2nd Edition

# Differential Equations and Mathematical Biology

**Also available as eBook on:**

*Deepen students’ understanding of biological phenomena*

Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, **Differential Equations and Mathematical Biology, Second Edition** introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material.

**New to the Second Edition**

- A section on spiral waves
- Recent developments in tumor biology
- More on the numerical solution of differential equations and numerical bifurcation analysis
- MATLAB
^{®}files available for download online - Many additional examples and exercises

This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincaré phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator–prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor growth and the spread of infectious diseases.

**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 higher-order equations

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 *λ* − *ω* 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 Poincaré plane

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**

### 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.

"… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions."

—SIAM Review, Sept. 2010, Vol. 52, No. 3"… Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book … The authors succeed because they do a splendid job of integrating their treatment of differential equations with the applications, and they don’t try to do too much. … Each chapter comes with a collection of well-selected exercises, and plenty of references for further reading."

—MAA Reviews, April 2010

Praise for the First Edition"A strength of [this book] is its concise coverage of a broad range of topics. … It is truly remarkable how much material is squeezed into the slim book’s 400 pages."

—SIAM Review, Vol. 46, No. 1

"It is remarkable that without the classical scheme (definition, theorem, and proof) it is possible to explain rather deep results like properties of the Fitz–Hugh–Nagumo model … or the Turing model. … This feature makes the reading of this text pleasant business for mathematicians. … [This book] can be recommended for students of mathematics who like to see applications, because it introduces them to problems on how to model processes in biology, and also for theoretically oriented students of biology, because it presents constructions of mathematical models and the steps needed for their investigations in a clear way and without references to other books."

—EMS Newsletter

"The title precisely reflects the contents of the book, a valuable addition to the growing literature in mathematical biology from a deterministic modeling approach. This book is a suitable textbook for multiple purposes. … Overall, topics are carefully chosen and well balanced. …The book is written by experts in the research fields of dynamical systems and population biology. As such, it presents a clear picture of how applied dynamical systems and theoretical biology interact and stimulate each other—a fascinating positive feedback whose strength is anticipated to be enhanced by outstanding texts like the work under review."

—Mathematical Reviews, Issue 2004g