Algorithmic global criteria for excluding oscillations

Area of Science:

• Computational Biology
• Mathematical Biology
• Systems Biology

Background:

• Biological systems often exhibit complex dynamics, including oscillations.
• Understanding parameter ranges that support these oscillations is crucial for model validation and prediction.
• Algorithmic approaches are needed to analyze complex biological models.

Purpose of the Study:

• To develop and apply algorithmic methods for determining the existence of biologically relevant parameter ranges that yield oscillating trajectories in systems of ordinary differential equations.
• To investigate the connection between criteria for the absence of limit cycles and quantifier elimination problems.
• To apply these methods to models within the field of algebraic biology.

Main Methods:

• Formulation of the problem in terms of systems of parametric ordinary differential equations.
• Utilizing known criteria that exclude non-constant limit cycles.
• Transforming these criteria into quantifier elimination problems over the real numbers.
• Application to polynomial vector fields representing biological models.

Main Results:

• Established that criteria excluding non-constant limit cycles in polynomial systems correspond to quantifier elimination problems.
• Demonstrated the applicability of these methods to established models in algebraic biology.
• Provided a framework for algorithmic analysis of parameter-dependent oscillations in biological systems.

Conclusions:

• Algorithmic methods can effectively determine biologically meaningful parameter ranges for oscillating trajectories.
• Quantifier elimination provides a powerful tool for analyzing the existence of limit cycles in polynomial biological models.
• This approach enhances the analysis of dynamical behaviors in systems biology and algebraic biology.