Differential Equations and Mathematical Biology

About this book

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.

Contents

Introduction Population growth Administration of drugs Cell division Differential equations with separable variables Equations of homogeneous type Linear differential equations of the first order Numerical solution of first-order equations Symbolic computation in MATLAB Linear Ordinary Differential Equations with Constant Coefficients Introduction First-order linear differential equations Linear equations of the second order Finding the complementary function Determining a particular integral Forced oscillations Differential equations of order n Uniqueness Systems of Linear Ordinary Differential Equations First-order systems of equations with constant coefficients 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 Numerical solution of linear boundary value problems Modelling Biological Phenomena Introduction Heartbeat Nerve impulse transmission Chemical reactions Predator-prey models First-Order Systems of Ordinary Differential Equations Existence and uniqueness Epidemics The phase plane and the Jacobian matrix Local stability Stability Limit cycles Forced oscillations Numerical solution of systems of equations Symbolic computation on first-order systems of equations and Numerical solution of nonlinear boundary value problems Appendix: existence theory Mathematics of Heart Physiology The local model The threshold effect The phase plane analysis and the heartbeat model Physiological considerations of the heartbeat cycle A model of the cardiac pacemaker Mathematics of Nerve Impulse Transmission Excitability and repetitive firing Travelling waves Qualitative behavior of travelling waves Piecewise linear model Chemical Reactions Wavefronts for the Belousov-Zhabotinskii reaction Phase plane analysis of Fisher's equation Qualitative behavior in the general case Spiral waves and lambda - omega systems Predator and Prey Catching fish The effect of fishing The Volterra-Lotka model Partial Differential Equations Characteristics 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 Visualization of solutions Evolutionary Equations The heat equation Separation of variables Simple evolutionary equations Comparison theorems Problems of Diffusion Diffusion through membranes Energy and energy estimates Global behavior of nerve impulse transmissions Global behavior in chemical reactions Turing diffusion driven instability and pattern formation Finite pattern forming domains Bifurcation and Chaos Bifurcation Bifurcation of a limit cycle Discrete bifurcation and period-doubling Chaos Stability of limit cycles The Poincare Averaging Numerical Bifurcation Analysis Fixed points and stability Path-following and bifurcation analysis Following stable limit cycles Bifurcation in discrete systems Strange attractors and chaos Stability analysis of partial differential equations Growth of Tumors Introduction Mathematical model I of tumor growth Spherical tumor growth based on model I Stability of tumor growth based on model I Mathematical model II of tumor growth Spherical tumor growth based on model II Stability of tumor growth based on model II Epidemics The Kermack-McKendrick model Vaccination An incubation model Spreading in space Answers to Selected Exercises Index

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Biography

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.