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Stability analysis of an equation with two delays and application to the production of platelets

Loïs Boullu 1, 2, 3, 4, 5, * Laurent Pujo-Menjouet 1, 3, 4, 5 Jacques Bélair 2 
* Corresponding author
1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire, Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne]
3 MMCS - Modélisation mathématique, calcul scientifique
ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: x'(t) = −γx(t) + g(x(t − τ1)) − g(x(t − τ1 − τ2))e(−γ τ2). The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudo-phase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate γ or of a decrease of the survival time τ2 of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 1/day).
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Loïs Boullu, Laurent Pujo-Menjouet, Jacques Bélair. Stability analysis of an equation with two delays and application to the production of platelets. Discrete and Continuous Dynamical Systems - Series S, 2020, 13 (11), pp.3005-3027. ⟨10.3934/dcdss.2020131⟩. ⟨hal-02109546⟩



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