  We consider a model for gene regulatory networks that is a modification of
Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$
the number of nodes, $r =$ the number of inputs to each node, and $p =$ the
expected fraction of 1's in the Boolean functions at each node. Following a
standard practice in the physics literature, we use a threshold contact process
on a random graph on $n$ nodes, in which each node has in degree $r$, to
approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$,
then the threshold contact process persists for a long time, which correspond
to chaotic behavior of the Boolean network. Unfortunately, we are only able to
prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot
2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.
