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Delay model of hematopoietic stem cell dynamics: Asymptotic stability and stability switch. (English) Zbl 1186.34117

The author investigates a model of hematopoietic stem cells consisting of two delay differential equations \[ \begin{aligned}\frac{dN}{dt}(t)&=-\delta N(t)-\beta(S(t))N(t)+2e^{-\gamma\tau}\beta(S(t-\tau))N(t-\tau),\\ \frac{dP}{dt}(t)&=-\gamma P(t)+\beta(S(t))N(t)-e^{-\gamma\tau}\beta(S(t-\tau))N(t-\tau)\end{aligned} \] with \(S(t)=P(t)+N(t)\), where each equation above describes the evolution of a sub-population, nonproliferating and proliferating cells, whose densities at time \(t\) are respectively denoted by \(N(t)\) and \(P(t)\). The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. The existence and stability of steady states are studied. By constructing a suitable Lyapunov functional, the author obtains the global asymptotic stability of the trivial equilibrium provided it is the only steady state. If there exists a positive steady state, then it is unique. In this case, by means of a technique developed by [E. Beretta and Y. Kuang, SIAM J. Math. Anal. 33, No. 5, 1144–1165 (2002; Zbl 1013.92034)], the author analyzes the eigenvalues of a second degree exponential polynomial characteristic equation and obtains the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches is also carried out.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
92C37 Cell biology

Citations:

Zbl 1013.92034

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