Fundamentals of Mathematical Evolutionary Genetics

ISBN-13: 9789027727725 / Angielski / Twarda / 1990 / 396 str.

Yuri M. Svirezhev;V.P. Passekov

Opis

One service mathematics has rendered the ~Et moi, ..., si j'avait su comment en revenir, human race. It has put common sense back je riy serais point aile.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. o'; 'One service logic has rendered com- puter science .. o'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

1. Deterministic Models in Mathematical Genetics.- 1. Brief Outline of Microevolution Theory with Some Facts from Genetics.- 1.1. History and Personalia.- 1.2. Conceptual Model of Microevolution.- 1.3. Elementary Evolutionary Structure and Elementary Evolutionary Phenomenon.- 1.4. Elementary Evolutionary Material.- 1.5. Elementary Evolutionary Factors.- 1.6. An Introduction to Principles of Inheritance.- 1.7. Notes and Bibliography.- 2. Basic Equations of Population Genetics.- 2.1. Description of a Population.- 2.2. Sexless Population.- 2.3. Equations for Populations in Evolution.- 2.4. Evolution of Populations and Integral Renewal Equations.- 2.5. Panmixia and Other Systems of Mating.- 2.6. Principles of Inheritance.- 2.7. Multi-Allele Autosomal Gene: Equations of Evolution.- 2.8. Equations of Evolution with Specific Demographic Functions.- 2.8.1. Global Panmixia, Multiplicative Fecundity.- 2.8.2. Global Panmixia, Additive Fecundity.- 2.8.3. Local Panmixia.- 2.9. Equations of Evolution: Fecundity of a Couple is Determined by that of the Female.- 2.10. Equal Fecundity, Different Mortality: Another Form for Evolutionary Equations.- 2.11. Semelparity: Models with Discrete Time.- 2.12. More Realistic Assumptions About the Particular Form of Fecundity and Mortality Functions.- 2.13. Some Generalizations of Classical Equations in Population Genetics. Another Way to Derive these Equations.- 2.14. Discrete-time Equations of Evolution.- 2.15. On the Relationship Between Continuous and Discrete Models.- 2.16. Notes and Bibliography.- 3. Simplest Population Models.- 3.1. Introduction.- 3.2. Equations of Evolution.- 3.3. Existence Conditions for Polymorphism.- 3.4. Sufficient Conditions for Stability of Limiting States of a Population.- 3.5. Population Without Age Structure. Continuous Model.- 3.6. Population Without Age Structure. Discrete Model.- 3.7. Polymorphism. Experiments and Theory. What Are the Malthusian Parameters or Genotype Fitnesses?.- 3.8. Genetico-Ecological Models.- 3.9. Special Cases of Genetico-Ecological Models.- 3.10. Passage from Genetico-Ecological Models to Models in Frequency Form.- 3.11. Notes and Bibliography.- 4. Multiple Alleles.- 4.1. Introduction.- 4.2. State of Genetic Equilibrium. Polymorphism.- 4.3. Mean Population Fitness. Fisher’s Fundamental Theorem.- 4.4. Mean Fitness as a Lyapunov Function.- 4.5. Adaptive Topography of a Population.- 4.6. The Case of Three Alleles. Search for Domains of Asymptotic Stability.- 4.7. Necessary and Sufficient Existence Conditions for Polymorphism.- 4.8. Theorem About Limited Variations and Another Form of Existence Conditions for Polymorphism.- 4.9. Elimination of Alleles and Theorem About Dominance.- 4.10 Simple Necessary Conditions of Existence for Polymorphic and Pure Equilibria.- 4.11. Population Trajectory as a Trajectory of Steepest Ascent.- 4.11.1. Introduction of a New Metric Space.- 4.11.2. Equations of Evolution and Local Extremal Principle.- 4.12. Another Form for Equations of Evolution.- 4.13. Notes and Bibliography.- 5. Sex-Limited and Sex-Linked Characters. Models Taking Account of Sex Distinctions.- 5.1. Introduction.- 5.2. Model Taking Account of Sex Distinctions.- 5.2.1. Autosomal Gene. Continuous Model.- 5.3. New Types of Polymorphism and Their Stability.- 5.4. Model Taking Account of Sex Distinctions.- 5.4.1. Sex-Linked Gene. Continuos Model.- 5.5. Sex-Linked Gene. Discrete Model.- 5.6. Sex-Linked Gene. Multiple Alleles.- 5.7. Minimax Properties of the Mean Fitness Function in a Model of an Age-Structured Population.- 5.8. Notes and Bibliography.- 6. Populations With Deviations from Panmixia.- 6.1. Introduction.- 6.2. Preference in Crossing and Preference Matrix.- 6.3. Model of Population with Deviations from Panmixia Caused by Preference in Crossing.- 6.4. Evolution and Stability of Deviations from Hardian Equilibrium. Inbreeding.- 6.5. Preference in Crossing. Discrete Model.- 6.6. Evolution of Genetic Structure of Population Under Inbreeding. Discrete Model.- 6.7. Isolation by Distance and Deviations from Panmixia.- 6.8. Models with Particular Functions of Deviations from Panmixia.- 6.9. Notes and Bibliography.- 7. Systems of Linked Populations. Migration.- 7.1. Introduction.- 7.2. Migration Between Populations of Different Sizes.- 7.3. Migration Between Population Occupying Two Similar Ecological Niches.- 7.4. On ‘Fast’ and ‘slow’ Variables in Systems of Linked Populations.- 7.5. Genetic Interpretation. Why Stable Divergence is Important in Systems of Linked Populations.- 7.6. Systems of Weakly Linked Populations.- 7.7. Populations with Continuous Area (Spatially Distributed Populations).