Differential equations and mathematical biology

The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century. Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics. This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.

IntroductionPopulation Growth Administration of Drugs Cell Division Differential Equations with Separable Variables General Properties Equations of Homogeneous Type Linear Differential Equations of the First Order Linear ordinary differential equations with constant coefficientsIntroduction First-Order Linear Differential Equations Linear Equations of the Second Order Finding the Complementary Function Determining a Particular Integral Forced Oscillations Differential Equations of the Order n Uniqueness Appendix: Existence Theory Simultaneous equations with constant coefficientsSimultaneous Equations of the First Order Replacement of One Differential Equation by a System The General System The Fundamental System Matrix Notation Initial and Boundary Value Problems Solving the Inhomogeneous Differential Equation Appendix: Symbolic Computation Modelling biological phenomenaIntroduction Heart Beat Blood Flow Nerve Impulse Transmission Chemical Reactions Predator-Prey Models First-order systems of ordinary differential equationsExistence and Uniqueness Epidemics The Phase Plane Local Stability Stability Limit Cycles Forced Oscillations Appendix: Existence Theory Appendix: Computing Trajectories Mathematics of heart physiologyThe Local Model The Threshold Effect Phase Plane Analysis and the Heart Beat Model Physiological Considerations of the Heart Beat Cycle A Model of the Cardiac Pacemaker Mathematics of nerve impulse transmissionExcitability and Repetitive Firing Travelling Waves Qualitative Behaviour of Travelling Waves Chemical reactionsWavefronts for the Belousov-Zhabotinskii Reaction Phase Plane Analysis of Fisher's Equation Qualitative Behaviour in the General Case Predator and preyCatching Fish The Effect of Fishing The Volterra-Lotka Model Partial differential equationsCharacteristics for Equations of the First Order Another View of Characteristics Linear Partial Differential Equations of the Second Order Elliptic Partial Differential Equations Parabolic Partial Differential Equations Hyperbolic Partial Differential Equations The Wave Equation Typical Problems for the Hyperbolic Equation The Euler-Darboux Equation Evolutionary equationsThe Heat Equation Separation of Variables Simple Evolutionary Equations Comparison Theorems Problems of diffusionDiffusion through Membranes Energy and Energy Estimates Global Behaviour of Nerve Impulse Transmissions Global Behaviour in Chemical Reactions Turing Diffusion Driven Instability and Pattern Formation Finite Pattern Forming Domains Bifurcation and chaosBifurcation Bifurcation of a Limit Cycle Discrete Bifurcation Chaos Stability The Poincaré Plane Averaging Appendix: Programs Growth of tumoursIntroduction A Mathematical Model of Tumour Growth A Spherical Tumour Stability EpidemicsThe Kermack-McKendrick Model Vaccination An Incubation Model Spreading in Space Answers to exercisesIndex Each chapter also contains Exercises.