Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods
Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods
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DOI：
10.1007/s00285-019-01357-0
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发表时间：
2019-04
影响因子：
1.9
通讯作者：
P. Getto;M. Gyllenberg;Y. Nakata;F. Scarabel
中科院分区：
文献类型：
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作者：
P. Getto;M. Gyllenberg;Y. Nakata;F. Scarabel
我们考虑一个数学模型描述的成熟过程的干细胞完全成熟的细胞。该模型被配制成一个微分方程与状态依赖延迟，成熟度被描述为一个连续变量。细胞的成熟率可以由成熟细胞的数量来调节，而且，它可能取决于细胞的成熟度：我们研究如何平衡的稳定性是受成熟率的选择。我们表明，该模型的线性化稳定性的原则，并制定了一些分析方法的调查固定时滞的特征方程。对于一个一般的成熟率，我们诉诸数值方法，我们扩展的伪谱离散技术近似的状态依赖的延迟方程与常微分方程系统。这是该技术的第一个应用程序的非线性状态依赖延迟方程，目前唯一的方法可用于研究平衡点的稳定性，通过建立的软件包进行分叉分析。在成熟率与成熟度无关的情况下，数值方法得到了验证，通过适当的时间变换，模型可以转化为固定时滞方程。我们利用解析和数值方法来研究参数平面上的稳定边界。我们的研究表明，一些激烈的定性变化的稳定边界的假设下的模型参数，这可能具有重要的生物意义。
We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.