## Karlin Theory On Growth and Mixing
Extended to Linear Differential Equations

### Lee Altenberg

2010.

**Abstract**

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** ≠ 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**^{T} **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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