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

• Liliana Puchuri ^[1] ; Orestes Bueno ^[2] ; Eduardo González-Olivares ^[3] ; Alejandro Rojas-Palma ^[4]
  1. [1] Pontificia Universidad Católica del Perú

     Pontificia Universidad Católica del Perú

     Perú

     
  2. [2] University of the Pacific

     University of the Pacific

     Estados Unidos

     
  3. [3] Pontificia Universidad Católica de Valparaíso

     Pontificia Universidad Católica de Valparaíso

     Valparaíso, Chile

     
  4. [4] Universidad Católica del Maule

     Universidad Católica del Maule

     Provincia de Talca, Chile

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
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  • 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.