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Summary: Resolvent Positive Linear Operators
Exhibit the Reduction Phenomenon
Lee Altenberg
altenber@hawaii.edu
September 2, 2011
Abstract
The spectral bound, s(A + V ), of a combination of
a resolvent positive linear operator A and an operator
of multiplication V , was shown by Kato to be convex
in R. This is shown here to imply, through an
elementary lemma, that s(A + V ) is also convex in
> 0, and notably, s(A + V )/ s(A) when it
exists. Diffusions typically have s(A) 0, so that for
diffusions with spatially heterogeneous growth or decay
rates, greater mixing reduces growth. Models of the
evolution of dispersal in particular have found this result
when A is a Laplacian or second-order elliptic operator,
or a nonlocal diffusion operator, implying selection for
reduced dispersal. These cases are shown here to be part
of a single, broadly general, `reduction' phenomenon.1
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