**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**