The spectral bound, s(αA + βV), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in β ∈ R. Kato's result is shown here to imply, through an elementary "dual convexity" lemma, that s(αA + βV) is also convex in α > 0, and notably, ∂s(αA + βV)/∂α ≤ s(A). 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.
%0 Journal Article
%1 altenberg2012resolvent
%A Altenberg, L
%D 2012
%J Proc Natl Acad Sci U S A
%K dispersal evolution_of_dispersal population_dynamics spatial_structure
%N 10
%P 3705-3710
%R 10.1073/pnas.1113833109
%T Resolvent positive linear operators exhibit the reduction phenomenon
%U http://www.ncbi.nlm.nih.gov/pubmed/22357763?dopt=Abstract
%V 109
%X The spectral bound, s(αA + βV), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in β ∈ R. Kato's result is shown here to imply, through an elementary "dual convexity" lemma, that s(αA + βV) is also convex in α > 0, and notably, ∂s(αA + βV)/∂α ≤ s(A). 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.
@article{altenberg2012resolvent,
abstract = {The spectral bound, s(αA + βV), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in β ∈ R. Kato's result is shown here to imply, through an elementary "dual convexity" lemma, that s(αA + βV) is also convex in α > 0, and notably, ∂s(αA + βV)/∂α ≤ s(A). 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.},
added-at = {2014-10-07T23:59:08.000+0200},
author = {Altenberg, L},
biburl = {https://www.bibsonomy.org/bibtex/2faf1dfc40484acd44bf823f419616e9d/peter.ralph},
doi = {10.1073/pnas.1113833109},
interhash = {2282d86a12ba3d9482a737bdf0c58182},
intrahash = {faf1dfc40484acd44bf823f419616e9d},
journal = {Proc Natl Acad Sci U S A},
keywords = {dispersal evolution_of_dispersal population_dynamics spatial_structure},
month = mar,
number = 10,
pages = {3705-3710},
pmid = {22357763},
timestamp = {2014-10-07T23:59:08.000+0200},
title = {Resolvent positive linear operators exhibit the reduction phenomenon},
url = {http://www.ncbi.nlm.nih.gov/pubmed/22357763?dopt=Abstract},
volume = 109,
year = 2012
}