Rey Alejandro D., Mackey Michael C.
Department of Chemical Engineering, McGill University, 3480 University Street, Montreal, Quebec H3A 2A7, CanadaDepartments of Physiology, Physics, and Mathematics, and Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, 3655 Drummond Street, Montreal H3G 1Y6, Canada.
Chaos. 1992 Apr;2(2):231-244. doi: 10.1063/1.165909.
Here cell population dynamics in which there is simultaneous proliferation and maturation is considered. The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters. The solution behavior depends on the initial function varphi(x) and a three component parameter vector P=(delta,lambda,r). For strictly positive initial functions, varphi(0) greater, similar 0, there are three homogeneous solutions of biological (i.e., non-negative) importance: a trivial solution u(t) identical with 0, a positive stationary solution u(st), and a time periodic solution u(p)(t). For varphi(0)=0 there are a number of different solution types depending on P: the trivial solution u(t), a spatially inhomogeneous stationary solution u(nh)(x), a spatially homogeneous singular solution u(s), a traveling wave solution u(tw)(t,x), slow traveling waves u(stw)(t,x), and slow traveling chaotic waves u(scw)(t,x). The regions of parameter space in which these solutions exist and are locally stable are delineated and studied.
在此，我们考虑细胞群体动力学，其中细胞同时进行增殖和成熟。由此产生的数学模型是关于细胞密度(u(t,x))的非线性一阶偏微分方程，该方程在时间变量((t))和成熟变量((x))上均存在延迟，并且包含三个参数。解的行为取决于初始函数(\varphi(x))和一个三分量参数向量(P = (\delta,\lambda,r))。对于严格正的初始函数，即(\varphi(0) \gt 0)，存在三种具有生物学（即非负）重要性的齐次解：平凡解(u(t) \equiv 0)、正的稳态解(u_{st})和时间周期解(u_p(t))。对于(\varphi(0) = 0)，根据(P)的不同存在多种不同类型的解：平凡解(u(t))、空间非齐次稳态解(u_{nh}(x))、空间齐次奇异解(u_s)、行波解(u_{tw}(t,x))、慢行波解(u_{stw}(t,x))以及慢行混沌波解(u_{scw}(t,x))。我们描绘并研究了这些解存在且局部稳定的参数空间区域。
