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Hopf bifurcation in host-parasitoid models.

H A Lauwerier1, J A Metz

  • 1Centre for Mathematics and Computer Science, Amsterdam, The Netherlands.

IMA Journal of Mathematics Applied in Medicine and Biology
|January 1, 1986
PubMed
Summary

This study simplifies complex host-parasitoid models using Arnold

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Dynamical Systems

Background:

  • Host-parasitoid models are crucial for understanding ecological dynamics.
  • Analyzing complex models often requires advanced mathematical techniques.
  • Bifurcation analysis reveals critical changes in model behavior.

Purpose of the Study:

  • To develop an explicit method for reducing host-parasitoid models to Arnold's normal form.
  • To analyze the Hopf bifurcation in these models and characterize limit curves.
  • To explore the rich bifurcation behavior and parameter space transitions.

Main Methods:

  • Explicit reduction of host-parasitoid models to Arnold's normal form.
  • Analysis of Hopf bifurcation to determine limit curve properties.
  • Numerical simulations to confirm theoretical predictions.

Main Results:

  • An explicit reduction method is established for a wide class of host-parasitoid models.
  • The shape and size of elliptic limit curves at Hopf bifurcation are derived.
  • Rich bifurcation phenomena, including supercritical and subcritical Hopf bifurcations, were observed.
  • A transition zone in the parameter plane exhibits coexisting stable and unstable limit curves.

Conclusions:

  • The explicit reduction method provides a powerful tool for analyzing host-parasitoid dynamics.
  • The study elucidates complex bifurcation behaviors and their implications for ecological stability.
  • Numerical experiments validate the theoretical framework and findings.

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