Algorithmic global criteria for excluding oscillations

In: Bulletin of Mathematical Biology (Apr. 2011), 73:4(899-917)
 

Abstract

We investigate algorithmic methods to tackle the following problem: Given a system of parametric ordinary differential equations built by a biological model, does there exist ranges of values for the model parameters and variables which are both meaningful from a biological point of view and where oscillating trajectories, can be found? We show that in the common case of polynomial vector fields known criteria excluding the existence of non-constant limit cycles lead to quantifier elimination problems over the reals.

We apply these criteria to various models that have been previously investigated in the context of algebraic biology.

Bibtex

@ARTICLE{WeberSturm2010a,
    author = {Weber, Andreas and Sturm, Thomas and Abdel-Rahman, Essam O.},
     pages = {899--917},
     title = {Algorithmic global criteria for excluding oscillations},
   journal = {Bulletin of Mathematical Biology},
    volume = {73},
    number = {4},
      year = {2011},
     month = apr,
      note = {Special issue on ``Algebraic Biology''.},
  abstract = {We investigate algorithmic methods to tackle the following problem:  Given a system of parametric
              ordinary differential equations built by  a biological model,   does there exist ranges of values
              for the model parameters and variables which are both meaningful  from a biological point of view
              and where oscillating trajectories,  can be found? We show that in the common case of polynomial
              vector fields known criteria excluding the existence of  non-constant limit cycles lead to
              quantifier elimination problems  over the reals.
              
               We apply these criteria to various models that have been previously  investigated in the context of
              algebraic biology.},
       doi = {10.1007/s11538-010-9618-0}
}