47 870, 21078 Dijon cedex, France

Citations (19)

DOI: https://doi.org/10.1070/RD2006v011n02ABEH000342

Abstract: We study the dynamics of a family of planar vector fields that models certain populations of predators and their prey. This model is adapted from the standard Volterra–Lotka system by taking into account group defense, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence. We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, we study the bifurcations between the various domains of structural stability. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. We find several codimension 3 bifurcations that form organizing centers for the global bifurcation set. Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.

Keywords: predator-prey dynamics, organizing center, bi-furcation, strange attractor.

Received: 01.08.2005
Accepted: 01.09.2005

Bibliographic databases:

Document Type: Article

MSC: 58K45, 34C23, 34C60, 37D45

Language: English

Citation: H. W. Broer, R. Roussarie, V. Naudot, K. Saleh, “A predator-prey model with non-monotonic response function”, Regul. Chaotic Dyn., 11:2 (2006), 155–165

Citation in format AMSBIB

\Bibitem{BroRouNau06}

\by H.~W.~Broer, R. Roussarie, V.~Naudot, K.~Saleh

\paper A predator-prey model with non-monotonic response function

\jour Regul. Chaotic Dyn.

\yr 2006

\vol 11

\issue 2

\pages 155--165

\mathnet{http://mi.mathnet.ru/rcd665}

\crossref{https://doi.org/10.1070/RD2006v011n02ABEH000342}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2245074}

\zmath{https://zbmath.org/?q=an:1164.37318}

Linking options:

https://www.mathnet.ru/eng/rcd665

https://www.mathnet.ru/eng/rcd/v11/i2/p155

This publication is cited in the following 19 articles:

Renato Huzak, Hildeberto Jardón-Kojakhmetov, Christian Kuehn, “Ergodicity in Planar Slow-Fast Systems Through Slow Relation Functions”, SIAM J. Appl. Dyn. Syst., 24:1 (2025), 317
Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin, “Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments”, Open Mathematics, 21:1 (2023)
Patidar K.C., Ramanantoanina A., “A Non-Standard Finite Difference Scheme For a Class of Predator-Prey Systems With Non-Monotonic Functional Response”, J. Differ. Equ. Appl., 27:9 (2021), 1310–1328
Lu M., Huang J., “Global Analysis in Bazykin'S Model With Holling II Functional Response and Predator Competition”, J. Differ. Equ., 280 (2021), 99–138
Jiang J., Zhang W., Yu P., “Tristable Phenomenon in a Predator-Prey System Arising From Multiple Limit Cycles Bifurcation”, Int. J. Bifurcation Chaos, 30:9 (2020), 2050129
Johan M. Tuwankotta, Eric Harjanto, Livia Owen, Springer Proceedings in Mathematics & Statistics, 295, Dynamical Systems, Bifurcation Analysis and Applications, 2019, 31
Dirk L. van Kekem, Alef E. Sterk, “Travelling waves and their bifurcations in the Lorenz-96 model”, Physica D: Nonlinear Phenomena, 367 (2018), 38
Amjad Khan, Lindi M. Wahl, Pei Yu, Springer Proceedings in Mathematics & Statistics, 259, Recent Advances in Mathematical and Statistical Methods, 2018, 375
Ram Chandra, Anupam Priyadarshi, AIP Conference Proceedings, 1975, 2018, 030023
Renato Huzak, “Regular and slow-fast codimension 4 saddle-node bifurcations”, Journal of Differential Equations, 262:2 (2017), 1119
Eric Harjanto, J.M. Tuwankotta, “Bifurcation of periodic solution in a Predator–Prey type of systems with non-monotonic response function and periodic perturbation”, International Journal of Non-Linear Mechanics, 85 (2016), 188
Chengzhi Li, Springer Proceedings in Mathematics & Statistics, 157, Mathematical Sciences with Multidisciplinary Applications, 2016, 301
Konstantinos Efstathiou, Xia Liu, Henk W. Broer, “The Boundary-Hopf-Fold Bifurcation in Filippov Systems”, SIAM J. Appl. Dyn. Syst., 14:2 (2015), 914
R. Huzak, P. De Maesschalck, F. Dumortier, “Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations”, Journal of Differential Equations, 255:11 (2013), 4012
K Saleh, FOO Wagener, “Semi-global analysis of periodic and quasi-periodic normal-internalk: 1 andk: 2 resonances”, Nonlinearity, 23:9 (2010), 2219
H.W. Broer, S.J. Holtman, G. Vegter, “Recognition of resonance type in periodically forced oscillators”, Physica D: Nonlinear Phenomena, 239:17 (2010), 1627
Henk W. Broer, Valery A. Gaiko, “Global qualitative analysis of a quartic ecological model”, Nonlinear Analysis: Theory, Methods & Applications, 72:2 (2010), 628
Yann Lamontagne, Caroline Coutu, Christiane Rousseau, “Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response”, J Dyn Diff Equat, 20:3 (2008), 535
Dibakar Ghosh, A. Roy Chowdhury, “On the Bifurcation Pattern and Normal Form in a Modified Predator–Prey Nonlinear System”, Journal of Computational and Nonlinear Dynamics, 2:3 (2007), 267

Citing articles in Google Scholar: Russian citations, English citations
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