Differential Equations and Mathematical Biology, Second Edition

by Jones; D.S.

• ISBN13: 9781420083576
• ISBN10: 1420083570
• Edition: 2nd
• Format: Hardcover
• Copyright: 2009-11-09
• Publisher: Chapman & Hall/

Summary

This second edition of a bestseller introduces the fundamental modeling and analytical techniques required to deepen understanding of biological phenomena. Keeping the successful format of the previous edition, it is still a dual purpose text suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology. The new edition includes a section on spiral waves, recent developments in tumor biology, and more significantly, covers much more of the numerical solutions of different equations and numerical bifurcation analysis. It contains many more examples and exercises and all MATLAB files found in the book are available for download.

Table of Contents

Preface to the First Edition	p. xiii
Preface to the Second Edition	p. xv
Introduction	p. 1
Population growth	p. 1
Administration of drugs	p. 4
Cell division	p. 9
Differential equations with separable variables	p. 11
Equations of homogeneous type	p. 14
Linear differential equations of the first order	p. 16
Numerical solution of first-order equations	p. 19
Symbolic computation in MATLAB“	p. 24
Notes	p. 27
Linear Ordinary Differential Equations with Constant Coefficients	p. 33
Introduction	p. 33
First-order linear differential equations	p. 35
Linear equations of the second order	p. 36
Finding the complementary function	p. 37
Determining a particular integral	p. 41
Forced oscillations	p. 50
Differential equations of order n	p. 52
Uniqueness	p. 55
Systems of Linear Ordinary Differential Equations	p. 61
First-order systems of equations with constant coefficients	p. 61
Replacement of one differential equation by a system	p. 64
The general system	p. 66
The fundamental system	p. 68
Matrix notation	p. 72
Initial and boundary value problems	p. 77
Solving the inhomogeneous differential equation	p. 82
Numerical solution of linear boundary value problems	p. 84
Modelling Biological Phenomena	p. 91
Introduction	p. 91
Heartbeat	p. 91
Nerve impulse transmission	p. 94
Chemical reactions	p. 100
Predator-prey models	p. 106
Notes	p. 109
First-Order Systems of Ordinary Differential Equations	p. 115
Existence and uniqueness	p. 115
Epidemics	p. 118
The phase plane and the Jacobian matrix	p. 119
Local stability	p. 121
Stability	p. 128
Limit cycles	p. 133
Forced oscillations	p. 139
Numerical solution of systems of equations	p. 143
Symbolic computation on first-order systems of equations and higher-order equations	p. 147
Numerical solution of nonlinear boundary value problems	p. 149
Appendix: existence theory	p. 153
Mathematics of Heart Physiology	p. 163
The local model	p. 163
The threshold effect	p. 166
The phase plane analysis and the heartbeat model	p. 168
Physiological considerations of the heartbeat cycle	p. 171
A model of the cardiac pacemaker	p. 173
Notes	p. 175
Mathematics of Nerve Impulse Transmission	p. 177
Excitability and repetitive firing	p. 177
Travelling waves	p. 185
Qualitative behaviour of travelling waves	p. 187
Piecewise linear model	p. 190
Notes	p. 194
Chemical Reactions	p. 197
Wavefronts for the Belousov-Zhabotinskii reaction	p. 197
Phase plane analysis of Fisher's equation	p. 198
Qualitative behaviour in the general case	p. 199
Spiral waves and ¿ - ¿ systems	p. 204
Notes	p. 207
Predator and Prey	p. 211
Catching fish	p. 211
The effect of fishing	p. 213
The Volterra-Lotka model	p. 215
Partial Differential Equations	p. 223
Characteristics for equations of the first order	p. 223
Another view of characteristics	p. 230
Linear partial differential equations of the second order	p. 232
Elliptic partial differential equations	p. 235
Parabolic partial differential equations	p. 239
Hyperbolic partial differential equations	p. 239
The wave equation	p. 240
Typical problems for the hyperbolic equation	p. 245
The Euler-Darboux equation	p. 250
Visualisation of solutions	p. 251
Evolutionary Equations	p. 259
The heat equation	p. 259
Separation of variables	p. 262
Simple evolutionary equations	p. 269
Comparison theorems	p. 277
Notes	p. 289
Problems of Diffusion	p. 293
Diffusion through membranes	p. 293
Energy and energy estimates	p. 299
Global behaviour of nerve impulse transmissions	p. 304
Global behaviour in chemical reactions	p. 308
Turing diffusion driven instability and pattern formation	p. 311
Finite pattern forming domains	p. 321
Notes	p. 325
Bifurcation and Chaos	p. 329
Bifurcation	p. 329
Bifurcation of a limit cycle	p. 334
Discrete bifurcation and period-doubling	p. 336
Chaos	p. 342
Stability of limit cycles	p. 346
The Poincaré plane	p. 350
Averaging	p. 355
Numerical Bifurcation Analysis	p. 367
Fixed points and stability	p. 367
Path-following and bifurcation analysis	p. 370
Following stable limit cycles	p. 376
Bifurcation in discrete systems	p. 378
Strange attractors and chaos	p. 380
Stability analysis of partial differential equations	p. 384
Notes	p. 385
Growth of Tumours	p. 389
Introduction	p. 389
Mathematical Model I of tumour growth	p. 392
Spherical tumour growth based on Model I	p. 395
Stability of tumour growth based on Model I	p. 399
Mathematical Model II of tumour growth	p. 401
Spherical tumour growth based on Model II	p. 404
Stability of tumour growth based on Model II	p. 406
Notes	p. 407
Epidemics	p. 411
The Kermack-McKendrick model	p. 411
Vaccination	p. 413
An incubation model	p. 414
Spreading in space	p. 418
Answers to Selected Exercises	p. 427
Index	p. 439
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