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