Dynamics of the Selkov oscillator

Area of Science:

• Mathematical Biology
• Biochemistry
• Dynamical Systems Theory

Background:

• The Selkov oscillator is a fundamental two-variable ordinary differential equation model describing glycolysis.
• Despite its importance, a complete rigorous analysis of its dynamics has been lacking.
• Understanding these dynamics is crucial for modeling metabolic oscillations.

Purpose of the Study:

• To provide a comprehensive mathematical analysis of the Selkov oscillator model.
• To investigate the behavior of solutions, including bounded and unbounded trajectories.
• To characterize the system's dynamics under different parameter regimes.

Main Methods:

• Analysis of the Poincaré compactification to study unbounded solutions.
• Investigation of the stability of the unique steady state.
• Characterization of bounded and unbounded solution convergence properties.

Main Results:

• Proved the existence of unbounded, eventually monotone solutions for all parameter values.
• Demonstrated that stable steady states attract all bounded solutions.
• Showed that unique periodic solutions attract all bounded non-steady-state solutions when the steady state is unstable.
• Established that all non-steady-state solutions are unbounded if no periodic solution exists.

Conclusions:

• The Selkov oscillator exhibits rich dynamics, including unbounded oscillations and convergence to stable steady states or unique periodic orbits.
• This rigorous analysis provides a deeper understanding of metabolic regulation modeled by glycolysis.
• The findings are essential for further theoretical and experimental studies of biological oscillations.