Two Little Known Theorems of Sam Karlin: Their implications for biological invasions, genetic system evolution, cultural evolution, and ideal free distributions in ecology
Ronin Institute, Ronin Institute
(October 15, 2013 10:20 AM - 11:15 AM)
Sam Karlin was one of the great mathematicians to contribute to evolutionary theory, both in the forward direction (dynamics) and the backward direction (bioinformatics). Karlin introduced two theorems in 1976 to analyze the effect of population subdivision on the protection of genetic diversity. Both could be summarized as the phenomenon that mixing reduces growth, with the consequence that greater dispersal in heterogenous environments reduces the survival of rare alleles.
They provide the basis to prove very generally that populations can always be invaded by genetic variants for information transmission (mutation and recombination) that better preserve information during reproduction --- the Reduction Principle initially discovered by Marc Feldman. The Reduction Principle has made an appearance in recent work on reaction diffusion models of dispersal in continuous space. Could a continuous-space version of Karlin's theorem be at work here? I will describe my recent extension of Karlin's theorem to infinite dimensional Banach spaces. This result unifies the reaction diffusion models showing that the slower disperser wins. It also applies to the generators of strongly continuous semigroups, elliptic operators, Schrivdinger operators, and local and nonlocal diffusions. The phenomenon that mixing reduces growth and hastens decay could be described as a universal phenomenon.