Stability of limit cycles in a pluripotent stem cell dynamics model

Mostafa Adimy (1) , Fabien Crauste (1) , Andrei Halanay (2) , Mihaela Neamtu (3) , Dumitru Opris (4)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Department of Mathematics
3 Faculty of economics
4 Department of Applied Mathematics

This paper is devoted to the study of the stability of limit cycles of a nonlinear delay differential equation with a distributed delay. The equation arises from a model of population dynamics describing the evolution of a pluripotent stem cells population. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We then investigate the stability of the limit cycles yielded by the bifurcation using the normal form theory and the center manifold theorem. We illustrate our results with some numerics.

Mots clés: Delay differential equations; distributed delay; Hopf bifurcation; stability; limit cycles; normal form; center manifold; blood production system; hematopoietic stem cells

Domaines: Equations aux dérivées partielles [math.AP]

Mostafa Adimy, Fabien Crauste, Andrei Halanay, Mihaela Neamtu, Dumitru Opris. Stability of limit cycles in a pluripotent stem cell dynamics model. Chaos, Solitons & Fractals, 2006, 27 (4), pp.1091-1107. ⟨10.1016/j.chaos.2005.04.083⟩. ⟨hal-00376002⟩