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:

                    MINUS

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

• $r$ is called the natural growth rate
• $K$ is called the carrying capacity of the environment, and represents the maximum sustainable population size.

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: