Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay describing the dynamics of blood cell production. This process takes place in the bone marrow where stem cells differentiate throughout divisions in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells on the introduction of cells in cycle, we are lead to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the population’s dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.

💡 Research Summary

The paper presents a rigorous mathematical investigation of a delayed two‑dimensional nonlinear system that models hematopoietic stem cell (HSC) dynamics in the bone marrow. The authors start by formulating the model: a population of quiescent (resting) stem cells (x(t)) and a population of cycling cells (y(t)). Quiescent cells die at rate (\delta) and enter the cell‑cycle at a rate that depends on the total HSC population (X(t)=x(t)+y(t)) through a saturating function (\beta/(1+X(t))). After a fixed maturation delay (\tau), cycling cells divide, producing two quiescent cells, while also experiencing a death rate (\gamma). The resulting system of delay differential equations (DDEs) captures the feedback of the overall stem‑cell pool on the recruitment of cells into division, a feature absent in many earlier models.

Two equilibria are identified. The trivial equilibrium (E_{0}=(0,0)) corresponds to extinction of the stem‑cell population, while a unique positive equilibrium (E^{}=(x^{},y^{})) exists under biologically realistic parameter ranges. The authors derive explicit algebraic relations for (E^{}) that link the recruitment rate (\beta) with the death rates (\delta) and (\gamma).

The first major contribution is a global asymptotic stability (GAS) result for (E_{0}). By constructing a Lyapunov–Krasovskii functional that incorporates both the current state and the delayed state, they show that (\dot V\le0) for all admissible solutions. Applying LaSalle’s invariance principle yields a necessary and sufficient condition for GAS: the recruitment strength must satisfy (\beta<\delta+\gamma). Under this inequality the coefficients of the characteristic equation remain positive for every delay (\tau\ge0), guaranteeing that all eigenvalues lie in the left half‑plane. Consequently, any trajectory starting with non‑negative initial data converges to extinction, regardless of the length of the maturation delay.

The second major contribution concerns the non‑trivial equilibrium (E^{}). Linearizing the system around (E^{}) leads to a characteristic equation of the form
\