The Malthusian population model is essentially one of exponential growth punctuated by periodic catastrophic collapse. It is a reasonable model over the short term, but does not usually do a good job of modeling long term population growth.

A somewhat more realistic model is the logistic growth model. In this model, growth is initially exponential, but as resources become scarce, the growth rate steadily decreases as the total population approaches the maximum sustainable value, called the carrying capacity of the environment in which the population lives; the behavior is more logarithmic as the population approaches the carrying capacity. The equation for the logistic model is $P\left(t\right)\=\frac{P_{0 }K }{P_0\+\left(K-P_0\right){\ⅇ}^{-r\cdot t}}$, where $K$, $P_0$, $r$ are carrying capacity, initial population, and growth rate, respectively, and $t$ represents time.

Use the sliders below to choose the carrying capacity, $K$, the initial population, $P_0$, and the rate of growth, $r$, and observe how the population growth model varies over ten time periods.