Use Euler's method by updating both populations from the same current state at each step. Calculate both rates first, then use those rates to obtain the two new population values.
This is AHL 5.16, an HL-only topic. It can appear on any GDC-active Math AI HL paper, especially in extended modelling questions.
For a coupled system, where and represent interacting quantities, and step size , Euler's recurrences are:
The key mechanism is local linear approximation. During one short interval, Euler's method treats both rates as constant at their values at . Because the equations are linked, each rate depends on the same pair of current populations.
Suppose a predator-prey model is given below, where is the prey population and is the predator population. Let the initial state be , , with .
First calculate both rates at the initial state:
Then update simultaneously using those two rates:
| Iteration | Current state | Next state |
|---|---|---|
For the second step, evaluate both rates at , not with one partly updated value. This gives and before producing the next state shown in the table.
The results have contextual meaning: during the first interval, prey decrease from 40 to 38.4 while predators increase from 9 to 9.9. Euler's method estimates this change; it does not give an exact solution, and a smaller step size generally improves accuracy.
The common misconception is a sequential update: calculating and immediately using it to calculate . This is incorrect. Standard Euler updating is simultaneous, so both new values must use , , and . Otherwise, you are applying a different numerical method and will obtain different values.
Exam technique: Show the first iteration explicitly, including both derivative evaluations and recurrence substitutions, before using a spreadsheet or GDC for later iterations. Keep full precision and round only the final answer. If asked to interpret, describe both populations over the relevant time interval.
