  For the reaction networks with one-dimensional stoichiometric subspaces, we
show the following results. (1) If the maximum number of positive steady states
is an even number N, then the maximum number of stable positive steady states
is N/2. (2) If the maximum number of positive steady states is an odd number N,
then we provide a condition on the network such that the maximum number of
stable positive steady states is (N-1)/2 if this condition is satisfied, and
this maximum number is (N+1)/2 otherwise.
