  For reaction networks arising in systems biology, the capacity for two or
more steady states, that is, multistationarity, is an important property that
underlies biochemical switches. Another property receiving much attention
recently is absolute concentration robustness (ACR), which means that some
species concentration is the same at all positive steady states. In this work,
we investigate the prevalence of each property while paying close attention to
when the properties occur together. Specifically, we consider a stochastic
block framework for generating random networks, and prove edge-probability
thresholds at which - with high probability - multistationarity appears and ACR
becomes rare. We also show that the small window in which both properties occur
only appears in networks with many species. Taken together, our results confirm
that, in random reversible networks, ACR and multistationarity together, or
even ACR on its own, is highly atypical. Our proofs rely on two prior results,
one pertaining to the prevalence of networks with deficiency zero, and the
other "lifting" multistationarity from small networks to larger ones.
