Here we will explore a model that describes the change of a population when the carrying capacity itself is modeled with a logistic function of time. We can create this if we know the initial and resulting capacities (K1 and K2) along with a parameter a used to tune the inflection point of the curve.
In particular, demonstrate this model on the following situation. In a population study of England from 1541 to 1975, starting with a population of about 1 million, early islanders appear to have a carrying capacity of around 5 million people. However, beginning around 1800 with the advent of the Industrial Revolution, the carrying capacity appears to have increased to about 50 million people. The change in the concavity from concave up to concave down for this new logistic appears to occur in about 1850 (Meyer and Ausubel 1999) (Shiflet and Shifflet, 2009).
Create a model of this situation using InsightMaker. Use sliders to allow for exploration of different values of a, and plot the Population and Carrying Capacity as a function of time.