Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY MATHEMATICAL METHODS
Ryszard Rudnicki
Marta Tyran-Kamińska
Piecewise
Deterministic
Processes in
Biological Models
123
SpringerBriefs in Applied Sciences
and Technology
Mathematical Methods
Series editor
Anna Marciniak-Czochra, Heidelberg, Germany
More information about this series at http://www.springer.com/series/11219
ń
Ryszard Rudnicki Marta Tyran-Kami ska
(cid:129)
Piecewise Deterministic
Processes in Biological
Models
123
Ryszard Rudnicki Marta Tyran-Kamińska
Institute of Mathematics Institute of Mathematics
Polish Academy of Sciences University of Silesia
Katowice Katowice
Poland Poland
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
SpringerBriefs inMathematical Methods
ISBN978-3-319-61293-5 ISBN978-3-319-61295-9 (eBook)
DOI 10.1007/978-3-319-61295-9
LibraryofCongressControlNumber:2017945225
©TheAuthor(s)2017
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Preface
The aim of this book is to give a short mathematical introduction to piecewise
deterministic Markov processes (PDMPs) and to present biological models where
they appear. The book is divided into six chapters. In the first chapter we present
some examples of biological phenomena such as gene activity and population
growth leading to different type of PDMPs: continuous-time Markov chains,
deterministicprocesseswithjumps,dynamicalsystemswithrandomswitchingand
point processes. The second chapter contains some theoretical results concerning
Markov processes and the construction of PDMPs. The next chapter is an intro-
duction to the theory of semigroups of linear operators which provide the primary
tools in the study of continuous-time Markov processes. In the fourth chapter we
introduce stochastic semigroups, provide some theorems on their existence and
findgeneratorsofsemigroupsrelatedtoPDMPsconsideredinthefirstchapter.The
next chapter is devoted to the long-time behaviour (asymptotic stability and
sweeping) of the stochastic semigroups induced by PDMPs. In the last chapter we
applythegeneralresults,especiallyconcerningasymptoticbehaviour,tobiological
models.
Thebookisdedicatedtobothmathematiciansandbiologists.Thefirstgroupwill
find here new biological models which lead to interesting and often new mathe-
matical questions. Biologists can observe how to include seemingly different bio-
logical processes into a unified mathematical theory and deduce from this theory
interesting biological conclusions. We try to keep the required mathematical
and biological background to a minimum so that the topics are accessible to
students.
v
vi Preface
Acknowledgements
This research was partially supported by the National Science Centre (Poland)
Grant No. 2014/13/B/ST1/00224. The authors are grateful to Michael C. Mackey
and Katarzyna Pichór who read the manuscript and made useful suggestions for
improvements.
Katowice, Poland Ryszard Rudnicki
May 2017 Marta Tyran-Kamińska
Contents
1 Biological Models ... .... ..... .... .... .... .... .... ..... .... 1
1.1 Introduction ... .... ..... .... .... .... .... .... ..... .... 1
1.2 Birth-Death Processes..... .... .... .... .... .... ..... .... 2
1.3 Grasshopper and Kangaroo Movement.... .... .... ..... .... 3
1.4 Velocity Jump Process .... .... .... .... .... .... ..... .... 4
1.5 Size of Cells in a Single Line... .... .... .... .... ..... .... 5
1.6 Two-Phase Cell Cycle Model... .... .... .... .... ..... .... 7
1.7 Stochastic Billiard as a Cell Cycle Model . .... .... ..... .... 9
1.8 Stochastic Gene Expression I... .... .... .... .... ..... .... 11
1.9 Stochastic Gene Expression II .. .... .... .... .... ..... .... 13
1.10 Gene Regulatory Models with Bursting ... .... .... ..... .... 16
1.11 Neural Activity. .... ..... .... .... .... .... .... ..... .... 17
1.12 Processes with Extra Jumps on a Subspace .... .... ..... .... 19
1.13 Size-Structured Population Model.... .... .... .... ..... .... 21
1.14 Age-Structured Population Model.... .... .... .... ..... .... 24
1.15 Asexual Phenotype Population Model .... .... .... ..... .... 25
1.16 Phenotype Model with a Sexual Reproduction.. .... ..... .... 26
1.17 Coagulation-Fragmentation Process in a Phytoplankton
Model.... .... .... ..... .... .... .... .... .... ..... .... 28
1.18 Paralog Families.... ..... .... .... .... .... .... ..... .... 29
1.19 Definition of PDMP. ..... .... .... .... .... .... ..... .... 30
2 Markov Processes ... .... ..... .... .... .... .... .... ..... .... 33
2.1 Transition Probabilities and Kernels.. .... .... .... ..... .... 33
2.1.1 Basic Concepts .... .... .... .... .... .... ..... .... 33
2.1.2 Transition Operators .... .... .... .... .... ..... .... 36
2.1.3 Substochastic and Stochastic Operators.. .... ..... .... 38
2.1.4 Integral Stochastic Operators.. .... .... .... ..... .... 39
2.1.5 Frobenius–Perron Operator ... .... .... .... ..... .... 41
2.1.6 Iterated Function Systems.... .... .... .... ..... .... 42
vii
viii Contents
2.2 Discrete-Time Markov Processes .... .... .... .... ..... .... 44
