Resolvent positive linear operators exhibit the reduction phenomenon.

Proceedings of the National Academy of Sciences of the United States of America
Lee Altenberg

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.

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Citations

Nov 28, 2014·Proceedings of the National Academy of Sciences of the United States of America·Oana CarjaMarcus W Feldman
Jul 29, 2015·Journal of Theoretical Biology·Tiago PaixãoBarbora Trubenová
Oct 22, 2014·Journal of Biological Dynamics·King-Yeung LamFrithjof Lutscher
Feb 21, 2019·Journal of Mathematical Biology·Gabriel MacielFrithjof Lutscher
Jun 13, 2020·AoB Plants·Sebastian J Schreiber, Noelle G Beckman
Mar 8, 2017·Proceedings of the National Academy of Sciences of the United States of America·Lee AltenbergMarcus W Feldman
Nov 15, 2016·Mathematical Biosciences·Robert Stephen CantrellSebastian J Schreiber
Mar 16, 2021·Journal of Mathematical Biology·Emeric BouinNichole Slover

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