Differential Equations and Mathematical Biology

ISBN-13: 9789401159722 / Angielski / Miękka / 2012 / 340 str.

Douglas Samuel Jones

Opis

Over the past decade, mathematics has made a considerable impact as a tool with which to model and understand biological phenomena. In return, biology has confronted the mathematician with a variety of challenging problems which have stimulated developments in the theory of nonlinear differential equations. This book is the outcome of the need to introduce undergraduates of mathematics, the physical and biological sciences to some of those developments. It is primarily directed towards students with a mathematical background up to and including that normally taught in a first-year physical science degree of a British university (sophomore year in a North American university) who are interested in the application of mathematics to biological and physical situations. Chapter 1 is introductory, showing how the study of first-order ordinary differential equations may be used to model the growth of a population, monitoring the administration of drugs and the mechanism by which living cells divide. In Chapter 2, a fairly comprehensive account of linear ordinary differential equations with constant coefficients is given. Such equations arise frequently in the discussion of the biological models encountered throughout the text. Chapter 3 is devoted to modelling biological pheno- mena and in particular includes (i) physiology of the heart beat cycle, (ii) blood flow, (iii) the transmission of electrochemical pulses in the nerve, (iv) the Belousov-Zhabotinskii chemical reaction and (v) predator-prey models.

Spis treści

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 the Belousov-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.