An asymptotic maximum principle for essentially linear evolution models

Details An asymptotic maximum principle for essentially linear evolution models An asymptotic maximum principle for essentially linear evolution models

2003-11-13

arxiv.org/abs/q-bio/0311020v2

J. Math. Biol. 50 (2005), 83--114

Biology

Quantitative Biology

Populations and Evolution

31 pages, 2 figures. Thorough revision, additional material included. J. Math. Biol., in press

Scientific paper

10.1007/s00285-004-0281-7

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 to infinity (where N is the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible N by 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.

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