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Changjin Xu1, Peiluan Li2

  • 1Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China.

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Summary

This study analyzes a delayed predator-prey model, revealing that Hopf bifurcations and periodic solutions emerge at critical time delays. These findings are crucial for understanding population dynamics and ecological stability.

Keywords:
Hassell–Varley functional responseHopf bifurcationPredator-prey modelStability

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Dynamical Systems

Background:

  • Predator-prey models are fundamental in ecology.
  • Time delays can significantly alter population dynamics.
  • Hassell-Varley functional response models predator feeding behavior.

Purpose of the Study:

  • Investigate a delayed predator-prey model with Hassell-Varley functional response.
  • Determine conditions for Hopf bifurcation.
  • Analyze the stability and global existence of periodic solutions.

Main Methods:

  • Analysis of the characteristic equation roots in the complex plane.
  • Application of normal form theory and center manifold theorem.
  • Utilizing global Hopf bifurcation results for functional differential equations.

Main Results:

  • Existence of a bifurcation parameter point identified.
  • Hopf bifurcation occurs at critical time delay values (τ).
  • Periodic solutions are shown to exist globally.

Conclusions:

  • Time delays are critical in inducing population oscillations.
  • The model predicts complex population dynamics through bifurcations.
  • Results offer insights into ecological stability and persistence.