When studying a colony of bugs, a scientist found that the birth rate of the bugs is inversely propo

Question

When studying a colony of bugs, a scientist found that the birth rate of the bugs is inversely proportional to its population and the death rate is proportional to its population. The population of the bugs (in thousands) at time $t$ days after they were first observed is denoted by $P$. It was found that when the population is $2000$, it remains constant. Assuming that $P$ and $t$ are continuous variables, show that $\frac{\mathrm{d}P}{\mathrm{d}t}=k\left( \frac{4}{P}-P \right)$, where $k$ is a constant.[3] Given that the initial population of the bugs was $4000$, and that the population was decreasing at the rate of $3000$ per day at that instant, find $P$ in terms of $t$.[4] Sketch the graph of $P$ against $t$, giving the equation of any asymptote(s). State what happens to the population of the bugs in the long run.[2] Another population of bugs, $N$(in thousands) in time $t$ days can be modelled by the differential equation $\frac{\mathrm{d}N}{\mathrm{d}t}=4+\frac{N}{t}$ for $t\ge 1$. Using the substitution $u=\frac{N}{t}$, solve this equation, given that the population was $1000$ when $t=1$.[3]

Accepted Answer

(a)(i) $\dfrac{\mathrm dP}{\mathrm dt}=k\left(\dfrac4P-P\right)$, $k>0$ (ii) $P=2\sqrt{1+3e^{-2t}}$ (iii) Horizontal asymptote $P=2$, so the population decreases towards $2000$ (b) $N=4t\ln t+t$

2019 HCI P2 Q4

2019 HCI P2 Q4

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