publication . Preprint . Article . Other literature type . 2012

Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon

Lee Altenberg;
Open Access English
  • Published: 22 Feb 2012
Abstract
The spectral bound, s(a A + b 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 b \in R. This is shown here, through an elementary lemma, to imply that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) / \partial a <= 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 selecti...
Subjects
free text keywords: Mathematics - Spectral Theory, Quantitative Biology - Populations and Evolution, 47B60, G.1.8, Physical Sciences, Multidisciplinary, Mathematical analysis, Regular polygon, Elliptic operator, Convexity, Laplace operator, Linear map, Lemma (mathematics), Mathematics, Resolvent, Operator (computer programming)
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