Global dynamics of a ratio-dependent predator-prey system

Area of Science:

• Mathematical Biology
• Ecology
• Dynamical Systems Theory

Background:

• Ratio-dependent predator-prey models are increasingly favored for interactions involving significant predator search time.
• These models exhibit undefined dynamics near the origin, posing a significant analytical challenge.
• Understanding behavior near the origin is crucial for comprehending the global dynamics of these ecological models.

Purpose of the Study:

• To investigate the qualitative behavior of a class of ratio-dependent predator-prey systems specifically at the origin.
• To analyze the topological structures present in the vicinity of the origin.
• To determine conditions for the existence and non-existence of limit cycles within the model.

Main Methods:

• Qualitative analysis of the predator-prey system's behavior at the origin in the first quadrant.
• Identification and characterization of various topological structures (parabolic, elliptic, hyperbolic orbits) near the origin.
• Global qualitative analysis incorporating all model parameters.
• Computer simulations to validate theoretical findings.

Main Results:

• The origin is confirmed as a higher-order critical point in these systems.
• Diverse topological structures, including combinations of parabolic, elliptic, and hyperbolic orbits, exist near the origin.
• Conditions governing the existence and absence of limit cycles have been established.
• Simulations visually confirm the derived theoretical conclusions.

Conclusions:

• The complex dynamics near the origin significantly influence the global behavior of ratio-dependent predator-prey models.
• The study provides a comprehensive understanding of the system's qualitative behavior across all parameter ranges.
• The findings offer valuable insights for refining ecological models and predicting population dynamics.