- 7.8. Genic Waves in Spatially Distributed Populations.- 7.9. Notes and Bibliography.- 8. Population Dynamics in Changing Environment.- 8.1. Introduction.- 8.2. Seasonal Oscillations in Coefficients of Relative Viability. Discrete Model.- 8.3. Polymorphism in Populations of Adalia Bipunctata.- 8.4. Environments Changing with Time. Continuous Model.- 8.5. How Variations in the Total Size of a Population Affect its Genetic Dynamics.- 8.6. How Periodic Variations in Coefficients of Relative Viability Affect the Total Population Size.- 8.7. Changing Environment. Adaption and Adaptability.- 8.8. Notes and Bibliography.- 9. Multi-Locus Models.- 9.1. Discrete Two-Locus Model of Segregation-Recombination and its Continuous Approximation.- 9.2. Continuous One- and Two-Locus Models with no Selection. Equations for Numbers and Frequencies, Fast and Slow Variables.- 9.3. Formalization of Recombination-Segregation in a Discrete-Time Multi-Locus System. Equations of Dynamics, Equilibria.- 9.4. Recombination-Segregation Model in a Multi-Locus Continuous-Time System.- 9.5. Additivity of Interaction Between Selection and Recombination-Segregation in Multi-Locus Models Presented by Differential Equations.- 9.6. Selection of Zygotes and Gametes in a Discrete-Time Model and its Continuous Approximation.- 9.7. Equations of Dynamics Under the Combined Action of Selection and Recombination-Segregation in Discrete- and Continuous-Time Models.- 9.8. Comparing the Dynamics in One-Locus and Multi-Locus Systems in the Presence of Selection.- 9.9. Model of Additive Selection in a Multi-Locus System.- 9.10. Models of Multiplicative and Additively Multiplicative Selection.- 9.11. Notes and Bibliography.- 2. Stochastic Models of Mathematical Genetics.- 10. Diffusion Models of Population Genetics.- 10.1. Types of Random Processes Relevant to Models of Population Genetics.- 10.2. Fundamental Problems in the Analysis of Stochastic Models.- 10.3. Forward and Backward Kolmogorov Equations.- 10.4. Diffusion Approximation of the Fisher-Wright and the Moran Models.- 10.5. Classification of Boundaries in Diffusion Models.- 10.6. Multidimensional Diffusion Models.- 10.7. Solutions to the Kolmogorov Equations by the Fourier Method. Transformations of Diffusion Processes. The Steady-State Density.- 10.8. Search for Moments of Some Functionals on Diffusion Processes.- 10.9. An Approach to Search for the Mean of a Function Defined on States of a Process.- 10.10. Notes and Bibliography.- 11. Random Genetic Drift in the Narrow Sense.- 11.1. The Kolmogorov Equations for a Single-Locus Model of Random Genetic Drift.- 11.2. Approximating the Random Genetic Drift Process within Small Intervals of Time.- 11.3. Asymptotics of the Fundamental Solution for the Random Genetic Drift Process When t ? ?.- 11.4. Boundary Attainment Probabilities.- 11.5. Characteristics of the Boundary Attainment Time.- 11.6. Probability Density Function for the Sojourn Time and the Age of an Allele.- 11.7. Moments of the Random Genetic Drift Process.- 11.8. Fundamental Solution to the Kolmogorov Equations.- 11.9. A Random Genetic Drift Model with Two Loci.- 11.10. Notes and Bibliography.- 12. Properties of Single-Locus Models under Several Microevolutionary Pressures.- 12.1. Kolmogorov Equations in Case of Several Microevolutionary Conditions.- 12.2. Probabilities of Allele Fixation.- 12.3. Characteristics of the Homozygosity Attainment Time.- 12.4. Steady-State Probability Density Function for the Case of a Single Diallelic Locus.- 12.5. Investigation of the Steady-State Probability Density Function for a Single Diallelic Locus.- 12.6. Steady-State Density Function and the Adaptive Landscape in the Two-Allele Case.- 12.7. Derivation of a Steady-State Density Function in the Case of Multiple Alleles.- 12.8. Contribution to the Steady-State Density Caused by Selection.- 12.9. Contribution Caused by Migrations and Mutations. The General Form of a Steady-State Density Function.- 12.10. Investigation of the Steady-State Probability Density Function for Concentrations of Multiple Alleles. A Multi-Locus Case.- 12.11. Steady-state Density and Objective Functions in Case of Multiple Alleles.- 12.12. Relation of Objective Functions to the Sphere Motion Potential. Mechanical Interpretation of Single-Locus Genetic Processes in Terms of Motion in a Force Field.- 12.13. Notes and Bibliography.- 13. Random Genetic Drift in Subdivided Populations.- 13.1. Generating Operator for the Random Genetic Drift Process in a Subdivided Finite-Sized Population with Migrations of the “Island” Type.- 13.2. Dynamics of Expected Allele Frequencies in a Subdivided Population.- 13.3. Behavior of Expected Heterozygosity Indices.- 13.4. Dynamics of Expected Indices of Linkage Disequilibrium in Case of Two Loci.- 13.5. Model of a Hierarchically Subdivided Population.- 13.6. Investigation of the Asymptotic Rate of Decrease in Heterozygosity in the Hierarchical Model.- 13.7. Model of Isolation by Distance.- 13.8. Properties of the Random Genetic Drift Process in a Subdivided Population with Migrations of the General Type.- 13.9. Notes and Bibliography.- Conclusion.- Short Glossary of Genetic Terms.