Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model

Zirui Zhu and Xingbo Liu

Mathematics and Computers in Simulation (MATCOM), 2025, vol. 227, issue C, 477-499

Abstract: This article establishes a bifurcation analysis of a singularly perturbed piecewise-smooth continuous predator–prey system with a sufficiently small parameter. The bifurcation that can generate limit cycles here is our main concern. To achieve this goal, we have developed a lemma that is used to determine the parameter region that can generate limit cycles. Further conclusions indicate that the existence of a 2-shaped critical manifold is required. Based on the Poincaré-Bendixon lemma, Fenichel’s theory and geometric singular perturbation theory, we demonstrate the possibility of generating smooth and nonsmooth bifurcations. In fact, nonsmooth bifurcations only occur in piecewise-smooth systems. More specifically, different types of nonsmooth bifurcations are also presented in this article, including nonsmooth Hopf bifurcation, Hopf-like bifurcation and super-explosion. In addition, this article discusses the existence of crossing limit cycles and explains whether the crossing limit cycle is characterized by canard cycles without head, canard cycles with head or relaxation oscillations. Furthermore, the coexistence of two relaxation oscillations, the coexistence of two canard cycles without head, and the coexistence of one relaxation oscillation and one canard cycle without head are investigated. Moreover, the one-parameter bifurcation diagram is also presented in this paper through numerical simulations.

Keywords: Piecewise-smooth predator–prey model; Geometric singular perturbation theory; Nonsmooth bifurcation; Super-explosion; Crossing limit cycle (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378475424003197
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:227:y:2025:i:c:p:477-499

DOI: 10.1016/j.matcom.2024.08.017

Access Statistics for this article

Mathematics and Computers in Simulation (MATCOM) is currently edited by Robert Beauwens

More articles in Mathematics and Computers in Simulation (MATCOM) from Elsevier
Bibliographic data for series maintained by Catherine Liu ().