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E-grāmata: Multiscale Problems in the Life Sciences: From Microscopic to Macroscopic

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  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1940
  • Izdošanas datums: 08-Apr-2008
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540783626
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Multiscale Problems in the Life Sciences: From Microscopic to MacroscopicPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1940
  • Izdošanas datums: 08-Apr-2008
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540783626
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The aim of this volume that presents lectures given at a joint CIME and Banach Center Summer School, is to offer a broad presentation of a class of updated methods providing a mathematical framework for the development of a hierarchy of models of complex systems in the natural sciences, with a special attention to biology and medicine. Mastering complexity implies sharing different tools requiring much higher level of communication between different mathematical and scientific schools, for solving classes of problems of the same nature. Today more than ever, one of the most important challenges derives from the need to bridge parts of a system evolving at different time and space scales, especially with respect to computational affordability. As a result the content has a rather general character; the main role is played by stochastic processes, positive semigroups, asymptotic analysis, kinetic theory, continuum theory, and game theory.



Positivity in Natural Sciences
Jacek Banasiak
1
1 Introduction
1
1.1 What can go Wrong?
3
1.2 And if Everything Seems to be Fine?
3
2 Spectral Properties of Operators
4
2.1 Operators
5
2.2 Spectral Properties of a Single Operator
7
3 Banach Lattices and Positive Operators
13
3.1 Defining Order
13
3.2 Banach Lattices
15
3.3 Positive Operators
19
3.4 Relation Between Order and Norm
20
3.5 Complexification
23
3.6 Spectral Radius of Positive Operators
24
4 First Semigroups
25
4.1 Around the Hille—Yosida Theorem
27
4.2 Dissipative Operators
28
4.3 Long Time Behaviour of Semigroups
29
4.4 Positive Semigroups
37
4.5 Generation Through Perturbation
39
4.6 Positive Perturbations of Positive Semigroups
42
5 What can go Wrong?
45
5.1 Applications to Birth-and-Death Type Problems
52
5.2 Chaos in Population Theory
59
6 Asynchronous Growth
61
6.1 Essential Growth Bound
61
6.2 Peripheral Spectrum of Positive Semigroups
63
6.3 Compactness, Positivity and Irreducibility of Perturbed Semigroups
67
7 Asymptotic Analysis of Singularly Perturbed Dynamical Systems
75
7.1 Compressed Expansion
77
References
87
Rescaling Stochastic Processes: Asymptotics
V. Capasso and D. Morale
91
1 Introduction
91
1.1 First Examples of Resealing
95
2 Stochastic Processes
97
2.1 Processes with Independent Increments
100
2.2 Martingales
100
2.3 Markov Processes
103
2.4 Brownian Motion and the Wiener Process
109
3 Ito Calculus
110
3.1 The Ito Integral
110
3.2 The Stochastic Differential
112
3.3 Stochastic Differential Equations
113
3.4 Kolmogorov and Fokker-Planck Equations
115
3.5 The Multidimensional Case
117
4 Deterministic Approximation of Stochastic Systems
118
4.1 Continuous Approximation of Jump Population Processes
118
4.2 Continuous Approximation of Stochastic Interacting Particle Systems
120
4.3 Convergence of the Empirical Measure
122
5 A Specific Model for Interacting Particles
128
5.1 Asymptotic Behavior of the System for Large Populations: A Heuristic Derivation
130
5.2 Asymptotic Behavior of the System for Large Populations: A Rigorous Derivation
134
6 Long Time Behavior: Invariant Measure
137
A Proof of the Identification of the Limit p
141
References
144
Modelling Aspects of Cancer Growth: Insight from Mathematical and Numerical Analysis and Computational Simulation
Mark A.J. Chaplain
147
1 Introduction
147
1.1 Macroscopic Modelling
148
1.2 Cancer Growth and Development
149
2 Nlodelling Avascillar Solid Tumour Growth
150
2.1 Introduction
150
2.2 Linearised Stability Theory
151
2.3 The Role of Pre-Pattern Theory in Solid Tumour Growth and Invasion
153
2.4 Model Extension: Application to a Growing Spherical Tumour
156
2.5 Discussion and Conclusions
157
3 Mathematical Modelling of T-Lymphocyte Response to a Solid Tumour
160
3.1 Introduction
160
3.2 The Mathematical Model
161
3.3 Travelling Wave Analysis
173
3.4 Discussion
178
4 Mathematical Modelling of Cancer Invasion
180
4.1 Introduction
180
4.2 Cancer Invasion of Tissue and Metastasis
182
4.3 Proteolysis and Extracellular Matrix Degradation
182
4.4 The Mathematical Model of Proteolysis and Cancer Cell Invasion of Tissue
184
4.5 Nondimensionalisation of the Model Equations
187
4.6 Model Analysis
188
4.7 Spatially Uniform Steady States
188
4.8 Taxis-Driven Instability and Dispersion Curves
188
4.9 Numerical Results
189
4.10 Numerical Technique
190
4.11 Computational Simulation Results
191
4.12 Discussion and Conclusions
191
5 Summary
195
References
195
Lins Between Microscopic and Macroscopic Descriptions
Miroslaw Lachowicz
201
1 Introduction
201
2 Microscopic (Stochastic) Systems
205
3 Generalized Kinetic Models
213
4 Diffusive Limit
227
5 Links in the Space—Homogeneous Case
231
6 Coagulation—Fragmentation Equations
243
7 The Space—Inhomogeneous Case: Reaction—Diffusion Equations
245
8 React ion—Diffusion—Chemot axis Equations
252
References
262
Evolutionary Game Theory and Population Dynamics
Jacek Mickisz
269
1 Short Overview
269
2 Introduction
270
3 A Crash Course in Game Theory
273
4 Replicator Dynamics
277
5 Replicator Dynamics with Migration
280
6 Replicator Dynamics with Time Delay
285
6.1 Social-Type Time Delay
285
6.2 Biological-Type Time Delay
288
7 Stochastic Dynamics of Finite Populations
290
8 Stochastic Dynamics of Well-Mixed Populations
292
9 Spatial Games with Local Interactions
298
9.1 Nash Configurations and Stochastic Dynamics
298
9.2 Ground States and Nash Configurations
300
9.3 Ensemble Stability
303
9.4 Stochastic Stability in Non-Potential Games
306
9.5 Dominated Strategies
310
10 Review of Other Results
311
References
312
List of Participants 317
Index 319
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