Simultaneous Hopf and Bogdanov–Takens Bifurcations on a Leslie–Gower Type Model with Generalist Predator and Group Defence

Liliana Puchuri, Orestes Bueno, Eduardo González-Olivares, Alejandro Rojas-Palma

Research output: Contribution to journalArticle in a journalpeer-review

Abstract

In this work, we analyze a two-dimensional continuous-time differential equations system derived from a Leslie–Gower predator–prey model with a generalist predator and prey group defence. For our model, we fully characterize the existence and quantity of equilibrium points in terms of the parameters, and we use this to provide necessary and sufficient conditions for the existence and the explicit form of two kinds of equilibrium points: both a degenerate one with associated nilpotent Jacobian matrix, and a weak focus. These conditions allows us to determine whether the system undergoes Bogdanov–Takens and Hopf bifurcations. Consequently, we establish the existence of a simultaneous Bogdanov–Taken and Hopf bifurcation. With this double bifurcation, we guarantee the existence of a new Hopf bifurcation curve and two limit cycles on the system: an infinitesimal and another non-infinitesimal.

Original languageEnglish
Article number255
JournalQualitative Theory of Dynamical Systems
Volume23
Issue numberSuppl 1
DOIs
StatePublished - Nov 2024

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

Keywords

  • Bogdanov–Takens bifurcation
  • Hopf bifurcation
  • Non-monotonic functional response
  • Predator–prey model

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