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Bifurcations in a discrete time model composed of Beverton-Holt function and Ricker function
Jin Shang1, Bingtuan Li1, Michael R Barnard1
1Department of Mathematics, University of Louisville, Louisville, KY 40292, United States.
Mathematical Biosciences
|March 14, 2015
Summary
This study analyzes a discrete-time population model combining Ricker and Beverton-Holt functions. We found a period-doubling bifurcation curve, separating stable and unstable regions, with chaos intermingled with periodic cycles.
Area of Science:
- Mathematical Biology
- Population Dynamics
- Ecological Modeling
Background:
- Population models are crucial for understanding ecological dynamics.
- Discrete-time models offer insights into population fluctuations.
- The Ricker and Beverton-Holt models are foundational in population ecology.
Purpose of the Study:
- To rigorously analyze a discrete-time population model integrating Ricker and Beverton-Holt functions.
- To investigate the stability and dynamical behaviors of the proposed model.
- To identify conditions leading to period-doubling bifurcations and chaos.
Main Methods:
- Analytical investigation of the discrete-time model.
- Identification and characterization of period-doubling bifurcation curves.
- Numerical bifurcation analysis to explore parameter space.
- Assessment of global stability properties.
Main Results:
- The model exhibits a period-doubling bifurcation curve.
- This curve delineates regions of stability and instability in the parameter space.
- Numerical simulations reveal intermingled regions of periodic cycles and chaotic dynamics.
- Global stability of the model was investigated.
Conclusions:
- The discrete-time Ricker-Beverton-Holt model displays complex dynamics, including chaos.
- Period-doubling bifurcations play a significant role in transitions to complex dynamics.
- The model provides a framework for studying population stability and fluctuations.
- Further research can explore extensions and applications of this model.
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