Building a mathematical model

A mathematical model begins with a biological question. The mathematics should be chosen to answer that question, rather than added before the biological problem is clear.

Question → System boundary → Mathematical representation → Analysis → Interpretation → Comparison with evidence

1. Define the question

State precisely what the model should investigate. For example: how will a population change through time under specified conditions?

2. Decide what the model contains

Choose the biological components that are relevant and define the boundary of the system. Details that do not materially affect the question may be omitted.

3. Choose mathematical quantities

Represent the important biological quantities mathematically. For a population, \(N(t)\) may represent population size. Initial information must also be specified, for example

\[N(0)=N_0.\]

4. Represent the biological mechanism

Translate the proposed mechanism into mathematical relationships. If population change is assumed to be proportional to population size, a possible representation is

\[\frac{dN}{dt}=rN.\]

The equation is not the biological system itself; it is a mathematical representation of a stated mechanism.

5. Analyse or simulate the model

The model can then be studied analytically or computationally to determine its consequences, such as growth, decline, equilibria or other patterns.

6. Interpret and evaluate

Results must be translated back into biological language. Where observations or data are available, model behaviour can be compared with them. Poor agreement may indicate that parameter values, mechanisms or the model structure need reconsideration.

Key principle. Model building is an iterative process. Biological reasoning determines the model, mathematics determines its consequences, and evidence helps determine whether the representation is useful for its intended purpose.