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.

  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.

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

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

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

\[ln(P)- ln(P_0-P)=P_0(kt+C)\]







  is an arbitrary constant.
\[t \rightarrow m\infty , \: P \rightarrow P_0\]

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