of these inequalities is violated, the mean-field dynamics exhibit bistability. Given that the first one does not hold, besides the ordered solution with $r^*=0$, a second stable inhomogeneous state appears. The higher degree population is in the excited state and the lower degree population remains in the non-excited one (cf. Fig. \[fig:1k2\_1k8\]). In contrast, if the second inequality is violated bistability occurs between the homogeneous non-excited states and the homogeneous excited ones (cf. Fig. \[fig:1k2\_1k3\]). Finally, if both inequalities do not hold, the solution has five intersections according to a tristable solution between the two homogeneous situations and the one non-homogeneous one (cf. Fig. \[fig:1k2\_1k4\]).

Solutions with finite noise and simulations {#subsec:sim}
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Examples of the qualitative behavior of the steady states for various noise levels $D$ are presented in Figs. \[fig:1k2\_1k3\]-\[fig:1k2\_1k4\]. The graphs have been obtained by numerical solution of and . The main difference of these graphs is the ratio $\alpha=k_1 / k_2$. In Fig. \[fig:1k2\_1k3\], $\alpha$ equals $3 / 2$ and for high noise the disordered state with mean-field $r^*=1
/ 2$ is stable. The latter bifurcates for lower noise to the known bistability of ordered states [@Kouvaris2010] which approach $(0,0)$, $(1,1)$ as $D \to 0$. These 