Singular homoclinic bifurcations in tritrophic food chains

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• The Rosenzweig-MacArthur model is a fundamental ecological model describing predator-prey dynamics.
• Understanding the long-term stability and behavior of ecological models is crucial for predicting population dynamics.
• Homoclinic orbits represent complex dynamic behaviors in ecological systems.

Purpose of the Study:

• To rigorously prove the existence of homoclinic orbits in the Rosenzweig-MacArthur food chain model.
• To identify specific parameter conditions that lead to the formation of these orbits.
• To present a methodology applicable to other complex dynamical systems.

Main Methods:

• A two-step approach combining geometric singular perturbation theory and numerical analysis.
• Identification of singular homoclinic orbits through a geometric approach.
• Numerical verification of nonsingular homoclinic orbits for perturbed parameter values.

Main Results:

• The existence of singular homoclinic orbits in the Rosenzweig-MacArthur model is mathematically proven.
• Nonsingular homoclinic orbits are demonstrated numerically under slightly varied parameter conditions.
• The findings are robust and do not rely on overly specific model properties.

Conclusions:

• The Rosenzweig-MacArthur food chain model exhibits homoclinic orbits, indicating complex dynamical behaviors.
• The methodology provides a framework for analyzing such orbits in ecological and other dynamical systems.
• This research enhances the understanding of stability and long-term dynamics in ecological modeling.