Chapter 1 An Introduction to Modelling and Differential Equations

“The profound study of nature is the most fertile source of mathematical discoveries.” - Joseph Fourier (1768-1830)

Why does a hot cup of coffee cool quickly at first, then more and more slowly? Why do some bridges shake themselves apart in the wind? Why do two competing species sometimes coexist, and sometimes drive each other to extinction?

Every one of these questions has the same secret hiding inside it: in each case, we rarely know the quantity we care about directly — what we know is how fast it is changing. That single idea, that a rate of change is often easier to state than the quantity itself, is the seed from which the entire subject of differential equations grows, and it’s what this course is really about.

Across this course we’ll follow the same rhythm again and again, applied to progressively richer situations:

• Remark (The modelling cycle).
• Motivate — start from a real question about a real system.
• Formulate — translate the situation into a differential equation, stating your assumptions honestly.
• Solve — bring the right mathematical technique to bear.
• Interpret — bring the solution back to the real world, and ask whether it’s plausible.

This chapter lays the conceptual groundwork for that cycle: what do we actually mean by a model, and what makes differential equations the natural language for describing change?

1.1 Modelling

Mathematical Modelling is a general procedure in the Mathematical Sciences wherein ideas about observed processes are presented and interrogated using mathematical principles and techniques. There may exist many different conceptual formulations of observed processes and so there may be many equivalent models for those processes. In the sections that follow, we shall discuss some of the conceptual aspects of building models that describe physical phenomena.

The Modelling Process

To gain an understanding of the processes involved in mathematical modelling, consider two worlds which can be depicted with the properties below:

• Real-world system
  • Observed behaviour
• Mathematical World
  • Models
  • Mathematical operations and rules
  • Mathematical conclusions

Suppose we want to understand some behaviour or phenomenon in the real world. We may wish to make predictions about that behaviour in the future and analyze the effects that various situations have on it.

For example, when studying the populations of two interacting species, we may wish to know if the species can coexist within their environment or if one species will eventually dominate and drive the other to extinction. In the case of the administration of a drug to a person, it is important to know the correct dosage and the time between doses to maintain a safe and effective level of the drug in the bloodstream.

In order to construct and use models in the mathematical world to help us better understand real-world systems, it is important to gain an understanding of how we link the two worlds together. To begin, we would first have to define what we mean by a real-world system.

Definition 1.1 (System) A system is an assemblage of objects with some interaction or interdependence.

Definition 1.2 (Model) A model is a conceptual representation of a given object or process that captures some collection of aspects of that object or process.

The modeller is interested in understanding how a particular system works, which features cause changes in the system, and how sensitive the system is to certain changes. They are also interested in predicting what changes might occur and when they occur.

Remark. Physical systems can be very complicated, with subtle intricacies that are either not well understood, or difficult to describe mathematically. Therefore, the impetus of mathematical modelling is to produce an approximation that is sufficient to describe the important parts of the system. This means that the domain of applicability of a model can be very narrow compared to the conceptually perfect mathematical representation of such a system.

Suppose we want to draw conclusions about some observed phenomenon. The most direct approach is to run experiments and observe the effects for ourselves. But this isn’t always possible, for a few reasons:

• Cost. Experiments can be prohibitively expensive, financially or socially — we can’t ethically determine the fatal concentration of a drug by testing it on people.
• Feasibility. Some experiments simply can’t be run — we can’t test how a specific change in the ionosphere’s composition affects the polar ice caps.
• Generalisability. Even a successful experiment only tells us about the exact conditions we tested. A trial run in Johannesburg at \(32^{\circ}C\) and 42% humidity may not explain why the behaviour occurred, or what happens under different conditions.

These limitations motivate an indirect approach: building a mathematical model instead. We can’t account for every factor influencing a real system, so we make simplifying assumptions — deciding, for instance, that humidity doesn’t matter much to modelling the aftermath of a power plant failure, and leaving it out. From the factors we keep, we propose tentative relationships between them, and use those relationships to construct a model. Finally, we apply mathematical analysis to the model.

