Ideal for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, this bestselling text introduces the fundamental modeling and analytical techniques used to understand biological phenomena. It discusses the modeling of biological behavior, including biochemical reactions, nerve pulses, predator-prey models, and epidemics. This edition includes a section on spiral waves, developments in tumor biology, and additional examples and exercises. With downloadable MATLAB® files available online, it presents numerical solutions of differential equations and numerical bifurcation analysis.…mehr
Ideal for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, this bestselling text introduces the fundamental modeling and analytical techniques used to understand biological phenomena. It discusses the modeling of biological behavior, including biochemical reactions, nerve pulses, predator-prey models, and epidemics. This edition includes a section on spiral waves, developments in tumor biology, and additional examples and exercises. With downloadable MATLAB® files available online, it presents numerical solutions of differential equations and numerical bifurcation analysis.
D.S. Jones, FRS, FRSE is Professor Emeritus in the Department of Mathematics at the University of Dundee in Scotland. M.J. Plank is a senior lecturer in the Department of Mathematics and Statistics at the University of Canterbury in Christchurch, New Zealand. B.D. Sleeman, FRSE is Professor Emeritus in the Department of Applied Mathematics at the University of Leeds in the UK.
Inhaltsangabe
Introduction. Linear Ordinary Differential Equations with Constant Coefficients. Systems of Linear Ordinary Differential Equations. Modelling Biological Phenomena. First-Order Systems of Ordinary Differential Equations. Mathematics of Heart Physiology. Mathematics of Nerve Impulse Transmission. Chemical Reactions. Predator and Prey. Partial Differential Equations. Evolutionary Equations. Problems of Diffusion. Bifurcation and Chaos. Numerical Bifurcation Analysis. Growth of Tumors. Epidemics. Answers to Selected Exercises. Index.
1 Introduction.- 1.1 Population growth.- 1.2 Administration of drugs.- 1.3 Cell division.- 1.4 Differential equations with separable variables.- 1.5 General properties.- 1.6 Equations of homogeneous type.- 1.7 Linear differential equations of the first order.- Notes.- Exercises.- 2 Linear ordinary differential equations with constant coefficients.- 2.1 Introduction.- 2.2 First-order linear differential equations.- 2.3 Linear equations of the second order.- 2.4 Finding the complementary function.- 2.5 Determining a particular integral.- 2.6 Forced oscillations.- 2.7 Differential equations of order n.- 2.8 Simultaneous equations of the first order.- 2.9 Replacement of one differential equation by a system.- 2.10 The general system.- 2.11 The fundamental system.- 2.12 Matrix notation.- 2.13 Initial and boundary value problems.- 2.14 Solving the inhomogeneous differential equation.- Exercises.- 3 Modelling biological phenomena.- 3.1 Introduction.- 3.2 Heart beat.- 3.3 Blood flow.- 3.4 Nerve impulse transmission.- 3.5 Chemical reactions.- 3.6 Predator-prey models.- Notes.- Exercises.- 4 First-order systems of ordinary differential equations.- 4.1 Existence and uniqueness.- 4.2 Epidemics.- 4.3 The phase plane.- 4.4 Local stability.- 4.5 Stability.- 4.6 Limit cycles.- 4.7 Forced oscillations.- 4.8 Appendix: existence theory.- Exercises.- 5 Mathematics of heart physiology.- 5.1 The local model.- 5.2 The threshold effect.- 5.3 The phase plane analysis and the heart beat model.- 5.4 Physiological considerations of the heart beat cycle.- 5.5 A model of the cardiac pacemaker 139 Notes.- Exercises.- 6 Mathematics of nerve impulse transmission.- 6.1 Phase plane methods.- 6.2 Qualitative behaviour of travelling waves.- Notes.- Exercises.- 7 Chemical reactions.- 7.1 Wavefronts for theBelousov-Zhabotinskii reaction.- 7.2 Phase plane analysis of Fisher's equation.- 7.3 Qualitative behaviour in the general case.- Notes.- Exercises.- 8 Predator and prey.- 8.1 Catching fish.- 8.2 The effect of fishing.- 8.3 The Volterra-Lotka model.- Exercises.- 9 Partial differential equations.- 9.1 Characteristics for equations of the first order.- 9.2 Another view of characteristics.- 9.3 Linear partial differential equations of the second order.- 9.4 Elliptic partial differential equations.- 9.5 Parabolic partial differential equations.- 9.6 Hyperbolic partial differential equations.- 9.7 The wave equation.- 9.8 Typical problems for the hyperbolic equation.- 9.9 The Euler-Darboux equation.- Exercises.- 10 Evolutionary equations.- 10.1 The heat equation.- 10.2 Separation of variables.- 10.3 Simples evolutionary equations.- 10.4 Comparison theorems.- Notes.- Exercises.- 11 Problems of diffusion.- 11.1 Diffusion through membranes.- 11.2 Energy and energy estimates.- 11.3 Global behaviour of nerve impulse transmissions.- 11.4 Global behaviour in chemical reactions.- Notes.- Exercises.- 12 Catastrophe theory and biological phenomena.- 12.1 What is a catastrophe?.- 12.2 Elementary catastrophes.- 12.3 Biology and catastrophe theory.- Exercises.- 13 Growth of tumours.- 13.1 Introduction.- 13.2 A mathematical model of tumour growth.- 13.3 A spherical tumour.- Notes.- Exercises.- 14 Epidemics.- 14.1 The Kermack-McKendrick model.- 14.2 Vaccination.- 14.3 An incubation model.- 14.4 Spreading in space.- Exercises.- Answers to exercises.
