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Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect.

Zuolin Shen1, Junjie Wei

  • 1Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China.

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Summary

This study analyzes a predator-prey model with delays and surplus killing. It establishes conditions for stability and Hopf bifurcation, revealing complex population dynamics.

Keywords:
Hopf bifurcationdiffusive predator-prey systemsurplus killing effecttime delay

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Dynamical Systems

Background:

  • Predator-prey models are fundamental in ecology.
  • Incorporating delays and surplus killing introduces complex dynamics.
  • Neumann boundary conditions are relevant for spatially distributed populations.

Purpose of the Study:

  • To analyze a delayed diffusive predator-prey system with surplus killing.
  • To investigate the stability of the steady state and the occurrence of Hopf bifurcation.
  • To determine the direction and stability of bifurcating periodic solutions.

Main Methods:

  • Analysis of a partial differential equation model.
  • Stability analysis of the steady state using eigenvalue distribution.
  • Hopf bifurcation analysis using normal form theory and center manifold theorem.
  • Numerical simulations to validate analytical findings.

Main Results:

  • Established conditions for the stability of the positive steady state when delay is zero.
  • Proved the existence of Hopf bifurcation when delay is non-zero.
  • Derived an algorithm to determine the direction of bifurcation and stability of periodic solutions.
  • Numerical simulations confirmed the theoretical predictions.

Conclusions:

  • The delay and surplus killing significantly influence the stability and dynamics of the predator-prey system.
  • Hopf bifurcation leads to the emergence of periodic oscillations in population densities.
  • The study provides a comprehensive mathematical framework for understanding complex ecological interactions.