Structural stability of invasion graphs for Lotka-Volterra systems

Area of Science:

• Mathematical Biology
• Ecology
• Dynamical Systems Theory

Background:

• The Lotka-Volterra system is a fundamental model in population dynamics.
• Understanding the global attractor is crucial for analyzing system long-term behavior.
• Structural stability in ecological models requires robust dynamical structures.

Purpose of the Study:

• To analyze the global attractor structure of Lotka-Volterra systems with Volterra-Lyapunov stable matrices.
• To establish the role of the invasion graph in representing heteroclinic connections.
• To define and investigate structural stability in ecology based on mathematical principles.

Main Methods:

• Detailed analysis of the global attractor for the Lotka-Volterra system.
• Utilizing the invasion graph framework (Hofbauer & Schreiber, 2022).
• Investigating parameter perturbation effects on the system's structure.

Main Results:

• The edges of the invasion graph precisely map all heteroclinic connections between system equilibria.
• The geometrical structure governing transient and asymptotic dynamics is robust under parameter perturbation.
• A novel definition of structural stability in ecology is proposed, aligning with mathematical concepts.

Conclusions:

• The invasion graph provides a complete and stable representation of Lotka-Volterra system dynamics.
• The proposed definition of structural stability offers a rigorous framework for ecological modeling.
• This work enhances the understanding of ecological system robustness and predictability.