Karlin增长与混合理论推广到线性微分方程
Karlin Theory On Growth and Mixing Extended to Linear Differential Equations
摘要：
Karlin（1982）定理5.2表明，在增长和混合阶段之间交替的线性系具有较低的渐近增长和较大的混合。在这里，这一结果被扩展到线性微分方程，结合网站特定的增长或衰减率，和网站之间的混合，显示的光谱横坐标的矩阵D + M A减少与m，其中D不等于C我是一个真正的对角矩阵，A是一个不可约矩阵与非负的非对角元素（ML-或基本上非负矩阵），和m >= 0。其结果是基于不等式：u'Av < r（A），其中u和v是矩阵D + A的左Perron向量和右Perron向量，r（A）是A的谱横坐标和Perron根.该结果为以往的研究工作提供了一个分析解，这些研究工作依赖于对微生物生长和混合模型的两点或数值模拟，如源和库生态模型，或微生物生长的多组织室模型。所得结果可应用于非线性系统扰动的李雅普诺夫稳定性。
英文摘要：
Karlin's (1982) Theorem 5.2 shows that linear systems alternating between growth and mixing phases have lower asymptotic growth with greater mixing. Here this result is extended to linear differential equations that combine site-specific growth or decay rates, and mixing between sites, showing that the spectral abscissa of a matrix D + m A decreases with m, where D does-not-equal c I is a real diagonal matrix, A is an irreducible matrix with non-negative off-diagonal elements (an ML- or essentially non-negative matrix), and m >= 0. The result is based on the inequality: u' A v < r(A), where u and v are the left and right Perron vectors of the matrix D + A, and r(A) is the spectral abscissa and Perron root of A. The result gives an analytic solution to prior work that relied on two-site or numerical simulation of models of growth and mixing, such as source and sink ecological models, or multiple tissue compartment models of microbe growth. The result has applications to the Lyapunov stability of perturbations in nonlinear systems.
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