Piecewise deterministic processes in biological models
This book presents a concise introduction to piecewise deterministic Markov processes (PDMPs), with particular emphasis on their applications to biological models. Further, it presents examples of biological phenomena, such as gene activity and population growth, where different types of PDMPs appea...
| Main Author: | Rudnicki, Ryszard. |
|---|---|
| Other Authors: | Tyran-Kamińska, Marta., SpringerLink (Online service) |
| Format: | Electronic |
| Language: | English |
| Published: |
Cham :
Springer,
2017.
|
| Physical Description: |
1 online resource. |
| Series: |
SpringerBriefs in applied sciences and technology. Mathematical methods.
|
| Subjects: |
Table of Contents:
- Preface; Acknowledgements; Contents; 1 Biological Models; 1.1 Introduction; 1.2 Birth-Death Processes; 1.3 Grasshopper and Kangaroo Movement; 1.4 Velocity Jump Process; 1.5 Size of Cells in a Single Line; 1.6 Two-Phase Cell Cycle Model; 1.7 Stochastic Billiard as a Cell Cycle Model; 1.8 Stochastic Gene Expression I; 1.9 Stochastic Gene Expression II; 1.10 Gene Regulatory Models with Bursting; 1.11 Neural Activity; 1.12 Processes with Extra Jumps on a Subspace; 1.13 Size-Structured Population Model; 1.14 Age-Structured Population Model; 1.15 Asexual Phenotype Population Model.
- 1.16 Phenotype Model with a Sexual Reproduction1.17 Coagulation-Fragmentation Process in a Phytoplankton Model; 1.18 Paralog Families; 1.19 Definition of PDMP; 2 Markov Processes; 2.1 Transition Probabilities and Kernels; 2.1.1 Basic Concepts; 2.1.2 Transition Operators; 2.1.3 Substochastic and Stochastic Operators; 2.1.4 Integral Stochastic Operators; 2.1.5 Frobenius
- Perron Operator; 2.1.6 Iterated Function Systems; 2.2 Discrete-Time Markov Processes; 2.2.1 Markov Processes and Transition Probabilities; 2.2.2 Random Mapping Representations; 2.2.3 Canonical Processes.
- 2.3 Continuous-Time Markov Processes2.3.1 Basic Definitions; 2.3.2 Processes with Stationary and Independent Increments; 2.3.3 Markov Jump-Type Processes; 2.3.4 Generators and Martingales; 2.3.5 Existence of PDMPs; 2.3.6 Transition Functions and Generators of PDMPs; 3 Operator Semigroups; 3.1 Generators and Semigroups; 3.1.1 Essentials of Banach Spaces and Operators; 3.1.2 Definitions and Basic Properties; 3.1.3 The Resolvent; 3.2 Basic Examples of Semigroups; 3.2.1 Uniformly Continuous Semigroups; 3.2.2 Multiplication Semigroups; 3.2.3 Translation Semigroups.
- 3.3 Generators of Contraction Semigroups3.3.1 The Hille
- Yosida Theorem; 3.3.2 The Lumer
- Phillips Theorem; 3.3.3 Perturbations of Semigroups; 3.3.4 Perturbing Boundary Conditions; 4 Stochastic Semigroups; 4.1 Aspects of Positivity; 4.1.1 Positive Operators; 4.1.2 Substochastic Semigroups; 4.1.3 Resolvent Positive Operators; 4.1.4 Generation Theorems; 4.1.5 Positive Perturbations; 4.1.6 Positive Unbounded Perturbations; 4.1.7 Adjoint and Transition Semigroups; 4.2 Stochastic Semigroups for PDMPs; 4.2.1 Jump-Type Markov Processes; 4.2.2 Semigroups for Semiflows; 4.2.3 PDMPs Without Boundaries.