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Discrete-time predator-prey model with flip bifurcation and chaos control.
A Q Khan1, I Ahmad2, H S Alayachi3
1Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan.
Mathematical Biosciences and Engineering : MBE
|October 30, 2020
Summary
This study investigates a predator-prey model with an Allee effect, revealing complex dynamics like chaos and bifurcations. Numerical simulations confirm theoretical findings, highlighting intricate population behaviors.
Area of Science:
- Ecology
- Mathematical Biology
- Dynamical Systems
Background:
- Predator-prey models are fundamental in ecology.
- The Allee effect describes reduced per capita growth at low population densities.
- Understanding complex dynamics is crucial for ecological stability.
Purpose of the Study:
- To analyze the local dynamics and bifurcations of a predator-prey model with a prey Allee effect.
- To investigate chaos control and the existence of periodic points within the model.
- To numerically demonstrate complex dynamical behaviors and confirm theoretical findings.
Main Methods:
- Theoretical analysis of local dynamics and bifurcations.
- Numerical simulations to observe population dynamics.
- Calculation of maximum Lyapunov exponents and fractal dimensions to identify chaos.
Main Results:
- The model exhibits flip bifurcation and chaotic behavior.
- Numerical simulations reveal complex orbits (e.g., period-2, 8, 11, 17, 20, 22).
- Maximum Lyapunov exponents and fractal dimensions confirm the presence of chaos.
Conclusions:
- The predator-prey model with a prey Allee effect displays rich and complex dynamics.
- Chaos control is a relevant aspect for managing such ecological systems.
- Numerical and theoretical approaches effectively characterize the model's behavior.
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