For the Logistic Model: [math]P_{\left\{n+1\right\}}=rP_n\left(1-P_n\right)[/math]  [br][br]Adjust the value of r using the black slider to see the effect that the parameter has on the shape of the curve.[br][br]The initial population, relative to the maximum sustainable population, is represented by the red dot on the horizontal axis.  You can adjust the initial population size by dragging the red dot along the horizontal axis.[br][br]The blue slider for n controls the number of iterates represented in the cobweb diagram.

Suppose [math]r=3.15[/math].  Then the fixed points for the model [math]P_{n+1}=3.15P_n\left(1-P_n\right)[/math][br]are given by [math]k=0[/math] and[math]k=1-\frac{1}{3.15}=\frac{43}{63}\approx0.6825[/math] .[br][br]The orbit of the initial population [math]P_0=0.6[/math] under the model [math]P_{n+1}=3.15P_n\left(1-P_n\right)[/math] is:[br][br][math]0.6\quad\longrightarrow\quad0.756\quad\longrightarrow\quad0.581062\quad\longrightarrow\quad0.766801\quad\longrightarrow\quad0.563274\quad\longrightarrow\quad0.774889\quad[/math][br][math]\longrightarrow\quad0.579474\quad\longrightarrow\quad0.779790\quad\longrightarrow\quad0.540911\quad\longrightarrow\quad0.782228\quad[/math][br][math]\longrightarrow\quad0.536594\quad\longrightarrow\quad0.783282\quad\longrightarrow\quad\cdots[/math] etc.[br][br]If we go out far enough, successive values cycle between the approximate values of 0.533494 and 0.783966.[br][br]This tells us that the population tends to a [b]periodic pattern[/b], namely a two-cycle.[br][br]You can verify this using the cobwebbing tool above by setting [math]r=3.15[/math] (using the slider tool for r) and by sliding the red dot to the value of 0.6 to represent [math]P_0=0.6[/math]. The two-cycle becomes visible when the number of iterates, n, nears the max value of the slider.  The cobweb will start to bounce between two points on the curve for x-values near 0.53 and 0.78.[br][br]Note: the two-cycle is even more visible on the cobweb plot if you pick your initial population to be approximately [math]P_0=0.78[/math].