BioSystems, 2605 Lioholo Place, Kihei, HI 96753-7118, USA.
Proc Natl Acad Sci U S A. 2012 Mar 6;109(10):3705-10. doi: 10.1073/pnas.1113833109. Epub 2012 Feb 22.
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.
Kato 证明了，对于一个正则线性算子 A 和一个乘法算子 V 的组合，其谱边界 s(αA + βV)在 β ∈ R 上是凸的。Kato 的结果通过一个基本的“对偶凸性”引理表明，s(αA + βV)在 α > 0 上也是凸的，特别地，∂s(αA + βV)/∂α ≤ s(A)。扩散过程通常有 s(A) ≤ 0，因此对于空间异质生长或衰减率的扩散过程，更大的混合会减少生长。特别地，在 A 是拉普拉斯算子或二阶椭圆算子或非局部扩散算子的扩散模型中，已经发现了这一结果，这意味着选择减少扩散。这里证明了这些情况是单个广泛的“简化”现象的一部分。
