Question

A population of size P (in millions) varies in time t (in years) according to a logistic equation of the form

dP/dt = P(4 − P) If the initial population size is P(0) = 1 find the population for all time

Answer 1

The ODE is separable, with

$\dfrac{\mathrm dP}{\mathrm dt}=P(4-P)\implies\dfrac{\mathrm dP}{P(4-P)}=\mathrm dt$

Split up the left side into partial fractions:

$\dfrac1{P(4-P)}=\dfrac14\left(\dfrac1P+\dfrac1{4-P}\right)$

Then integrating both sides gives

$\displaystyle\frac14\int\left(\frac1P-\frac1{P-4}\right)\,\mathrm dP=\int\mathrm dt$

$\dfrac14(\ln|P|-\ln|P-4|)=t+C$

$\dfrac14\ln\left|\dfrac P{P-4}\right|=t+C$

$\dfrac P{P-4}=Ce^{4t}$

$P=\dfrac{4e^{4t}}{e^{4t}-C}$

Given that $P(0)=1$, we find

$1=\dfrac4{1-C}\implies C=-3$

so that the population is given by the model

$\boxed{P(t)=\dfrac{4e^{4t}}{e^{4t}+3}}$

or

$\boxed{P(t)=\dfrac4{1+3e^{-4t}}}$