What Is a Mathematical Model?

Module 1: Low-Dimensional Dynamics

Reading

Recommended: Strogatz (2024), Chapter 1.
Course material: Syllabus and How the Course Works.

A mathematical model is a deliberate simplification of a real system. You keep a few variables and a few rules that you believe control the behaviour you care about, and you discard the rest. The model then lets you reason about the system with the tools of mathematics and computation.

Variables, parameters and rules

Most models in this course have three parts.

State variables describe the system at a given time: a population $x(t)$, the fractions $S, I, R$ of an epidemic, two chemical concentrations.
Parameters are numbers that stay fixed during one experiment: a growth rate, a carrying capacity, a transmission rate.
Rules say how the state changes. In Part I of the course the rules are ordinary differential equations $\dot{\mathbf x} = \mathbf f(\mathbf x; \theta)$ (Strogatz 2024).

The same structure reappears later with random rules (Module 3), spatial rules (Module 4), networks (Module 5) and discrete rules (Module 6).

All models are wrong

George Box wrote that “since all models are wrong the scientist cannot obtain a ‘correct’ one by excessive elaboration” (box1976science?). A few years later he titled a section “All models are wrong but some are useful” (box1979robustness?). The point is practical. You cannot make a model true by adding detail, so you should ask whether it is useful for a stated purpose.

Two consequences follow.

A model that fits the data well can still be useless, for example when it has so many parameters that it would fit any data.
A model that gets the numbers wrong can still be useful, for example when it correctly predicts that a system has two stable states.

Parsimony

Box also argued for parsimonious models: use the fewest parameters that still capture the effect you study (box1976science?). Each extra parameter must be estimated, and each one makes it harder to see which mechanism produces the behaviour.

The spruce budworm model of Session 2 is a good example. Ludwig, Jones and Holling reduced a detailed forest simulation to three differential equations, and a single equation for the fast variable already explains the outbreak pattern (ludwig1978qualitative?). They state that their conclusions depend on the qualitative shape of the terms, not on their exact form (ludwig1978qualitative?).

When is a model good enough?

There is no general rule, but you can ask the same questions every time.

Purpose: what question should the model answer? Explanation, prediction and control need different levels of detail.
Assumptions: can you list them? Which one is the weakest?
Sensitivity: does the conclusion survive small changes in the parameters and in the form of the terms?
Data: does the model reproduce the observations it was not fitted to?
Limits: does it give the right answer in simple limiting cases you already understand?

Exercise

Take the logistic model $\dot N = rN(1 - N/K)$ for a population. List three assumptions it makes about the real population. For each one, name a situation where it fails.

Solution

Possible answers:

The population is large enough to treat $N$ as a continuous variable. It fails for small or endangered populations, where random births and deaths matter.
The per capita growth rate decreases linearly with $N$. It fails when individuals need a minimum density to reproduce (Allee effect).
$r$ and $K$ are constant. They fail when food supply or climate changes on the same time scale as the population.

The next page turns these questions into a workflow you will use in every session.

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "What Is a Mathematical Model?" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- ::: {.callout-note} ## Reading - **Recommended:** @strogatz2024nonlinear, Chapter 1. - **Course material:** [Syllabus](../../syllabus.qmd) and [How the Course Works](how-the-course-works.qmd). ::: A mathematical model is a deliberate simplification of a real system. You keep a few variables and a few rules that you believe control the behaviour you care about, and you discard the rest. The model then lets you reason about the system with the tools of mathematics and computation. ## Variables, parameters and rules Most models in this course have three parts. - **State variables** describe the system at a given time: a population $x(t)$, the fractions $S, I, R$ of an epidemic, two chemical concentrations. - **Parameters** are numbers that stay fixed during one experiment: a growth rate, a carrying capacity, a transmission rate. - **Rules** say how the state changes. In Part I of the course the rules are ordinary differential equations $\dot{\mathbf x} = \mathbf f(\mathbf x; \theta)$ [@strogatz2024nonlinear]. The same structure reappears later with random rules (Module 3), spatial rules (Module 4), networks (Module 5) and discrete rules (Module 6). ## All models are wrong George Box wrote that "since all models are wrong the scientist cannot obtain a 'correct' one by excessive elaboration" [@box1976science, p. 792]. A few years later he titled a section "All models are wrong but some are useful" [@box1979robustness, p. 202]. The point is practical. You cannot make a model true by adding detail, so you should ask whether it is useful for a stated purpose. Two consequences follow. - A model that fits the data well can still be useless, for example when it has so many parameters that it would fit any data. - A model that gets the numbers wrong can still be useful, for example when it correctly predicts that a system has two stable states. ## Parsimony Box also argued for parsimonious models: use the fewest parameters that still capture the effect you study [@box1976science]. Each extra parameter must be estimated, and each one makes it harder to see which mechanism produces the behaviour. The spruce budworm model of Session 2 is a good example. Ludwig, Jones and Holling reduced a detailed forest simulation to three differential equations, and a single equation for the fast variable already explains the outbreak pattern [@ludwig1978qualitative]. They state that their conclusions depend on the qualitative shape of the terms, not on their exact form [@ludwig1978qualitative, p. 317]. ## When is a model good enough? There is no general rule, but you can ask the same questions every time. - **Purpose:** what question should the model answer? Explanation, prediction and control need different levels of detail. - **Assumptions:** can you list them? Which one is the weakest? - **Sensitivity:** does the conclusion survive small changes in the parameters and in the form of the terms? - **Data:** does the model reproduce the observations it was not fitted to? - **Limits:** does it give the right answer in simple limiting cases you already understand? ::: {.callout-note} ## Exercise Take the logistic model $\dot N = rN(1 - N/K)$ for a population. List three assumptions it makes about the real population. For each one, name a situation where it fails. ::: ::: {.callout-tip collapse="true"} ## Solution Possible answers: - The population is large enough to treat $N$ as a continuous variable. It fails for small or endangered populations, where random births and deaths matter. - The per capita growth rate decreases linearly with $N$. It fails when individuals need a minimum density to reproduce (Allee effect). - $r$ and $K$ are constant. They fail when food supply or climate changes on the same time scale as the population. ::: ## Next The next page turns these questions into a workflow you will use in every session.