Stability and Hopf Bifurcation for a Cell Population Model with State-Dependent Delay

Abstract

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.

Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

“…One can show (see Adimy et al [7], Theorem 6.1) that the positive steady state N = N * undergoes a transcritical bifurcation when µ = µ. Moreover, when µ = 0, N * is locally asymptotically stable, since the only characteristic root of ∆(λ, 0) is λ = β ′ (N * (0))N * (0) < 0.…”

“…Existence and uniqueness of solutions of (5.1) are not straightforwardly obtained, yet this can be shown (see [7]) using Mallet-Paret et al [68] and Walther [89]. Moreover, (5.1) has a trivial steady state, N = 0, and a positive steady state N = N * provided that The positive steady state value is then given by…”

“…Assuming η and σ are decreasing on the interval [0, β −1 (δ)], and η β −1 (δ) < −2δ, then there exists a unique µ * ∈ (0, µ) such that condition (5.7) is satisfied if and only if µ ∈ [0, µ * ) (see Adimy et al [7]). …”

“…However, using the Implicit Function Theorem, one can show that N * is a decreasing continuously differentiable function of µ [7]. We will show that Equation (5.1) can have oscillating solutions following a Hopf bifurcation at the positive steady state, and we will use µ as the bifurcation parameter.…”

“…Bernard et al [21] used the Mackey's model to study the existence of oscillations in cyclic neutropenia. Adimy et al [3][4][5][6][7][8][9][10][11][12] analyzed various versions of Mackey's model and investigated the effect of perturbations of cell cycle duration, the rate of apoptosis, differentiation rate, and reintroduction rate from the quiescent compartment to the proliferating one on the behavior of the cell population.…”