# Math Insight

### Applet: Logistic and exponential growth

Illustration of how logistic and exponential growth agree for small population sizes and diverge as the population gets larger. The blue circles show the solution to the dynamical system \begin{align*} P_{t+1} & = P_t + r P_t \left( 1 - \frac{P_t}{M} \right), \end{align*} and the red X's show the solution to the dynamical system \begin{align*} W_{t+1} & = W_t + r W_t. \end{align*} Both dynamical systems use the same initial condition $W_0=P_0$, which you can change by typing in a value in the box or dragging the purple diamond with your mouse. The carrying capacity $M$ is illustrated by the horizontal line. You can change $M$ and the low density growth rate $r$ by typing in values in the corresponding boxes. You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: logistic_and_exponential_growth.ggb