Question

7. (EC) A population P(t) of animals has a birth rate of B .1 and an unknown constant death rate δ. If there are 60 of these animals initially and we know that three are born in the first month (P(O) -3), how long (how many months) will it take for this population to double?

Answer 1

$\frac{dP(t)}{dt}=(\beta-\delta)P(t)=(0.1-\delta)P(t)$

Integrating, we get $P(t)=C\exp\left ( (0.1-\delta)t \right )$ is the general equation for the population

And $P(0)=60\Rightarrow C=60$ as well as

$3=P'(0)=(0.1-\delta)P(0)=60(0.1-\delta )$

Meaning $3=6-60\delta\Rightarrow 60\delta=3\Rightarrow \delta=\frac{3}{60}=$$0.05$ is the death rate

And so the required general solution is $P(t)=C\exp((0.1-\delta)t)$ equals

$P(t)=60\exp((0.1-0.05)t)$

That is, $P(t)=60\exp(0.05t)$

For doubling, we must have $P(t_{d})=60\exp(0.05t_d)=120$ so $\exp(0.05t_d)=2\Rightarrow 0.05t_d=\ln(2)\Rightarrow t_d=\frac{\ln(2)}{0.05}$

Therefore, $t_d\approx 13.863$ months is the required time for the population to double