The refinement of some of the simple mechanics models obtained in Part A is to be undertaken in this chapter and the next. Thus, for example, in Chapter 2 the problem of free fall under gravity was considered. The medium through which the object moves was completely ignored and so was the size and shape of the falling object. In this chapter these features will be included and it will be seen that two new forces become relevant – the drag force and the buoyant force. These forces provide the mechanism for some phenomena not present in the model of Chapter 2: the decrease in acceleration of all free falling objects, and the ability of some objects such as balloons to rise rather than fall.

The differential equations obtained in this chapter are first-order linear with constant coefficients or first-order separable. Knowledge of Section 11.1 of Chapter 11 is therefore required.

Some basic fluid mechanics

As an object moves through a fluid, a force is exerted by the fluid on the object which is in the opposite direction to the motion of the object. This force is called the drag force. To gain an understanding of the quantities influencing the drag force in a fluid, it is necessary first to discuss two fundamental properties of a fluid: the viscosity and Reynolds' number.

Viscosity

Gases and liquids are collectively known as fluids since they can both be made to flow if a force is applied.

In the previous chapter, difference equations were used to predict the change in the total population of a species from generation to generation. This leads naturally to the question of predicting the change in a particular characteristic of the individuals in the population. Since the heredity units which determine characteristics are the genes, this question comes under the heading of population genetics.

The formulation of models in population genetics requires a knowledge of the fundamentals of the theory of genetics. This chapter begins by presenting the required theory, before formulating particular models in terms of difference equations. These models differ from those obtained in the previous two chapters in that they are based on some firmly established laws. In this respect they are similar to the models obtained in mechanics. The laws of genetics, however, are expressed as probabilities and hence are only relevant, in practice, to populations which are sufficiently large.

The models in population genetics give rise to non-linear difference equations. This chapter assumes the elements of probability theory.

Some background genetics

An ability to predict the most probable characteristics of offspring from knowledge of the characteristics of the parents is a skill of prime importance to plant and animal breeders as well as to medical scientists. In many cases this problem can be reduced to the study of difference equations – but only after the fundamentals of the theory of genetics have been learned. This section presents the necessary background theory.

This book provides an introduction to modelling with both differential and difference equations. Our approach to mathematical modelling is to emphasize what is involved by looking at specific examples from a variety of disciplines. From each discipline enough background is provided to enable students to understand both the assumptions and the predictions of the models. Exercises have been included at the end of each section. They are intended to provide a balanced development of some of the main skills used in mathematical modelling, and hence they are an essential part of the book.

The main mathematical tools used in the book are differential and difference equations. Differential equations have their origins in mechanics: Newton's laws of motion lead to differential equations whose solutions can be used to predict the position of a body at some later time. Differential equations have been closely associated with the rise of physical science in previous centuries and they are now being used as models for real world problems in a variety of other disciplines. Difference equations are the discrete analogues of differential equations. They have risen to prominence in the last decade, during which it has become generally known that solutions of even very simple difference equations can exhibit complex chaotic behaviour.

To allow time for the development of other modelling skills besides solving the equations arising from the models, we have selected only models involving differential equations which are relatively easy to solve.

This chapter is about mechanical systems in which particles are attached to a rope which passes around some pulleys. To keep the mathematical model simple, we shall ignore friction, together with the masses of the rope and the pulleys. Newton's laws can then be used to find the tension in the rope and the acceleration of the particles.

On the basis of this mathematical model, it is possible to explain the operation of the ‘block and tackle’, which is often used in factories to raise heavy loads. It will be shown how a small force exerted by a workman pulling on one end of a rope can be converted by the pulleys into a large force acting on the heavy load.

The mathematical model also provides the theory underlying Atwood's machine, a contraption which is sometimes used to measure the acceleration due to gravity.

The main skill which this chapter aims to develop is that of using Newton's laws to derive equations of motion. The chapter also provides incidental practice at solving systems of simultaneous linear equations, solving differential equations by antidifferentiation, and manipulation of inequalities.

First, however, a model will be constructed for the forces acting in the rope.

Tension in the rope

The dynamical role of a piece of rope may be illustrated by what happens in a tug of war. By pulling on the rope, one team is able to exert a force on the other team, even though the teams are some distance apart.

In this chapter we look at simple models in which two quantities interact with each other. The models lead to pairs of simultaneous differential equations for the two quantities and, because of the interaction, the equations are coupled.

The models considered will be simple enough to produce differential equations which are first-order linear, with constant coefficients. A systematic method for uncoupling – and hence solving – such equations will be explained. It involves eliminating one of the two quantities to give a second-order differential equation of the type studied in Chapter 15.

The ideas will be illustrated by a mixing model, similar to the one in Chapter 13, but involving a pair of vats. Two models from physiology are then presented: the first models the glucose-insulin homeostasis in the bloodstream, while the second models the mother-fetus exchange of nutrients via the placenta.

Two-compartment mixing

From now on we shall be considering models which lead to a pair of simultaneous differential equations, rather than a single differential equation. The equations will involve a pair of quantities, rather than a single quantity, which are to be expressed as functions of the time, say. In this section we show how an extension of the mixing problem discussed in Chapter 13 leads to such a pair of equations.

