This applet allows to understnd which limits must be imposed on the values of the parameter [math]p[/math] of the logistic map [math]f(x)=px(1-x)[/math] in order that its values are comprised between 0 and 1, [math]0\leq f(x)\leq 1[/math], whenever [math]0\leq x\leq 1[/math], i.e. that the function maps the closed interval [math][0,1][/math] into itself: this is necessary to interpret [math]f(x_n)[/math] again as an environmental occupancy percentage when [math]x_n[/math] is.
In the Graphics view change values to [math]p[/math] in order to see that its values can not exceed some value: which one? Of course, [math]p\geq 0[/math] otherwise [math]f(x)[/math] could become negative. Verify, by hand computation, that the maximum value of [math]f(x)[/math] for [math]p\geq 0[/math] is [math]f(\frac{1}{2})[/math], i.e. that [math]f(x)<f(\frac{1}{2})[/math] for any [math]x\neq\frac{1}{2}[/math] when [math]p\geq 0[/math]. Confirm this result using the CAS view of GeoGebra. Find, by hand computation, which is the maximal range of possible values of [math]p[/math] that allows values of [math]f(x)[/math] to belong to the closed interval [math][0,1][/math] when [math]x\in[0,1][/math]. Confirm this result using the CAS view of GeoGebra.