title: Resolvent positive linear operators exhibit the reduction phenomenon.
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.
authors.affiliation: BioSystems, 2605 Lioholo Place, Kihei, HI 96753-7118, USA. altenber@hawaii.edu
authors.forename: Lee
authors.initials: L
authors.lastname: Altenberg
mesh_terms: D000465:Algorithms; D000483:Alleles; D001703:Biophysics / Q000379:methods*; D019295:Computational Biology / Q000379:methods*; D004058:Diffusion; D019143:Evolution, Molecular; D014644:Genetic Variation; D005823:Genetics; D005828:Genetics, Population; D008954:Models, Biological; D008957:Models, Genetic; D008962:Models, Theoretical; D010825:Physics / Q000379:methods; D011995:Recombination, Genetic