Mixing with two vats

Consider two interconnected vats, each containing a mixture of dye and water, as in Figure 18.1.1. Dye runs into the first vat.

Compartment modelling is a means of constructing a differential equation for a complicated process by considering just the inputs and outputs of the process, during a small time interval. The basic ideas are developed in the context of a model describing the mixing of a dye and water. Compartment models are then formulated for the pollution in a lake and the temperature of a domestic hot water system. The latter model uses ideas about the flow of heat from Chapter 12. The differential equations obtained are mainly of the first-order linear constant-coefficient type.

A mixing problem

One of the aims of modelling is to isolate the most important factors in a problem and ignore those which may not be important. Even very complicated processes can initially be analysed using very simple mathematical models which may later be extended to more complex and realistic models by incorporating more features. In problems involving the mixing of two or more substances, simple models may be formulated by considering the input and output to a compartment containing the quantity of interest.

The following problem will be used to illustrate these ideas. The problem is illustrated in Figure 13.1.1.

Statement of problem

In a dye factory a large vat is used to mix dye and water. The water flows in at a rate of 6 litres/minute and the dye flows in at a rate of 2 litres/minute.

The aim of mechanics is to explain and predict the motion of bodies. Early in the history of mankind the motion of celestial objects — sun, moon, stars, planets, comets — became a source of curiosity and wonder. At the terrestrial level, questions arising from the motion of falling bodies and of projectiles — whether stones or arrows — attracted the interest of some of the greatest thinkers of antiquity.

The system of mechanics devised by Newton in the seventeenth century made it possible to explain, for the first time, motion of both celestial and terrestrial objects with the one set of postulates, or laws. Newtonian mechanics proved to be one of the most successful mathematical models ever devised and it showed conclusively the value of mathematics in understanding nature. For these reasons, its advent is generally regarded as one of the turning points in the history of human thought.

This chapter introduces Newton's laws and sets them against the background of the set of postulates for mechanics due to the ancient Greek philosopher Aristotle. It also refers to the results discovered by Galileo and Kepler, which showed the inadequacy of the model proposed by Aristotle and thereby set the stage for Newton.

The outstanding success of Newtonian mechanics should not be allowed to blind us to the fact that it is only a model of what really happens. In the present century it has been found inadequate to explain motion at the subatomic level and at speeds close to that of light, which occur in astronomy.

The quantities involved in mechanics — such as displacement, velocity and acceleration — are typically related to time by smooth functions defined on an entire interval. Problems in mechanics lead, via Newton's second law, to differential equations. By way of contrast, the mathematical models to be studied in this part of the course involve quantities whose values are known only at certain specified times, equally spaced. Such quantities are expressed as functions of the time via sequences. The assumptions in the models can then be expressed as difference equations for these sequences. The difference between models leading to differential equations and those leading to difference equations is often expressed by saying that the former are continuous whereas the latter are discrete.

This chapter introduces the idea of a difference equation via a problem involving rabbit populations. Basic ideas regarding the solutions of these equations are then explained.

Introductory example

Leonardo of Pisa, or Fibonacci as he was better known, is often claimed to be the greatest mathematician of the Middle Ages. His book, Liber abaci, completed in 1202, took advantage of the Hindu–Arabic numerals. Among the problems which the book contains, the one of greatest interest to later mathematicians is as follows:

Linear difference equations have the advantage that a closed-form solution can be easily obtained. But, in many cases, the behaviour of linear difference equation models is not consistent with observation. This is true in many areas of biology, and particularly in studies of populations, where non-linear models are better.

In this chapter non-linear models are developed which describe how a population of individuals grows over time. Difference equation models are appropriate when a species has a distinct breeding season. The simplest non-linear model, the ‘logistic equation’ is studied in detail. Similar ideas are also used to model a measles epidemic. This involves iterating a pair of simultaneous difference equations for the number of those who can infect others and for the number of those susceptible to being infected.

Closed-form solutions usually cannot be found for non-linear difference equations. Thus to interpret the models one has to resort to numerical simulation or devise approximate closed-form solutions to the equations. Both approaches are developed here. The ideas rely on concepts developed in Chapters 7 and 8.

Linear models for population growth

Many people are very interested in the way populations grow and in determining what factors influence their growth. Knowledge of this kind is important in studies of bacterial growth, wildlife management, ecology and harvesting. In this section a very simple model is formulated for a population which breeds at fixed time intervals.

In this chapter some problems of growth and decay will be studied for which differential equations, rather than difference equations, are the appropriate mathematical models. Such problems include:

• the growth of large populations in which breeding is not restricted to specific seasons,

• the absorption of drugs into the body tissues,

• the decay of radioactive substances.

The differential equations which arise from the above problems are all of the first order. The two methods of solution which we explain are sufficient to solve all the differential equations which arise in the next three chapters. The theoretical background for these two methods is contained in Chapter 5. The first of the two methods, which applies only to linear differential equations, is very similar to the method already given in Section 8.1 for solving linear difference equations. The continuous models used in this chapter are similar to the discrete models discussed in Chapter 9.

First-order differential equations

The two types of differential equations which you need to be able to solve in this chapter are called linear with constant coefficients and variables separable differential equations. The former arise from problems of unrestricted growth, while the latter appear when the growth is restricted. How to recognize and solve the two types of differential equations will now be explained.