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Remarks on food chain dynamics

Y A Kuznetsov1, S Rinaldi

  • 1Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands.

Mathematical Biosciences
|May 1, 1996
PubMed
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This study explores a food chain model with logistic prey and Holling type II predator/superpredator, revealing complex dynamics and chaotic behaviors through bifurcation analysis.

Area of Science:

  • Ecology
  • Mathematical Biology
  • Theoretical Ecology

Background:

  • Food chain dynamics are crucial for ecosystem stability.
  • Predator-prey interactions can lead to complex population fluctuations.
  • Holling Type II functional response is common in predator-prey systems.

Purpose of the Study:

  • To analyze the dynamical behaviors of a food chain model.
  • To investigate the conditions leading to chaos in the ecosystem.
  • To understand the role of bifurcations in ecological models.

Main Methods:

  • Bifurcation analysis using normal form and numerical continuation.
  • Examination of a three-species food chain model (logistic prey, Holling Type II predator and superpredator).
  • Analysis of equilibrium points and limit cycles.

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Main Results:

  • The two-parameter bifurcation diagram includes Hopf, fold, and transcritical bifurcations for equilibria and limit cycles.
  • Chaos emerges from a Hopf bifurcation and a degenerate homoclinic bifurcation in the prey-predator subsystem.
  • The chaotic region exhibits a unique boundary structure.

Conclusions:

  • The food chain model exhibits rich dynamics, including chaos.
  • Bifurcation theory is essential for understanding ecological complexity.
  • The specific structure of the chaotic boundary warrants further investigation.