One important caution: any conclusions this process produces are conclusions about the model, not directly about reality. Since the model rests on simplifying assumptions and imperfect observations, its predictions are only ever as good as those assumptions. Roughly, the procedure looks like this:

1. Through observation, identify the primary factors involved in the real-world behaviour.
2. Conjecture tentative relationships among the factors.
3. Apply mathematical analysis to the resultant model.
4. Interpret mathematical conclusions in terms of the real-world problem.

We shall consider several scenarios in detail using this approach in Chapters 2, ??, and ??.

Mathematical Models

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?” - Albert Einstein (1879-1955)

Let us begin with a useful definition.

Definition 1.3 (Mathematical Model) A mathematical model is a mathematical construct designed to study a particular real-world system or phenomenon.

There are two approaches we can take when modelling a phenomenon mathematically. We can either take pre-existing models of well studied phenomenon and build on these models, or we can construct a completely new model for our special phenomenon.

Sometimes a constructed model becomes so complex that it’s effectively impossible to analyse or solve — think of trying to capture the global effects of population growth, resource use, and pollution in a single system of equations. When a model becomes this intractable, it loses its usefulness. In cases like these, we may fall back on replicating the phenomenon directly: gathering data through experiments and analysing it statistically, for example via curve-fitting.

It is important to note that there is a trade-off among factors in selecting a specific type of model. Three of these factors are listed below.

• Fidelity: The preciseness of a model’s representation of reality.
• Cost: The total cost of the modelling process.
• Flexibility: The ability to change and control conditions affecting the model as required data are gathered.

The table below summarises how the five approaches we’ve discussed — real-world observation, experiments, simulations, constructed models, and pre-existing models — typically compare on these three factors.

Approach	Fidelity	Cost	Flexibility
Real-world observation	Highest — a direct record of reality	Highest — data collection is expensive	Lowest — we can’t change the environment we’re observing
Experiments	High — a scaled-down version of the real phenomenon	High — equipment and setup are costly	Limited by available technology
Simulations	Moderate — may miss intricacies of the real system	High — equipment and setup are costly	Limited by available technology
Constructed models	Tailored to the problem — often outperforms generic models	Moderate — cost of tailoring to the specific problem	Highest — easy to vary and adjust
Pre-existing models	Lower — generic, not tailored	Lowest — readily available	Moderate — can be used as a base and built upon

The general pattern: methods closer to direct observation of reality give the highest fidelity, but cost the most and are the least flexible. Constructed models trade away some fidelity in exchange for being cheap to build and easy to adapt.

In the beginning of Chapter 2, we will use these elementary discussions around the modelling process and the concept of a mathematical model to discuss the technical aspects of model building by unpacking the steps to follow in order to construct a mathematical model. Before we can do that though, we need a deeper understanding on the ways in which we can mathematically model change.

1.2 Differential Equations

“It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations.” - Felix Klein (1849-1925)

Before we do anything, it is important to understand the key differences between an algebraic equation and a differential equation.

Now that we know this key difference, the question remains: what actually is a differential equation?

Why study differential equations? Because they arise naturally in models across science, engineering, and economics — and they arise naturally because physical, biological, and economic systems are all marked by the same thing: change. Differential equations model these systems by describing how they change. So we’re interested in describing a system’s dynamics (compare to statics).

What do I mean by this?

Exercise:

- Complete the task at the end of the above video.
- Summarize the concepts and contents of this Chapter.

Summary

At the end of this Chapter, you should:

• understand the difference between a system and a model;
• be able to explain the concept of a mathematical model;
• know the difference between an algebraic and differential equation;
• understand what a differential equation is, and why it is useful; and
• be able to conceptualize differential equations as dynamic models.

With these foundations in place, the rest of the course tells one continuous story in three acts. Chapter 2 builds first-order equations and their classic applications — growth, decay, and cooling. Chapter ?? moves to second-order equations and the oscillatory systems they describe — springs, circuits, damping, and resonance. Chapter ?? brings these ideas together into systems of interacting equations — predator-prey dynamics, epidemics, and coupled systems. Every step follows the same modelling cycle we introduced at the start of this chapter: motivate, formulate, solve, interpret.