grows monotonically and is bounded between values of the interval $[0,1]$. Hence solutions $r^*$ are also in this interval. Solutions with one, three or five intersections can be found. Bifurcations between these monostable, bistable of tristable behavior are saddle-node bifurcations or, if these coincide, a pitchfork bifurcation.

If the number of degrees would be increased, the number of steps will also increase in the same manner, giving rise to even higher multistable states.

Solutions with vanishing noise {#subsec:D0}
------------------------------

It is illustrative to look first in detail at the case of vanishing noise. Then the r.h.s. of equation vanishes $\propto \exp(-1/D)$ as $r \to 0$ and approaches unity for large values of $r$. In between, the r.h.s. makes two jumps with magnitude $\nu k_1/\langle k \rangle $ and $(1-\nu)k_2/\langle k
\rangle$, respectively. These steps are located at $r_1=1/(\sigma
k_1)$ and $r_2=1/(\sigma k_2)$. For the r.h.s. to possess one intersection (monostability) with the straight line $r$, we obtain the following conditions by using that $k_1 >k_2$: $$\label{eq:twosteps}
\frac{1}{\sigma k_1}\, > \, \frac{\nu k_1}{\nu k_1+(1-\nu)
    k_2},\,\qquad\frac{1}{\sigma k_2} >  1.$$ For vanishing noise the monostable state is always the ordered one with $r^*=0$, i.e. neither of the two populations is excited.. If one 