An asymptotic maximum principle for essentially linear evolution models
- Baake, Ellen
- Baake, Michael
- Bovier, Anton
- Klein, Markus
2010 Mathematics Subject Classification
- 15A18 95D15 60J80.
- asymptotics of leading eigenvalue, reversibility, mutation-selection models, ancestral distribution, lumping
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 variational principle in the limit $N to infty$ (where $N$ is the number of types). In order to generalize these results, we consider here a large family of reversible $N times N$ matrices 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.
- J. Math. Biol. (2004). Math. Biol., 50, no. 1, pp. 83-114, 2005, and, DOI 10.1007/s00285-004-0281-7J .