We revisit the multi-allelic mutation-fitness balance problem especially when fitnesses are multiplicative. Using ideas arising from quasi-stationary distributions, we analyze the qualitative differences between the fitness-first and mutation-first models, under various schemes of the mutation pattern. We give some stochastic domination relations between the equilibrium states resulting from these models.

W. J. Ewens, “Mathematical Population Genetics. I. Theoretical Introduction,” 2nd Edition, Interdisciplinary Applied Mathematics, Springer-Verlag, New York, Vol. 27, 2004.

R. Bürger, “The Mathematical Theory of Selection, Recombination, and Mutation,” Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

S. Karlin, “Mathematical Models, Problems, and Controversies of Evolutionary Theory,” Bulletin of the American Mathematical Society (N.S.), Vol. 10, No. 2, 1984, pp. 221-273.

J. F. C. Kingman, “Mathematics of Genetic Diversity,” CBMS-NSF Regional Conference Series in Applied Mathematics, 34. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1980.

J. Hermisson, O. Redner, H. Wagner and E. Baake “Mutation-Selection Balance: Ancestry, Load and Maximum Principle,” Theoretical Population Biology, Vol. 62, No. 1, 2002, pp. 9-46. doi:10.1006/tpbi.2002.1582

E. Baake and H.-O. Georgii, “Mutation, Selection, and Ancestry in Branching Models: A Variational Approach,” Journal of Mathematical Biology, Vol. 54, No. 2, 2007, pp. 257-303. doi:10.1007/s00285-006-0039-5

G. Sella and A. E. Hirsh, “The Application of Statistical Physics to Evolutionary Biology,” Proceedings of the National Academy of Sciences, Vol. 102, No. 27, 2005, pp. 9541-9546. doi:10.1073/pnas.0501865102

N. Champagnat, R. Ferrière and S. Méleard, “From Individual Stochastic Processes to Macroscopic Models in Adaptive evolution,” Stochastic Models, Vol. 24, Suppl. 1, 2008, pp. 2-44. doi:10.1080/15326340802437710

S. Shashahani, “A New Mathematical Framework for the Study of Linkage and Selection,” Memoirs of the American Mathematical Society, Vol. 17, No. 211, 1979, pp.1-34.

Y. M. Svirezhev, “Optimum Principles in Genetics,” Studies on Theoretical Genetics, V. A. Ratner (Ed.), USSR Academy of Science, Novosibirsk, 1972, pp. 86-102.

J. N. Darroch and E. Seneta, “On Quasi-Stationary Distributions in Absorbing Discrete-Time Finite Markov chains,” Journal of Applied Probability, Vol. 2, No. 1, 1965, pp. 88-100. doi:10.2307/3211876

S. Martínez, “Quasi-Stationary Distributions for Birth-death Chains. Convergence Radii and Yaglom Limit,” Cellular Automata and Cooperative Systems (Les Houches, 1992), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 396, Kluwer Academic Publishers, Dordrecht, 1993, pp. 491-505.

S. Martínez and M. E. Vares, “A Markov Chain Associated with the Minimal Quasi-Stationary Distribution of Birth-Death Chains,” Journal of Applied Probability, Vol. 32, No. 1, 1995, pp. 25-38. doi:10.2307/3214918