2.2.1 Markov Processes and Transition Probabilities ..... .... 44
2.2.2 Random Mapping Representations . .... .... ..... .... 46
2.2.3 Canonical Processes .... .... .... .... .... ..... .... 47
2.3 Continuous-Time Markov Processes.. .... .... .... ..... .... 49
2.3.1 Basic Definitions... .... .... .... .... .... ..... .... 49
2.3.2 Processes with Stationary and Independent
Increments... ..... .... .... .... .... .... ..... .... 51
2.3.3 Markov Jump-Type Processes. .... .... .... ..... .... 52
2.3.4 Generators and Martingales... .... .... .... ..... .... 54
2.3.5 Existence of PDMPs.... .... .... .... .... ..... .... 56
2.3.6 Transition Functions and Generators of PDMPs .... .... 59
3 Operator Semigroups .... ..... .... .... .... .... .... ..... .... 63
3.1 Generators and Semigroups .... .... .... .... .... ..... .... 63
3.1.1 Essentials of Banach Spaces and Operators... ..... .... 63
3.1.2 Definitions and Basic Properties ... .... .... ..... .... 65
3.1.3 The Resolvent..... .... .... .... .... .... ..... .... 67
3.2 Basic Examples of Semigroups.. .... .... .... .... ..... .... 69
3.2.1 Uniformly Continuous Semigroups. .... .... ..... .... 69
3.2.2 Multiplication Semigroups ... .... .... .... ..... .... 70
3.2.3 Translation Semigroups.. .... .... .... .... ..... .... 71
3.3 Generators of Contraction Semigroups.... .... .... ..... .... 73
3.3.1 The Hille–Yosida Theorem... .... .... .... ..... .... 73
3.3.2 The Lumer–Phillips Theorem . .... .... .... ..... .... 76
3.3.3 Perturbations of Semigroups.. .... .... .... ..... .... 78
3.3.4 Perturbing Boundary Conditions... .... .... ..... .... 80
4 Stochastic Semigroups.... ..... .... .... .... .... .... ..... .... 83
4.1 Aspects of Positivity. ..... .... .... .... .... .... ..... .... 83
4.1.1 Positive Operators.. .... .... .... .... .... ..... .... 83
4.1.2 Substochastic Semigroups.... .... .... .... ..... .... 84
4.1.3 Resolvent Positive Operators.. .... .... .... ..... .... 86
4.1.4 Generation Theorems ... .... .... .... .... ..... .... 87
4.1.5 Positive Perturbations ... .... .... .... .... ..... .... 88
4.1.6 Positive Unbounded Perturbations.. .... .... ..... .... 91
4.1.7 Adjoint and Transition Semigroups. .... .... ..... .... 94
4.2 Stochastic Semigroups for PDMPs... .... .... .... ..... .... 96
4.2.1 Jump-Type Markov Processes. .... .... .... ..... .... 96
4.2.2 Semigroups for Semiflows ... .... .... .... ..... .... 99
4.2.3 PDMPs Without Boundaries.. .... .... .... ..... .... 101
4.2.4 Dynamical Systems with Jumps ... .... .... ..... .... 103
4.2.5 Randomly Switched Dynamical Systems. .... ..... .... 104
Contents ix
4.2.6 Jumps from Boundaries.. .... .... .... .... ..... .... 106
4.2.7 Semigroups for the Stein Model ... .... .... ..... .... 109
5 Asymptotic Properties of Stochastic Semigroups—General
Results .... .... .... .... ..... .... .... .... .... .... ..... .... 115
5.1 Asymptotic Stability and Sweeping .. .... .... .... ..... .... 115
5.1.1 Definitions of Asymptotic Stability and Sweeping .. .... 115
5.1.2 Lower Function Theorem .... .... .... .... ..... .... 116
5.1.3 Partially Integral Semigroups and Asymptotic
Stability. .... ..... .... .... .... .... .... ..... .... 117
5.1.4 Sweeping via the Existence of r-finite Invariant
Function .... ..... .... .... .... .... .... ..... .... 118
5.2 Asymptotic Decomposition of Stochastic Semigroups ..... .... 120
5.2.1 Theorem on Asymptotic Decomposition. .... ..... .... 120
5.2.2 Sweeping and the Foguel Alternative ... .... ..... .... 122
5.2.3 Asymptotic Stability .... .... .... .... .... ..... .... 124
5.2.4 Hasminskiĭ Function.... .... .... .... .... ..... .... 125
6 Asymptotic Properties of Stochastic Semigroups—Applications ........ 129
6.1 Continuous-Time Markov Chains.... .... .... .... ..... .... 129
6.1.1 Foguel Alternative.. .... .... .... .... .... ..... .... 129
6.1.2 Non-explosive Markov Chains and Asymptotic
Stability. .... ..... .... .... .... .... .... ..... .... 130
6.1.3 Markov Chain with an Absorbing State . .... ..... .... 131
6.1.4 Asymptotics of Paralog Families... .... .... ..... .... 132
6.1.5 Applications to Pure Jump-Type Markov Processes . .... 133
6.2 Dynamical Systems with Random Switching ... .... ..... .... 135
6.2.1 General Formulation.... .... .... .... .... ..... .... 135
6.2.2 Applications to Stochastic Gene Expression Models. .... 137
6.3 Cell Maturation Models ... .... .... .... .... .... ..... .... 140
6.3.1 Introductory Remarks ... .... .... .... .... ..... .... 140
6.3.2 Flows with Jumps.. .... .... .... .... .... ..... .... 141
6.3.3 Size-Structured Model... .... .... .... .... ..... .... 142
6.3.4 Two-Phase Cell Cycle Model. .... .... .... ..... .... 144
6.3.5 Lebowitz–Rubinow’s Model.. .... .... .... ..... .... 146
6.3.6 Stein’s Model ..... .... .... .... .... .... ..... .... 147
Appendix A: Measure and Probability Essentials.. .... .... ..... .... 149
References.... .... .... .... ..... .... .... .... .... .... ..... .... 159
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 165
