If we let $P(t)$ represent the population of a species at time $t$, then a population model for $P(t)$ can be created by using the general balance equation:

Rate at which population $P(t)$ changes = Rate at which individuals are added (birth, immigration)
                    MINUS
Rate at which individuals are removed (death, emigration)

Using this balance law, we can develop the Logistic Model for population growth. For this model, we assume that we add population at a rate proportional to how many are already there. (The more adults, the more babies.) We also assume that individuals die at a rate proportional to the square of the population (say, because of competition for food). That is $$\frac {dP}{dt} = rP-bP^2, \qquad P(0)=P_0,$$ where $r$ and $b$ are constants describing the fertility of the species and the competition for food. After simplifying, this becomes $$P'=rP(1-(b/r)P)=rP\left (1-\frac{P}{K}\right ),$$ where $K=\frac{r}{b}$. The equation $P'=rP\left (1-\frac{P}{K}\right )$ is called the logistic equation for single species population growth, where

We can easily modify our Matlab m-file Euler to provide a numerical solution for the logistic IVP $$P'=rP\left (1-\frac{P}{K}\right ), \qquad P(0)=P_0.$$ The Matlab function Logistics (available on the MATLAB page) users Euler's Method to solve the Logistic IVP. It produces the following plot: