## Modelling Population - The Logistic Equation

The logistic equation is used to model populations. It is suppose that the rate of increase of a the population of a species is proportional to the population and to difference between some maximum stable population and the population.
$\frac{dP}{dt}=kP(P_0-P)$

where
$\frac{dP}{dt}$
is the rte of population increase
$k \ge 0$
is constant of proportionality
$P \ge 0$
is the population
$P_0 \ge 0$
is the maximum stable population
We can solve this equation by separation of variables.
$\frac{dP}{(P(P_0-P)}=dt$

Write nbsp;
$\frac{dP}{(P(P_0-P)}$
as partial fractions
$\frac{dP}{(P(P_0-P)}=\frac{1/P_0}{P}+ \frac{1/P_0}{P_0-P}$

Now integrate.
$\int \frac{1/P_0}{P}+\frac{1/P_0}{P_0-P} dP = \int dt$

$\frac{1}{P_0}ln(P)- \frac{1}{P_0} ln(P_0-P)=kt+C$

$ln(P)- ln(P_0-P)=P_0(kt+C)$

$ln(\frac{P}{P_0-P})=P_0(kt+C)$

$\frac{P}{P_0-P}=e^{P_0(kt+C)}=Ae^{kt}$

$P=(P_0-P)Ae^{kt}=P_0Ae^{kt}-PAe^{kt}$

$P+PAe^{kt}=P_0Ae^{kt}$

$P(1+Ae^{kt})=P_0Ae^{kt}$

$P=\frac{P_0Ae^{kt}}{1+Ae^{kt}}=\frac{AP_0}{e^{-kt}+A}$

$A$
is an arbitrary constant.
As
$t \rightarrow m\infty , \: P \rightarrow P_0$
.