This applet allows to understnd which limits must be imposed on the values of the parameter of the logistic map in order that its values are comprised between 0 and 1, , whenever , i.e. that the function maps the closed interval into itself: this is necessary to interpret again as an environmental occupancy percentage when is.

In the Graphics view change values to in order to see that its values can not exceed some value: which one? Of course, otherwise could become negative.
Verify, by hand computation, that the maximum value of for is , i.e. that for any when . Confirm this result using the CAS view of GeoGebra.
Find, by hand computation, which is the maximal range of possible values of that allows values of to belong to the closed interval when . Confirm this result using the CAS view of GeoGebra.