Figure 8.

Plots of the mapping F for k = 2 and $\alpha =f\left(\frac{1}{2}\right)=\frac{1}{2}$. (a) $r+\frac{3}{2}s>1$. (b) $r+\frac{3}{2}s\ge 1$. If r - s > 1/2 (which implies M ≥ 2) then there is a neutral fixed point at 1/2 that represents an isolated 2 cluster cyclic solution. If the second segment intersects the diagonal, then there is an isolated 2 cluster cyclic solutions that is stable for α < 0 and unstable for α > 0. There also may exist neutral fixed points for r - s < 1/2 (M = 1) and certain conditions on the parameters where the third piecewise segment of F intersects the diagonal line. These fixed points represent 2 cluster cyclic solutions that are not isolated, but yet are neutral.