Stability and Hopf bifurcation analysis in a delayed three-node circuit involving interlinked positive and negative...

Area of Science:

• Systems Biology
• Molecular Biology
• Biophysics

Background:

• Gene regulatory networks (GRNs) utilize feedback loops for dynamic control.
• Time delays are inherent in GRNs due to spatial separation of transcription and translation.
• Understanding delay effects is crucial for predicting network behavior.

Purpose of the Study:

• To systematically investigate the impact of time delays on the dynamics of a three-node gene regulatory circuit.
• To derive conditions for stability and oscillatory behavior in the presence of multiple time delays.
• To analyze the stability and direction of Hopf bifurcations induced by time delays.

Main Methods:

• Linear stability analysis
• Bifurcation theory
• Hopf bifurcation analysis
• Normal form method
• Center manifold theorem

Main Results:

• Sufficient conditions for the stability of equilibria were established.
• Conditions for oscillatory behaviors via Hopf bifurcation were derived.
• The stability and direction of Hopf bifurcations were analyzed with respect to time delays and feedback strengths.
• Analytical results were illustrated with numerical examples.

Conclusions:

• Time delays significantly influence the stability and oscillatory dynamics of gene regulatory circuits.
• Bifurcation theory provides a robust framework for analyzing delay-induced behaviors in biological networks.
• The findings offer insights into the design principles and robustness of genetic control systems.