Hopf and Generalized Hopf Bifurcations in a Delayed Host-Parasite Model

Yucen Jin; Pei Yu

Ayuda

Hopf and Generalized Hopf Bifurcations in a Delayed Host-Parasite Model

Yucen Jin ^[1] ; Pei Yu ^[1]
1. [1] Western University
  
  Western University
  
  Azerbaiyán
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
Idioma: inglés
Enlaces
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Resumen
- This paper investigates the qualitative dynamics of a delayed host-parasite-competitor system obtained by extending a previously studied three-dimensional ordinary differential model through the inclusion of a biologically motivated maturation delay in the parasite population. The resulting delay differential equation exhibits rich dynamical behaviour as the delay parameter varies. In particular, we identify Hopf and generalized Hopf (Bautin) bifurcations that organize transitions between stable equilibria and periodic oscillations. Using the method of multiple time scales, the normal form is derived up to fifth order, providing a systematic analytical framework for detecting and classifying codimension-two bifurcations without resorting to center manifold reduction. The analysis reveals regions of bistability characterized by the coexistence of a stable steady state and a stable periodic orbit. These results contribute to the qualitative understanding of nonlinear delay systems and clarify how intrinsic time delays shape oscillatory dynamics in host-parasite interactions.
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