Hopf bifurcation in delayed nutrient-microorganism model with network structure.

Area of Science:

• Mathematical Biology
• Dynamical Systems
• Network Science

Background:

• Nutrient-microorganism models are crucial for understanding ecological dynamics.
• Hopf bifurcation analysis is essential for studying stability and oscillations in biological systems.
• Investigating delayed systems with random network structures presents unique challenges.

Purpose of the Study:

• To introduce and analyze a delayed nutrient-microorganism model with a random network structure.
• To determine the direction of the Hopf bifurcation using time delay as the critical parameter.
• To address the rarity of Hopf bifurcation direction results in random network models.

Main Methods:

• Employing the method of multiple time scales (MTS).
• Deriving an amplitude equation to analyze the system's dynamics.
• Utilizing time delay (τ) as the primary bifurcation parameter.
• Conducting numerical experiments to validate theoretical findings.

Main Results:

• The delayed random network nutrient-microorganism model can exhibit both supercritical and subcritical Hopf bifurcations.
• The direction of the Hopf bifurcation was successfully determined using the MTS method.
• Theoretical predictions were corroborated by numerical simulations.

Conclusions:

• The study provides a novel analysis of Hopf bifurcation direction in delayed random network models.
• The findings contribute to a deeper understanding of stability and oscillatory behaviors in complex ecological networks.
• The MTS method proves effective for analyzing such intricate systems.