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Huillet, Thierry; Martínez, Servet; Möhle, Martin

On polymorphism for discrete evolutionary dynamics driven either by selection or segregation distortion. (English) Zbl 1397.92499

Comput. Appl. Math. 37, No. 2, 1352-1368 (2018).
Summary: We revisit some problems arising in the context of multiallelic discrete-time and deterministic evolutionary dynamics driven first by fitness differences and then by segregation distortion. In the model with fitness, we describe classes of fitness matrices exhibiting polymorphism. In the segregation case, still in search for conditions of polymorphism, we focus on a class of skew-symmetric matrices with a unique strictly positive kernel vector. Our main results reduce the study of these cases to the analysis of stochastic matrices.

MSC:

92D15 Problems related to evolution
15B51 Stochastic matrices
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Keywords:

species frequencies dynamics; selection; segregation; polymorphism; stability; potential and stochastic matrices
PDFBibTeX XMLCite
\textit{T. Huillet} et al., Comput. Appl. Math. 37, No. 2, 1352--1368 (2018; Zbl 1397.92499)
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References:

[1] Atkinson, FV; Watterson, GA; Moran, PAP, A matrix inequality, Quart J Math Oxford Ser, 11, 137-140, (1960) · Zbl 0136.24905 · doi:10.1093/qmath/11.1.137
[2] Bapat RB, Raghavan TES (1997) Nonnegative Matrices and Applications. Cambridge University Press, vol 64 · Zbl 1041.92021
[3] Bürger R (2000) The mathematical theory of selection, recombination, and mutation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester · Zbl 0959.92018
[4] Castilloux, AM; Lessard, S, The fundamental theorem of natural selection in ewens’ sense (case of many loci), Theor Popul Biol, 48, 306-315, (1995) · Zbl 0840.92015 · doi:10.1006/tpbi.1995.1031
[5] Dellacherie, C; Martínez, S; San Martín, J, Description of the sub-Markov kernel associated to generalized ultrametric matrices. an algorithmic approach, Linear Algebra Appl, 318, 1-21, (2000) · Zbl 0967.65054 · doi:10.1016/S0024-3795(00)00193-2
[6] Ewens WJ (2004) Mathematical population genetics. I.Theoretical introduction. Second edition. Interdisciplinary Applied Mathematics, 27. Springer-Verlag, New York · Zbl 1151.92024
[7] Ganikhodzhaev R, Mukhamedov F, Rozikov U (2011) Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 14 (2), 279 · Zbl 1242.60067
[8] Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge · Zbl 0914.90287 · doi:10.1017/CBO9781139173179
[9] Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge · Zbl 0576.15001 · doi:10.1017/CBO9780511810817
[10] Karlin, S, Mathematical models, problems, and controversies of evolutionary theory, Bull Am Math Soc (N.S.), 10, 221-274, (1984) · Zbl 0552.92013 · doi:10.1090/S0273-0979-1984-15239-X
[11] Kimura, M, Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities, Genetics, 39, 280-295, (1954)
[12] Kingman, JFC, A mathematical problem in population genetics, Proc Camb Philos Soc, 57, 574-582, (1961) · Zbl 0104.38202 · doi:10.1017/S0305004100035635
[13] Kingman, JFC, On an inequality in partial averages, Quart J Math Oxford Ser, 12, 78-80, (1961) · Zbl 0099.24802 · doi:10.1093/qmath/12.1.78
[14] Kingman JFC (1980) Mathematics of genetic diversity. CBMS-NSF Regional Conference Series in Applied Mathematics, 34. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa · Zbl 0458.92009
[15] Lessard, S; Karlin, S, A criterion for stability-instability at fixation states involving an eigenvalue one with applications in population genetics, Theor Popul Biol, 22, 108-126, (1982) · Zbl 0518.92013 · doi:10.1016/0040-5809(82)90038-7
[16] Lewontin, RC; Ginzburg, LR; Tuljapurkar, SD, Heterosis as an explanation for large amounts of genic polymorphism, Genetics, 88, 149-170, (1978)
[17] Liberman, U, Mendelian segregation: a choice between ‘order’ and ‘chaos’, J Math Biol, 28, 435-449, (1990) · Zbl 0733.92011 · doi:10.1007/BF00178327
[18] Liberman, U, On the relation between the instability of ESS in discrete dynamics and segregation distortion, J Theor Biol, 150, 421-436, (1991) · doi:10.1016/S0022-5193(05)80437-0
[19] Lyubich, YI; Maistrovskii, GD; Olkhovskii, YG, Selection-induced convergence to equilibrium in a single-locus autosomal population, Probl Peredachi Inf, 16, 93-104, (1980) · Zbl 0429.92017
[20] Mandel, SPH, The stability of a multiple allelic system, Heredity, 13, 289-302, (1959) · doi:10.1038/hdy.1959.36
[21] Martínez, S; Michon, G; San Martín, J, Inverse of strictly ultrametric matrices are of Stieltjes type, SIAM J Matrix Anal Appl, 15, 98-106, (1994) · Zbl 0798.15030 · doi:10.1137/S0895479891217011
[22] McDonald, JJ; Neumann, M; Schneider, H; Tsatsomeros, M, Inverse M-matrix inequalities and generalized ultrametric matrices, Linear Algebra Appl, 220, 321-341, (1995) · Zbl 0824.15020 · doi:10.1016/0024-3795(94)00077-Q
[23] Nabben, R; Varga, RS, A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix, SIAM J Matrix Anal Appl, 15, 107-113, (1994) · Zbl 0803.15020 · doi:10.1137/S0895479892228237
[24] Nabben, R; Varga, RS, Generalized ultrametric matrices—a class of inverse M-matrices, Linear Algebra Appl, 220, 365-390, (1995) · Zbl 0828.15020 · doi:10.1016/0024-3795(94)00086-S
[25] Nagylaki T (1992) Introduction to theoretical population genetics. Biomathematics, 21. Springer-Verlag, Berlin · Zbl 0839.92011
[26] Okasha, S, Fisher’s fundamental theorem of natural selection—a philosophical analysis, Br J Philos Sci, 59, 319-351, (2008) · Zbl 1151.92024 · doi:10.1093/bjps/axn010
[27] Price, GR, Fisher’s fundamental theorem made clear, Ann Hum Genet Lond, 36, 129-140, (1972) · Zbl 0241.92011 · doi:10.1111/j.1469-1809.1972.tb00764.x
[28] Sanyal, DC; Sarkar, B, Interaction between selection and segregation ratios with redefined concept of fitness differentials, TWMS J App Eng Math, 3, 87-102, (2013)
[29] Shashahani S (1979) A new mathematical framework for the study of linkage and selection. Mem, Amer. Math. Soc. 17 · Zbl 0473.92008
[30] Tallis, GM, Equilibria under selection for \(k\) alleles, Biometrics, 22, 121-127, (1966) · Zbl 0139.14101 · doi:10.2307/2528219
[31] Weissing, FJ; Boven, M, Selection and segregation distortion in a sex-differentiated population, Theor Popul Biol, 60, 327-341, (2001) · Zbl 1041.92021 · doi:10.1006/tpbi.2001.1550
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