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9781402015540

Evolution of Biological Systems in Random Media

by ; ;
  • ISBN13:

    9781402015540

  • ISBN10:

    1402015542

  • Format: Hardcover
  • Copyright: 2003-10-01
  • Publisher: Kluwer Academic Pub
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List Price: $119.99

Summary

This is a new book in biomathematics, which includes new models of stochastic non-linear biological systems and new results for these systems. These results are based on the new results for non-linear difference and differential equations in random media. This book contains:-New stochastic non-linear models of biological systems, such as biological systems in random media: epidemic, genetic selection, demography, branching, logistic growth and predator-prey models; -New results for scalar and vector difference equations in random media with applications to the stochastic biological systems in 1); -New results for stochastic non-linear biological systems, such as averaging, merging, diffusion approximation, normal deviations and stability; -New approach to the study of stochastic biological systems in random media such as random evolution approach.

Table of Contents

Table of Contents
vii
Preface xiii
List of Notations
xvii
Random Media
1(28)
Markov Chains
1(6)
Ergodicity and Reducibility of Markov Chains
7(5)
Markov Renewal Processes
12(2)
Semi-Markov Processes
14(3)
Jump Markov Processes
17(1)
Wiener Processes and Diffusion Processes
18(1)
Martingales
19(1)
Semigroups of Operators and Their Generators
20(3)
Martingale Characterization of Markov and Semi-Markov Processes
23(4)
Martingale Characterization of Markov Chains
23(1)
Martingale Characterization of Markov Processes
24(1)
Martingale Characterization of Semi-Markov Processes
24(1)
Merging of Semi-Markov Processes
25(2)
General Representation and Measurability of Biological Systems in Random Media
27(2)
General Representation of the Biological Systems
27(1)
Measurability of the General Systems
27(2)
Limit Theorems for Difference Equations in Random Media
29(58)
Limit Theorems for Random Evolutions
31(17)
Definitions and Classifications of Random Evolutions
31(4)
Martingale Characterization of Random Evolutions
35(3)
Limit Theorems for Random Evolutions
38(10)
Averaging of Difference Equations in Random Media
48(6)
Averaging in Markov Random Media
48(3)
Averaging in Semi-Markov Random Media
51(3)
Diffusion Approximation of Difference Equations in Random Media
54(4)
Diffusion Approximation in Markov Media
54(3)
Diffusion Approximation in Semi-Markov Random Media
57(1)
Normal Deviations of Difference Equations in Random Media
58(4)
Merging of Difference Equations in Random Media
62(3)
Stability of Difference Equations in Averaging and Diffusion Approximation Schemes
65(8)
Stochastic Stability of Difference Equations in Averaging Scheme
65(4)
Stochastic Stability of Difference Equations in Diffusion Approximation Schemes
69(4)
Limit Theorems for Vector Difference Equations in Random Media
73(14)
Averaging of Vector Difference Equations
73(3)
Diffusion Approximation of Vector Difference Equations
76(2)
Normal Deviations of Vector Difference Equations
78(4)
Merging of Vector Difference Equations
82(5)
Epidemic Models
87(30)
Deterministic Epidemic Models
87(6)
Reed-Frost Model: S → I → R
87(1)
Kermack-McKendrick Model
88(1)
Continuous Epidemic Models
89(4)
Stochastic Epidemic Model (Epidemic Model in Random Media)
93(3)
Discrete Epidemic Model in Markov Random Media
93(1)
Continuous Epidemic Model in Markov Renewal Random Media
94(1)
Continuous Epidemic Model in Semi-Markov Random Media
95(1)
Averaging of Epidemic Model in Random Media
96(7)
Merging of Epidemic Models in Random Media
103(2)
Diffusion Approximation of Epidemic Models in Random Media
105(3)
Normal Deviations of Epidemic Model in Random Media
108(3)
Stochastic Stability of Epidemic Model
111(6)
Stochastic Stability of Epidemic Model in Averaging Scheme
111(2)
Stability of Epidemic Model in Diffusion Approximation Scheme
113(1)
Stability of Epidemic Model in Normal Deviations Scheme
114(3)
Genetic Selection Models
117(24)
Deterministic Genetic Selection Models
117(11)
Bacterial Genetics
118(4)
Human Genetics
122(4)
Continuous Genetic Selection Models
126(2)
Stochastic Genetic Selection Model (Genetic Selection Model in Random Media)
