title: Solution of classical evolutionary models in the limit when the diffusion approximation breaks down.
abstract: The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.
authors.affiliation: Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.|A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers St., Yerevan 375036, Armenia.
authors.forename: David B
authors.initials: DB
authors.lastname: Saakian
authors.affiliation: Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.|National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu 30013, Taiwan.|Business School, University of Shanghai for Science and Technology, Shanghai 200093, China.
authors.forename: Chin-Kun
authors.initials: CK
authors.lastname: Hu
mesh_terms: D005075:Biological Evolution; D005828:Genetics, Population; D008957:Models, Genetic; D009154:Mutation; D012641:Selection, Genetic; D013995:Time