IB Maths AI HL AI 5.3 practice question

Question 1

Paper 2Medium7 marks

In a reserve, the prey and predator populations are represented by indices $x$ and $y$ measured in hundreds of animals. The time $t$ is measured in years after monitoring begins. The populations are modelled by

\frac{\mathrm{d}x}{\mathrm{d}t}=0.8x-0.04xy, \qquad \frac{\mathrm{d}y}{\mathrm{d}t}=0.02xy-0.6y,

with $x(0)=20$ and $y(0)=5$ A researcher uses Euler’s method with a step size of $0.5$ years.

(a)

Show that the Euler estimate for $x(0.5)$ is $26$ and find the corresponding estimate for $y(0.5)$

[3]

(b)

Hence, use one further Euler step to estimate $x(1)$ and $y(1)$

[2]

(c)

A researcher claims that the predator population is decreasing at $t=1$ since the estimate for $y(1)$ is lower than that for $y(0.5)$ Use the model to assess this claim.

[2]

More AI 5.3 Differential equations questions