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A Degenerate Bifurcation Perspective on High Sensitivity in a Modified Leslie-Gower Model with Additive Allee Effect
Xiaoling Wang1, Kuilin Wu1, Lan Zou2
1School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, People's Republic of China.
Journal of Mathematical Biology
|July 25, 2026
Summary
Population dynamics are sensitive to parameters and initial conditions, leading to complex outcomes like coextinction or multistable states. This study reveals intricate bifurcations and up to five limit cycles, highlighting initial condition dependence.
Area of Science:
- Mathematical Biology
- Theoretical Ecology
- Dynamical Systems Theory
Background:
- Population dynamics are crucial for ecological understanding.
- The Leslie-Gower model is a foundational tool for population modeling.
- Allee effects introduce complexity, influencing population persistence and stability.
Purpose of the Study:
- To analyze population dynamics in a modified Leslie-Gower model incorporating an additive Allee effect.
- To investigate the impact of parameter sensitivity and initial conditions on model outcomes.
- To identify and characterize complex bifurcations and oscillatory behaviors.
Main Methods:
- Utilized a modified Leslie-Gower model with an additive Allee effect.
- Performed bifurcation analysis to understand system responses to parameter changes.
- Established the existence of codimension 4 nilpotent cusp and degenerate Bogdanov-Takens bifurcations.
- Investigated Hopf bifurcations and the number of bifurcated limit cycles.
- Employed numerical simulations to confirm theoretical findings, including heteroclinic loops and multiple limit cycles.
Main Results:
- Population dynamics exhibit high sensitivity to parameters and initial densities.
- Outcomes range from coextinction to sustained multistable steady states.
- Complex bifurcations, including codimension 4 nilpotent cusp and degenerate Bogdanov-Takens bifurcations, were identified.
- Up to five limit cycles were observed bifurcating from Hopf bifurcations, a rare finding in ecological models.
- Oscillatory regimes demonstrated a strong dependence on initial population conditions.
Conclusions:
- The modified Leslie-Gower model with an additive Allee effect displays rich and complex population dynamics.
- Bifurcation theory is essential for understanding the sensitivity and multistability of these dynamics.
- The high number of limit cycles and their dependence on initial conditions present novel insights into ecological oscillations.
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