Solving the Logistic Equation

As we saw in class, one possible model for the growth of a population is the logistic equation:

$$\frac{dP}{dt} = r P \left( 1 - \frac{P}{K} \right), \quad P(0) = P_0.$$

Here the number $P_0$ is the initial density of the population, $r$ is the intrinsic growth rate of the population (for given, finite initial resources available) and $K$ is the carrying capacity, or maximum potential population density. This equation is an Ordinary Differential Equation (ODE) because it is an equation which involves ordinary derivatives.