Systèmes dynamiques
Sensitive dependence and singular dynamics in gene network models.
Edwards, Roderick
University of Victoria

Active regulation in gene networks poses mathematical challenges regarding flow in transition regions that have led to conflicting approaches to analysis. That is, competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. We have argued that the spurious solutions of the Filippov approach could be avoided, and an extension, due to Artstein and coauthors, of the classical singular perturbation approach is an appropriate way to handle the most complex situation, where non-trivial dynamics of fast variables occurs in singular domains. Now we show that even in this context, it is possible for non-uniqueness to arise in such a system in the case of limiting step-function interactions, and thus cannot be avoided completely, even if we avoid the overly inclusive set-valued Filippov definition of solutions. Real gene networks have sigmoidal interactions, however, and in the examples considered here, it is shown that the corresponding behaviour in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different attractors.

Jeudi, 20 juin, 11h30
Salle Des Plaines A