Author | Maruyama, Takeo. author |
---|---|
Title | Stochastic Problems in Population Genetics [electronic resource] / by Takeo Maruyama |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1977 |
Connect to | http://dx.doi.org/10.1007/978-3-642-93065-2 |
Descript | VIII, 248 p. online resource |
1 Orientation -- 1.1 Discrete space, continuous time, random walk -- 1.2 Discrete space, discrete time -- 1.3 Circular space, continuous time -- 1.4 Continuous space, continuous time -- 1.5 Markov process -- 1.6 Dynkinโs formula -- 2 Population Genetics Models -- 2.1 Wrightโs model -- 2.2 Fellerโs model -- 2.3 Moranโs model -- 2.4 Variable population size -- 2.5 Wrightโs model with mutation -- 2.6 A model of irreversible mutation or a model of infinite alleles -- 2.7 A selection model -- 2.8 A model of dominance -- 2.9 Birth-and-death process -- 2.10 Density or frequency dependent process -- 2.11 Time inhomogeneous process -- 2.12 A model of random environment -- 3 Classification of Boundaries -- 3.1 Regular boundary -- 3.2 Exit boundary -- 3.3 Entrance boundary -- 3.4 Natural boundary -- 3.5 Nature of boundary -- 3.6 Examples -- 4 Expectation of Integration Along Sample Paths -- 4.1 Integration along sample paths -- 4.2 The boundary conditions -- 4.3 An example -- 4.4 Greenโs function for a pure random process -- 4.5 Computer simulation -- 4.6 Sum of heterozygotes -- 4.7 Process with reflecting boundary -- 4.8 Irreversible mutation model (or infinite alleles) -- 4.9 General form of Greenโs function -- 4.10 Probability of fixation -- 4.11 Behavior of sample paths near the origin -- 4.12 Higher moments -- 4.13 Nagylakiโs formula -- 5 Modification of Processes -- 5.1 Killing and creating paths -- 5.2 Selection of paths -- 5.3 Random drift and fixation time -- 5.4 A case of genic selection -- 5.5 A symmetric property of sample paths -- 5.6 A general formula -- 5.7 Age of sample paths -- 5.8 Number of affected individuals and genetic load -- 5.9 Sojourn time of conditional sample paths on the present-frequency -- 5.10 A conditional age -- 5.11 Computer simulation -- 5.12 Random time change -- 6 Numerical Integration of the Kolmogorov Backward Equation -- 6.1 Integration method -- 6.2 Examples -- 7 Eigenvalues and Eigenvectors of the KBE -- 7.1 Eigenvalues and eigenvectors -- 7.2 Pure random drift case -- 7.3 Mutation and random drift case -- 7.4 Irreversible mutation case -- 7.5 Hypergeometric differential equation -- 7.6 Orthogonality of eigenfunctions -- 7.7 Expansion by eigenfunctions -- 7.8 The steady-state distribution of gene frequencies -- 8 Approximation Methods -- 8.1 Perturbation -- 8.2 Examples -- 8.3 Numerical method -- 8.4 Singular perturbation -- 9 Geographical Structure of Populations -- 9.1 One-dimensional populations, discrete colonies -- 9.2 Continuous space -- 9.3 Two-dimensional populations -- 9.4 Two-dimensional continuous space -- 9.5 Higher order moments -- 9.6 Numerical analysis at equilibrium -- 9.7 A differential equation and asymptotic formulae -- 9.8 Random drift -- 9.9 Time to fixation -- 10 Geographically Invariant Properties -- 10.1 Discrete time model -- 10.2 Continuous time model -- 10.3 Markov process -- 10.4 Diffusion method -- 10.5 Computer simulation of gene frequency change -- 10.6 Invariance based on diffusion method -- 10.7 Computer simulation of heterozygote distribution and other invariant properties -- 11 Gene Frequency Distributions and Random Drift in Geographically Structured Populations -- 11.1 Gene frequency distribution (global) in a structured population -- 11.2 Distribution of local gene frequencies -- 11.3 Random drift in a structured population -- 12 Some Special Problems -- 12.1 Variance of homozygote probability for the infinite neutral allele model -- 12.2 Variance of homozygote probability in a geographically structured population -- 12.3 Number of alleles -- 12.4 Some properties of the stepwise mutation model -- A1.1 Mutation model I -- A1.2 Mutation model II -- A1.3 Derivation of Kolmogorov backward equations (KBE) -- Appendix II A Supplementary Note on the Existence and Uniqueness of the Solution for the Recurrence Equation (9.7) -- Appendix III Distribution of Stochastic Integrals -- References