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Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect
1Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China.
Mathematical Biosciences and Engineering : MBE
|November 1, 2018
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.
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.
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