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Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect.

Changyong Xu1, Qiang Li2, Tonghua Zhang3

  • 1College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China.

Mathematical Biosciences and Engineering : MBE
|December 31, 2020
PubMed
Summary

This study explores a delayed diffusive competition model. Increasing time delays can destabilize the system, leading to new periodic solutions, crucial for understanding ecological dynamics.

Keywords:
Hopf bifurcationcompetitive modeldiffusionstabilitytime delay

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems Theory
  • Ecological Modeling

Background:

  • Competition models are fundamental in ecology.
  • Time delays and saturation effects significantly influence population dynamics.
  • Understanding bifurcations is key to predicting ecosystem stability.

Purpose of the Study:

  • Investigate the dynamics of a delayed diffusive competition model with saturation.
  • Analyze the stability of the positive equilibrium.
  • Examine the occurrence and properties of Hopf bifurcations.

Main Methods:

  • Stability analysis of the positive equilibrium.
  • Analysis of Hopf bifurcations.
  • Derivation of formulas for bifurcation direction and properties.
  • Numerical simulations for validation.

Main Results:

  • The positive equilibrium is asymptotically stable under specific conditions.
  • A critical delay value exists, beyond which stability is lost.
  • Hopf bifurcations lead to spatially homogeneous or inhomogeneous periodic solutions.
  • Formulas for bifurcation direction and properties were derived.

Conclusions:

  • Time delays play a critical role in the stability of competition models.
  • Hopf bifurcations can generate complex periodic behaviors in ecological systems.
  • The model provides insights into the emergence of oscillations in competing populations.