Resolvent positive linear operators exhibit the reduction phenomenon

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 . 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.