Speaker: Lee Altenberg (ICS UH Mānoa)
TITLE: Application of Dual Convexity to the Spectral Bound of Resolvent Positive Operators
Donsker and Varadhan (1975) developed a variational expression for the spectral bound of generators of strongly continuous positive semigroups, which Karlin (1981) applied to finite matrices to show that the spectral radius of product [(1-m) I + m P]D decreases monotonically in m, where P is a stochastic matrix and D a positive diagonal matrix. This result has key applications in evolutionary and population dynamics. Simultaneously, Cohen (1981) showed that the spectral bound of essentially nonnegative matrices is convex in the diagonal elements. Kato (1982) extended Cohen’s result to resolvent positive operators. A “dual convexity” lemma shows that Cohen’s and Karlin’s results are actually equivalent, and via Kato’s theorem, allows extension of Karlin’s theorem to all resolvent positive operators, showing that d/dm s(L m + V) <= s(L), where s is the spectral bound, L is a resolvent positive operator, and V is an operator of multiplication. The motivation behind this work is that it unifies many results in reaction diffusion theory and shows generally that increased diffusion in the presence of local variation in decay or growth rates will decrease asymptotic growth rates.