Canard phenomenon in a slow-fast modified Leslie-Gower model

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Ecology

Background:

• Geometrical Singular Perturbation Theory (GSPT) effectively analyzes biological systems with disparate timescales.
• The Leslie-Gower model is a foundational predator-prey model in mathematical ecology.
• Previous analyses have not fully explored folded singularities in modified Leslie-Gower models.

Purpose of the Study:

• To apply GSPT to a modified Leslie-Gower predator-prey model featuring a significant prey reproduction advantage.
• To investigate the dynamics of a slow-fast system arising from a small parameter (ϵ) representing timescale differences.
• To analyze a novel folded singularity and its impact on system behavior.

Main Methods:

• Utilized geometrical singular perturbation theory (GSPT) to analyze the slow-fast system.
• Employed the blow-up technique to visualize and understand dynamics near the folded singularity.
• Applied classical regular and singular perturbation theory to analyze regions outside the fold.

Main Results:

• Quantified the attractive limit-cycle with a geometric approach, achieving an error of O(ϵ).
• Identified and analyzed a unique folded singularity not previously studied in this context.
• Demonstrated the presence of the canard phenomenon as the system trajectory crosses the folded singularity.

Conclusions:

• GSPT provides a robust framework for analyzing predator-prey models with significant timescale differences.
• The folded singularity and associated canard phenomenon significantly influence the model's dynamics.
• This analysis offers a precise quantification of ecological model behavior under perturbation.