Differential equations and mathematical biology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press, Taylor & Francis Group
[2010]
|
| Edition: | Second edition |
| Series: | Mathematical and computational biology series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 444 Seiten |
| ISBN: | 9781420083576 |
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Record in the Search Index
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| adam_text | Titel: Differential equations and mathematical biology
Autor: Jones, Douglas S
Jahr: 2010
Contents
Preface to the First Edition xiii
Preface to the Second Edition xv
1 Introduction 1
1.1 Population growth 1
1.2 Administration of drugs 4
1.3 Cell division 9
1.4 Differential equations with separable variables 11
1.5 Equations of homogeneous type 14
1.6 Linear differential equations of the first order 16
1.7 Numerical Solution of first-order equations 19
1.8 Symbolic computation in MATLAB® 24
1.9 Notes 27
2 Linear Ordinary Differential Equations with Constant
CoefRcients 33
2.1 Introduction 33
2.2 First-order linear differential equations 35
2.3 Linear equations of the second order 36
2.4 Finding the complementary function 37
2.5 Determining a particular integral 41
2.6 Forced oscillations 50
2.7 Differential equations of order n 52
2.8 Uniqueness 55
3 Systems of Linear Ordinary Differential Equations 61
3.1 First-order Systems of equations with constant coefficients . 61
3.2 Replacement of one differential equation by a system .... 64
3.3 The general system 66
3.4 The fundamental system 68
3.5 Matrix notation 72
3.6 Initial and boundary value problems 77
3.7 Solving the inhomogeneous differential equation 82
3.8 Numerical Solution of linear boundary value problems .... 84
91
91
91
94
100
106
109
115
115
118
119
121
128
133
139
143
147
149
153
163
163
166
168
171
173
175
177
177
185
187
190
194
197
197
198
199
204
207
Contents
Modelling Biological Phenomena
4.1 Introduction
4.2 Heartbeat
4.3 Nerve impulse transmission
4.4 Chemical reactions
4.5 Predator-prey models
4.6 Notes
First-Order Systems of Ordinary Differential Equations
5.1 Existence and uniqueness
5.2 Epidemics
5.3 The phase plane and the Jacobian matrix
5.4 Local stability
5.5 Stability
5.6 Limit cycles
5.7 Forced oscillations
5.8 Numerical Solution of Systems of equations
5.9 Symbolic computation on first-order Systems of equations and
higher-order equations
5.10 Numerical Solution of nonlinear boundary value problems . .
5.11 Appendix: existence theory
Mathematics of Heart Physiology
6.1 The local model
6.2 The threshold effect
6.3 The phase plane analysis and the heartbeat model
6.4 Physiological considerations of the heartbeat cycle
6.5 A model of the cardiac pacemaker
6.6 Notes
Mathematics of Nerve Impulse Transmission
7.1 Excitability and repetitive firing
7.2 Travelling waves
7.3 Qualitative behaviour of travelling waves
7.4 Piecewise linear model
7.5 Notes
Chemical Reactions
8.1 Wavefronts for the Be 1 ousov Zhabotinskii reaction
8.2 Phase plane analysis of Fisher s equation
8.3 Qualitative behaviour in the general case
8.4 Spiral waves and A — u Systems
8.5 Notes
Contents xi
9 Predator and Prey 211
9.1 Catching fish 211
9.2 The effect of fishing 213
9.3 The Volterra-Lotka model 215
10 Partial Differential Equations 223
10.1 Characteristics for equations of the first order 223
10.2 Another view of characteristics 230
10.3 Linear partial differential equations of the second order . . . 232
10.4 Elliptic partial differential equations 235
10.5 Parabolic partial differential equations 239
10.6 Hyperbolic partial differential equations 239
10.7 The wave equation 240
10.8 Typical problems for the hyperbolic equation 245
10.9 The Euler-Darboux equation 250
10.10 Visualisation of Solutions 251
11 Evolutionary Equations 259
11.1 The heat equation 259
11.2 Separation of variables 262
11.3 Simple evolutionary equations 269
11.4 Comparison theorems 277
11.5 Notes 289
12 Problems of Diffusion 293
12.1 Diffusion through membranes 293
12.2 Energy and energy estimates 299
12.3 Global behaviour of nerve impulse transmissions 304
12.4 Global behaviour in chemical reactions 308
12.5 Turing diffusion driven instability and pattern formation . . 311
12.6 Finite pattern forming domains 321
12.7 Notes 325
13 Bifurcation and Chaos 329
13.1 Bifurcation 329
13.2 Bifurcation of a limit cycle 334
13.3 Discrete bifurcation and period-doubling 336
13.4 Chaos 342
13.5 Stability of limit cycles 346
13.6 The Poincare plane 350
13.7 Averaging 355
xii Contents
14 Numerical Bifurcation Analysis 367
14.1 Fixed points and stability 367
14.2 Path-following and bifurcation analysis 370
14.3 Following stable limit cycles 376
14.4 Bifurcation in discrete Systems 378
14.5 Strange attractors and chaos 380
14.6 Stability analysis of partial differential equations 384
14.7 Notes 385
15 Growth of Tumours 389
15.1 Introduction 389
15.2 Mathematical Model I of tumour growth 392
15.3 Spherical tumour growth based on Model I 395
15.4 Stability of tumour growth based on Model I 399
15.5 Mathematical Model II of tumour growth 401
15.6 Spherical tumour growth based on Model II 404
15.7 Stability of tumour growth based on Model II 406
15.8 Notes 407
16 Epidemics 411
16.1 The Kermack-McKendrick model 411
16.2 Vaccination 413
16.3 An incubation model 414
16.4 Spreading in space 418
Answers to Selected Exercises 427
Index 439
|
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| author | Jones, D. S. |
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| dewey-tens | 570 - Biology |
| discipline | Biologie Mathematik |
| edition | Second edition |
| format | Book |
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| spelling | Jones, D. S. Verfasser (DE-588)141943092 aut Differential equations and mathematical biology D. S. Jones, M. J. Plank, B. D. Sleeman Second edition Boca Raton [u.a.] CRC Press, Taylor & Francis Group [2010] XVII, 444 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematical and computational biology series Biomathematics Differential equations Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Biomathematik (DE-588)4139408-2 gnd rswk-swf Biomathematik (DE-588)4139408-2 s Differentialgleichung (DE-588)4012249-9 s DE-604 Plank, Michael J. Sonstige (DE-588)141943122 oth Sleeman, Brian D. 1939- Sonstige (DE-588)10835105X oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022463652&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jones, D. S. Differential equations and mathematical biology Biomathematics Differential equations Differentialgleichung (DE-588)4012249-9 gnd Biomathematik (DE-588)4139408-2 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4139408-2 |
| title | Differential equations and mathematical biology |
| title_auth | Differential equations and mathematical biology |
| title_exact_search | Differential equations and mathematical biology |
| title_full | Differential equations and mathematical biology D. S. Jones, M. J. Plank, B. D. Sleeman |
| title_fullStr | Differential equations and mathematical biology D. S. Jones, M. J. Plank, B. D. Sleeman |
| title_full_unstemmed | Differential equations and mathematical biology D. S. Jones, M. J. Plank, B. D. Sleeman |
| title_short | Differential equations and mathematical biology |
| title_sort | differential equations and mathematical biology |
| topic | Biomathematics Differential equations Differentialgleichung (DE-588)4012249-9 gnd Biomathematik (DE-588)4139408-2 gnd |
| topic_facet | Biomathematics Differential equations Differentialgleichung Biomathematik |
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