  Boolean networks have been the object of much attention, especially since S.
Kauffman proposed them in the 1960's as models for gene regulatory networks.
These systems are characterized by being defined on a Boolean state space and
by simultaneous updating at discrete time steps. Of particular importance for
biological applications are networks in which the indegree for each variable is
bounded by a fixed constant, as was stressed by Kauffman in his original
papers.
  An important question is which conditions on the network topology can rule
out exponentially long periodic orbits in the system. In this paper we consider
cooperative systems, i.e. systems with positive feedback interconnections among
all variables, which in a continuous setting guarantees a very stable dynamics.
In Part I of this paper we presented a construction that shows that for an
arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional
Boolean cooperative networks in which both the indegree and outdegree of each
for each variable is bounded by two (bi-quadratic networks) and which
nevertheless contain periodic orbits of length at least c^n.
  In this part, we prove an inverse result showing that for sufficiently large
n and for 0<c<2 sufficiently close to 2, any n-dimensional cooperative,
bi-quadratic Boolean network with a cycle of length at least c^n must have a
large proportion of variables with indegree 1. Such systems therefore share a
structural similarity to the systems constructed in Part I.
