Preface ......................................................... V
1. Growth Models ................................................ 1
1.1. A Growth Model for one Population ....................... 1
1.2. Interacting Growth of two Populations ................... 9
1.3. Interacting Growth of n ≥ 2 Populations ................ 15
1.4. Discretization of the Time-Continuous Model ............ 23
1.4.1. The n-Population Model .......................... 23
1.4.2. The One-Population Model ........................ 33
1.5. Determination of Model Parameters from Data ............ 36
References .................................................. 39
2. A Game-Theoretic Evolution Model ............................ 41
2.1. Evolution-Matrix-Games for one Population .............. 41
2.1.1. The Game and Evolutionarily Stable Equilibria ... 41
2.1.2. Characterization of Evolutionarily Stable
Equilibria ...................................... 45
2.1.3. Evolutionarily Stable Equilibria for
2x2-Matrices .................................... 50
2.1.4. On the Detection of Evolutionarily Stable
Equilibria ...................................... 52
2.1.5. A Dynamical Treatment of the Game ............... 57
2.1.6. Existence and Iterative Calculation of
Nash Equilibria ................................. 62
2.1.7. Zero-Sum Evolution Matrix Games ................. 74
2.2. Evolution-Bi-Matrix-Games for two Populations .......... 79
2.2.1. The Game and Evolutionarily Stable
Equilibria ...................................... 79
2.2.2. A Dynamical Treatment of the Game ............... 83
2.2.3. Existence and Iterative Calculation of
Nash Equilibria ................................. 88
2.2.4. A Direct Method for the Calculation of
Nash Equilibria ................................. 93
References .................................................... 102
3. Four Models of Optimal Control in Medicine ................. 103
3.1. Controlled Growth of Cancer Cells ..................... 103
3.2. Optimal Administration of Drugs ....................... 111
3.2.1. A One-Compartment Model ........................ 112
3.2.2. A Two-Compartment Model ........................ 114
3.3. Optimal Control of Diabetes Mellitus .................. 119
3.3.1. The Model ...................................... 119
3.3.2. On the Approximate Solution of the Model
Problem ........................................ 121
3.3.3. A Time-Discrete Diabetes Model ................. 124
3.3.4. An Exact Solution of the Model Problem ......... 127
3.4. Optimal Control Aspects of the Blood Circulation
in the Heart .......................................... 130
3.4.1. Blood Circulation in the Heart ................. 130
3.4.2. A Model of the Left-Ventricular Ejection
Dynamics ....................................... 130
3.4.3. An Optimal Control Problem ..................... 132
3.4.4. Another Model of the Left-Ventricular
Ejection Dynamics .............................. 137
References .................................................... 139
4. A Mathematical Model of Hemodialysis ....................... 141
4.1. A One-Compartment Model ............................... 141
4.1.1. The Mass Transport in the Dialyzer ............. 141
4.1.2. The Temporal Development of the Toxin
Concentration in the Blood without
Ultrafiltration ................................ 143
4.1.3. The Temporal Development of the Toxin
Concentration in the Blood with
Ultrafiltration ................................ 148
4.2. A Two-Compartment Model ............................... 152
4.2.1. Derivation of the Model Equations .............. 152
4.2.2. Determination of the Clearance of the Cell
Membranes for Urea ............................. 154
4.3. Computation of Periodic Toxin Concentrations .......... 158
4.3.1. The General Method ............................. 158
4.3.2. The Case of Constant Clearance of the
Dialyzer ....................................... 162
4.3.3. Discretization of the Model Equations .......... 163
4.3.4. Numerical Results for Urea ..................... 167
4.3.5. The Influence of the Urea Generation Rate ...... 170
4.3.6. Determination of the Urea Generation Rate and
the Rest Clearance of the Kidneys .............. 171
4.4. A Three-Compartment Model ............................. 173
4.4.1. Motivation and Derivation of the
Model Equations ................................ 173
4.4.2. Determination of the Clearance of the Cell
Membranes of the Brain ......................... 175
4.4.3. Computation of Periodic Urea Concentration
Curves ......................................... 176
4.4.4. Numerical Results .............................. 182
References .................................................... 183
A. Appendix ................................................... 185
A.1. A Problem of Optimal Control .......................... 185
A.l.l. The Problem .................................... 185
A.1.2. Multiplier Rule ................................ 186
A.2. Existence of Positive Periodic Solutions in
a General Diffusion Model ............................. 189
A.2.1. The Model ...................................... 189
A.2.2. An Existence and Unicity Theorem ............... 190
A.3. Asymptotic Stability of Fixed Points .................. 195
Index ......................................................... 201
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