Building a mathematical model
A mathematical model begins with a biological question. We decide what is important, make simplifying assumptions, represent the biological process mathematically, and then study what the model predicts.
We will use one simple example throughout this page: the growth of a bacterial population.
1. Start with a biological question
This question tells us what we want the model to describe: population size through time.
2. Decide what to include
A real bacterial population is affected by nutrients, temperature, space, competition and many other factors. A first model does not need to represent all of them.
For our simple model, we include only the number of bacteria and their reproduction. We temporarily ignore other biological details.
3. State the assumptions
Assumptions make clear how the real biological system is being simplified. For this first model we assume:
- the environment remains suitable for growth;
- there is enough food and space during the period considered;
- bacteria are not removed from the population;
- each bacterium has the same average reproductive behaviour;
- the per-capita growth rate remains constant.
These assumptions are not claims that real bacterial populations always behave this way. They define the conditions represented by this particular model.
4. Define the variable and parameter
Let
\[ N(t)=\text{number of bacteria at time }t. \]Suppose initially there are 100 bacteria:
\[ N(0)=100. \]Let \(r\) be the per-capita growth-rate parameter. For example, we might use
\[ r=0.2\ \text{hour}^{-1}. \]The parameter \(r\) describes the average growth contribution per bacterium per unit time.
5. Translate the biological idea into mathematics
If there are more bacteria, there are more bacteria available to reproduce. Under our assumptions, it is therefore reasonable to suppose that the population's rate of change is proportional to its current size:
\[ \text{rate of population change}=r\times\text{current population}. \]Mathematically,
\[ \frac{dN}{dt}=rN. \]For example, when \(N=100\) and \(r=0.2\ \text{hour}^{-1}\),
\[ \frac{dN}{dt}=0.2\times100=20\ \text{bacteria per hour}. \]This does not mean that exactly 20 bacteria must appear during every hour. The differential equation describes a continuous deterministic approximation to the population's average growth behaviour.
6. Solve the model
The equation
\[ \frac{dN}{dt}=rN \]has solution
\[ N(t)=N(0)e^{rt}. \]Using \(N(0)=100\) and \(r=0.2\),
\[ N(t)=100e^{0.2t}. \]After 5 hours the model gives
\[ N(5)=100e^{1}\approx272. \]So, under the assumptions of the model, the predicted population after five hours is approximately 272 bacteria.
7. Interpret the result biologically
The mathematics predicts increasingly rapid growth because the model assumes that the same per-capita growth rate continues as the population becomes larger.
This is useful over a period when resources are effectively plentiful. Over a longer period, however, food or space may become limited. The assumptions would then become unrealistic and the exponential model would tend to overestimate growth.
That observation suggests a better model might include resource limitation. For example, logistic growth introduces a carrying-capacity parameter \(K\). Model building therefore develops naturally as biological understanding and evidence reveal which assumptions need improvement.
The complete modelling process
Assumptions: suitable environment, plentiful resources, no removal, constant per-capita growth rate.
Variable: \(N(t)\), population size.
Parameter: \(r\), per-capita growth rate.
Model: \(dN/dt=rN\).
Prediction: \(N(t)=N(0)e^{rt}\).
Evaluation: compare the prediction with biological observations and reconsider assumptions when necessary.