  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.
