An asymptotic maximum principle for essentially linear evolution models.
Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N-->infinity (where N, or N(d) with d> or =1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N(d) and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.
Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. ebaake@techfak.uni-bielefeld.de
Ellen
E
Baake
Michael
M
Baake
Anton
A
Bovier
Markus
M
Klein
D000818:Animals; D019143:Evolution, Molecular; D005828:Genetics, Population; D016014:Linear Models; D008390:Markov Chains; D008957:Models, Genetic; D009154:Mutation; D012641:Selection, Genetic; D013269:Stochastic Processes