Stability and Hopf bifurcation for a cell population model with state-dependent delay

Mostafa Adimy (1) , Fabien Crauste (1, 2) , Hassan Hbid (3) , Redouane Qesmi (4)

1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
2 ICJ - Institut Camille Jordan
3 Department of Mathematics [Marroco]
4 Department of Mathematics and Statistics [Toronto]

We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.

Mots clés: hematopoietic stem cells; functional differential equation; state-dependent delay; Lyapunov-Razumikhin function; Hopf bifurcation

Domaines: Equations aux dérivées partielles [math.AP]; Bio-Informatique, Biologie Systémique [q-bio.QM]; Biologie cellulaire

Mostafa Adimy, Fabien Crauste, Hassan Hbid, Redouane Qesmi. Stability and Hopf bifurcation for a cell population model with state-dependent delay. SIAM Journal on Applied Mathematics, 2010, 70 (5), pp.1611-1633. ⟨10.1137/080742713⟩. ⟨hal-00542655⟩