Introduction. Linear Ordinary Differential Equations with Constant Coefficients. Systems of Linear Ordinary Differential Equations. Modelling Biological Phenomena. First-Order Systems of Ordinary Differential Equations. Mathematics of Heart Physiology. Mathematics of Nerve Impulse Transmission. Chemical Reactions. Predator and Prey. Partial Differential Equations. Evolutionary Equations. Problems of Diffusion. Bifurcation and Chaos. Numerical Bifurcation Analysis. Growth of Tumors. Epidemics. Answers to Selected Exercises. Index.
1 Introduction.- 1.1 Population growth.- 1.2 Administration of drugs.- 1.3 Cell division.- 1.4 Differential equations with separable variables.- 1.5 General properties.- 1.6 Equations of homogeneous type.- 1.7 Linear differential equations of the first order.- Notes.- Exercises.- 2 Linear ordinary differential equations with constant coefficients.- 2.1 Introduction.- 2.2 First-order linear differential equations.- 2.3 Linear equations of the second order.- 2.4 Finding the complementary function.- 2.5 Determining a particular integral.- 2.6 Forced oscillations.- 2.7 Differential equations of order n.- 2.8 Simultaneous equations of the first order.- 2.9 Replacement of one differential equation by a system.- 2.10 The general system.- 2.11 The fundamental system.- 2.12 Matrix notation.- 2.13 Initial and boundary value problems.- 2.14 Solving the inhomogeneous differential equation.- Exercises.- 3 Modelling biological phenomena.- 3.1 Introduction.- 3.2 Heart beat.- 3.3 Blood flow.- 3.4 Nerve impulse transmission.- 3.5 Chemical reactions.- 3.6 Predator-prey models.- Notes.- Exercises.- 4 First-order systems of ordinary differential equations.- 4.1 Existence and uniqueness.- 4.2 Epidemics.- 4.3 The phase plane.- 4.4 Local stability.- 4.5 Stability.- 4.6 Limit cycles.- 4.7 Forced oscillations.- 4.8 Appendix: existence theory.- Exercises.- 5 Mathematics of heart physiology.- 5.1 The local model.- 5.2 The threshold effect.- 5.3 The phase plane analysis and the heart beat model.- 5.4 Physiological considerations of the heart beat cycle.- 5.5 A model of the cardiac pacemaker 139 Notes.- Exercises.- 6 Mathematics of nerve impulse transmission.- 6.1 Phase plane methods.- 6.2 Qualitative behaviour of travelling waves.- Notes.- Exercises.- 7 Chemical reactions.- 7.1 Wavefronts for theBelousov-Zhabotinskii reaction.- 7.2 Phase plane analysis of Fisher's equation.- 7.3 Qualitative behaviour in the general case.- Notes.- Exercises.- 8 Predator and prey.- 8.1 Catching fish.- 8.2 The effect of fishing.- 8.3 The Volterra-Lotka model.- Exercises.- 9 Partial differential equations.- 9.1 Characteristics for equations of the first order.- 9.2 Another view of characteristics.- 9.3 Linear partial differential equations of the second order.- 9.4 Elliptic partial differential equations.- 9.5 Parabolic partial differential equations.- 9.6 Hyperbolic partial differential equations.- 9.7 The wave equation.- 9.8 Typical problems for the hyperbolic equation.- 9.9 The Euler-Darboux equation.- Exercises.- 10 Evolutionary equations.- 10.1 The heat equation.- 10.2 Separation of variables.- 10.3 Simples evolutionary equations.- 10.4 Comparison theorems.- Notes.- Exercises.- 11 Problems of diffusion.- 11.1 Diffusion through membranes.- 11.2 Energy and energy estimates.- 11.3 Global behaviour of nerve impulse transmissions.- 11.4 Global behaviour in chemical reactions.- Notes.- Exercises.- 12 Catastrophe theory and biological phenomena.- 12.1 What is a catastrophe?.- 12.2 Elementary catastrophes.- 12.3 Biology and catastrophe theory.- Exercises.- 13 Growth of tumours.- 13.1 Introduction.- 13.2 A mathematical model of tumour growth.- 13.3 A spherical tumour.- Notes.- Exercises.- 14 Epidemics.- 14.1 The Kermack-McKendrick model.- 14.2 Vaccination.- 14.3 An incubation model.- 14.4 Spreading in space.- Exercises.- Answers to exercises.
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