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BK

2nd ed.

Boca Raton : CRC Press, c2010

xviii, 444 s. : il. ; 24 cm

ISBN 978-1-4200-8357-6 (váz.)

Chapman & Hall/CRC mathematical and computional biology series

Obsahuje rejstřík

000187655

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 Coefficients 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 // 4 Modelling Biological Phenomena 91 // 4.1 Introduction 91 // 4.2 Heartbeat 91 // 4.3 Nerve impulse transmission 94 // 4.4 Chemical reactions 100 // 4.5 Predator-prey models 106 // 4.6 Notes 109 // 5 First-Order Systems of Ordinary Differential Equations 115 // 5.1 Existence and uniqueness 116 // 5.2 Epidemics 118 // 5.3 The phase plane and the Jacobian matrix 119 // 5.4 Local stability 121 // 5.5 Stability 128 // 5.6 Limit cycles 183 // 5.7 Forced oscillations 139 // 5.8 Numerical solution of systems of equations 143 // 5.9 Symbolic computation on first-order systems of equations and higher-order equations 147 //

5.10 Numerical solution of nonlinear boundary value problems 149 // 5.11 Appendix: existence theory 153 // 6 Mathematics of Heart Physiology 163 // 6.1 The local model 163 // 6.2 The threshold effect 166 // 6.3 The phase plane analysis and the heartbeat model 168 // 6.4 Physiological considerations of the heartbeat cycle 171 // 6.5 A model of the cardiac pacemaker 173 // 6.6 Notes 1 5 // 7 Mathematics of Nerve Impulse Transmission 177 // 7.1 Excitability and repetitive firing 177 // 7.2 Travelling waves 185 // 7.3 Qualitative behaviour of travelling waves 187 // 7.4 Piecewise linear model 190 // 7.5 Notes 194 // 8 Chemical Reactions 197 // 8.1 Wavefronts for the Belousov-Zhabotinskii reaction 197 // 8.2 Phase plane analysis of Fisher’s equation 198 // 8.3 Qualitative behaviour in the general case 199 // 8.4 Spiral waves and Ŕ - a; systems 204 // 8.5 Notes 207 // 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 Poincaré plane 350 // 13.7 Averaging 355 // 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