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Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control
1School of Mathematics and Statistics, Central South University, Changsha, 410083, China. xqz19870705@163.com.
Mathematical Biosciences and Engineering : MBE
|August 18, 2015
Summary
This study analyzes predator-prey dynamics with impulsive harvesting and a prey Allee effect. Optimal harvesting rates are crucial for sustainable ecological development and preventing species decline.
Area of Science:
- Mathematical Biology
- Ecology
- Dynamical Systems
Background:
- Predator-prey models are fundamental in ecology.
- Allee effects in prey populations complicate population dynamics.
- Impulsive harvesting strategies introduce periodic perturbations.
Purpose of the Study:
- Analyze a predator-prey model with state feedback impulsive harvesting.
- Investigate the impact of a strong Allee effect on prey dynamics.
- Determine conditions for sustainable ecological development.
Main Methods:
- Geometric theory of differential equations for unperturbed systems.
- Theory of rotated vector fields for perturbed systems.
- Analysis of order-1 heteroclinic cycles and periodic solutions.
- Numerical simulations.
Main Results:
- Existence of order-1 heteroclinic cycles and positive periodic solutions in the unperturbed system.
- Characterization of periodic solutions and heteroclinic bifurcations in the perturbed system.
- Identification of a critical range for harvesting rates.
Conclusions:
- Impulsive harvesting strategies significantly influence predator-prey dynamics.
- A strong Allee effect necessitates careful management of harvesting.
- Maintaining harvesting rates within a reasonable range is essential for ecosystem sustainability.
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