128(2)
Discrete Genetic Selection Model in Markov Random Media
128(1)
Continuous Genetic Selection Model in Markov Renewal Random Media
129(1)
Continuous Genetic Selection Model in Semi-Markov Random Media
129(1)
Averaging of Slow Genetic Selection Model in Random Media
130(3)
Merging of Slow Genetic Selection Model in Random Media
133(3)
Diffusion Approximation of Slow Genetic Selection in Random Media
136(1)
Normal Deviations of Slow Genetic Selection Model in Random Media
137(1)
Stochastic Stability of Slow Genetic Selection Model
138(3)
Stability of Slow Genetic Selection Model in Averaging Scheme
138(1)
Stability of Slow Genetic Selection in Diffusion Approximation Scheme
139(2)
Branching Models
141(16)
Branching Models with Deterministic Generating Function
141(4)
The Galton-Watson-Fisher Model
141(2)
The Bellman-Harris Branching Process
143(2)
Branching Models in Random Media
145(1)
Bellman-Harris Branching Process in Markov Random Media
145(1)
Bellman-Harris Branching Process in Markov Renewal Random Media
146(1)
Bellman-Harris Branching Process in Semi-Markov Random Media
146(1)
Averaging of Branching Models in Random Media
146(2)
Merging of Branching Model in Random Media
148(1)
Diffusion Approximation of Branching Process in Random Media
148(2)
Normal Deviations of Branching Process in Random Media
150(1)
Stochastic Stability of Branching Model in Averaging and Diffusion Approximation Schemes
150(7)
Stability of Branching Model in Averaging Scheme
150(4)
Stability of Branching Model in Diffusion Approximation Scheme
154(3)
Demographic Models
157(18)
Deterministic Demographic Model
157(5)
Fibonacci Sequence
157(1)
Fibonacci Reproduction Matrix
158(2)
Leslie Reproduction Matrix
160(2)
Stochastic Demographic Models (Demographic Models in Random Media)
162(3)
Discrete Demographic Model in Markov Random Media
162(1)
Continuous Demographic Model in Markov Renewal Random Media
163(1)
Demographic Models on Infinite Time Interval
163(2)
Averaging of Demographic Models in Random Media
165(1)
Merging of Demographic Model
166(1)
Diffusion Approximation of Demographic Model
167(1)
Normal Deviations of Demographic Models in Random Media
168(1)
Stochastic Stability of Demographic Model in Averaging and Diffusion Approximation Schemes
168(7)
Stability of Demographic Model in Averaging Scheme
168(4)
Stability of Demographic Model in Diffusion Approximation Scheme
172(3)
Logistic Growth Models
175(12)
Deterministic Logistic Growth Model
175(2)
Discrete Logistic Growth Model
175(1)
Continuous Logistic Growth Model
176(1)
Stochastic Logistic Growth Model (Logistic Growth Model in Random Media)
177(1)
Discrete Logistic Growth Model in Markov Random Media
177(1)
Continuous Logistic Growth Model in Markov Renewal Random Media
177(1)
Continuous Logistic Growth Model in Semi-Markov Random Media
178(1)
Averaging of Logistic Growth Model in Random Media
178(2)
Merging of Logistic Growth Model in Random Media
180(1)
Diffusion Approximation of Logistic Growth Model in Random Media
180(1)
Normal Deviations of Logistic Growth Model in Random Media
181(1)
Stochastic Stability of Logistic Growth Model in Averaging and Diffusion Approximation Schemes
182(5)
Stability of Logistic Growth Model in Averaging Scheme
182(1)
Stability of Logistic Growth Model in Diffusion Approximation Scheme
183(4)
Predator-Prey Models
187(22)
Deterministic Predator-Prey Model
187(2)
Discrete Predator-Prey Model
187(1)
Continuous Predator-Prey Model
188(1)
Stochastic Predator-Prey Model (Predator-Prey Model in Random Media)
189(2)
Discrete Predator-Prey Model in Markov Random Media
189(1)
Continuous Predator-Prey Model in Markov Renewal Random Media
190(1)
Continuous Predator-Prey Model in Semi-Markov Random Media
190(1)
Averaging of Predator-Prey Model in Random Media
191(3)
Merging of Predator-Prey Model
194(2)
Diffusion Approximation of Predator-Prey Model
196(1)
Normal Deviations of Predator-Prey Model in Random Media
197(3)
Stochastic Stability of Predator-Prey Model in Averaging and Diffusion Approximation Schemes
200(9)
Stochastic Stability of Predator-Prey Model in Averaging Scheme
200(4)
Stability of Predator-Prey Model in Diffusion Approximation Scheme
204(2)
Stability of Predator-Prey Model in Normal Deviations Scheme
206(3)
Bibliography 209(6)
Index 215

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