01 · Lesson 01
Functions and graphs
A function is a rule connecting an input to one definite output. In mathematical biology, functions let us describe how a biological quantity depends on time, population size, dose, temperature or another measurable quantity.
Biological scenario
Suppose a bacterial population begins with 100 cells and doubles every hour. We want a rule that gives the population at any time, not a separate calculation for every hour.
Let \(t\) be time in hours and \(N(t)\) be the number of bacteria. The model is
The notation \(N(t)\) is read “\(N\) of \(t\).” It means that the population value depends on the selected time.
Input, rule and output
| Part | In the example | Meaning |
|---|---|---|
| Input | \(t\) | The time supplied to the function |
| Rule | \(100\,2^t\) | The mathematical operation applied |
| Output | \(N(t)\) | The predicted bacterial population |
For example,
A function must give only one output for each permitted input. Different inputs may produce the same output, but one input cannot be assigned two conflicting outputs by the same function.
Independent and dependent variables
The input is often called the independent variable. The output is the dependent variable because its value depends on the input.
In \(N(t)\), time \(t\) is the independent variable and population \(N\) is the dependent variable. This terminology describes the mathematical relationship; it does not prove that time biologically causes every observed change.
Domain and range
The domain is the set of allowed inputs. The range is the set of outputs produced.
Algebraically, \(100\,2^t\) is defined for every real \(t\). Biologically, if the experiment begins at \(t=0\) and ends at 10 hours, a suitable domain is \(0\le t\le10\). The biological context therefore restricts the mathematical domain.
The model predicts positive values, so its range over this experiment is \(100\le N(t)\le102400\). Whether such rapid growth remains realistic is a separate modelling question.
Tables and graphs show the same function
| Time \(t\) | Population \(N(t)\) |
|---|---|
| 0 | 100 |
| 1 | 200 |
| 2 | 400 |
| 3 | 800 |
| 4 | 1600 |
On a graph, inputs are normally placed on the horizontal axis and outputs on the vertical axis. Each point \((t,N(t))\) records one input–output pair.
A rising graph means the modelled quantity increases as the input increases. Its steepness indicates how rapidly it changes. The graph communicates the relationship visually; it does not add assumptions that are absent from the function.
Common functions in mathematical biology
| Function type | Example | Possible biological use |
|---|---|---|
| Linear | \(y=a+bx\) | Approximately constant change over a limited range |
| Exponential | \(N(t)=N_0e^{rt}\) | Unrestricted population growth or decay |
| Logistic | \(N(t)=K/(1+Ae^{-rt})\) | Growth limited by carrying capacity |
| Saturating | \(v(S)=V_{\max}S/(K_M+S)\) | Enzyme reaction rate versus substrate concentration |
| Periodic | \(y(t)=A\sin(\omega t)+c\) | Seasonal or circadian variation |
Parameters change the relationship
In \(N(t)=N_0e^{rt}\), \(t\) is the variable, while \(N_0\) and \(r\) are parameters. The initial value \(N_0\) moves the starting height. The growth-rate parameter \(r\) changes how quickly the curve rises or falls.
- \(r>0\): exponential growth;
- \(r=0\): constant population;
- \(r<0\): exponential decline.
A parameter comparison should change one declared assumption while keeping the others fixed, unless an interaction is deliberately being studied.
Conditions and limitations
- State the input, output and their units.
- Use a biologically meaningful domain.
- Do not interpret a curve outside the range where the model assumptions are reasonable.
- Distinguish a model prediction from an observed data point.
- Remember that a good fit does not by itself establish a biological mechanism.
What this lesson establishes
You can now identify a function’s input, output, rule, domain and range; read function notation; interpret tables and graphs; and explain how variables and parameters play different roles in a biological model.