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

`(\mathrm dP)/(\mathrm dt)=P(4-P)\implies(\mathrm dP)/(P(4-P))=\mathrm dt`

Split up the left side into partial fractions:

`\frac1{P(4-P)}=\frac14\left(\frac1P+\frac1{4-P}\right)`

Then integrating both sides gives

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

`\frac14(\ln|P|-\ln|P-4|)=t+C`

`\frac14\ln\left|\frac P{P-4}\right|=t+C`

`\frac P{P-4}=Ce^(4t)`

`P=(4e^(4t))/(e^(4t)-C)`

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

`1=\frac4{1-C}\implies C=-3`

so that the population is given by the model

`\boxed{P(t)=(4e^(4t))/(e^(4t)+3)}`

or

`\boxed{P(t)=\frac4{1+3e^(-4t)}}`