  We examine the behavior of a model which describes the melting of
double-stranded DNA chains. The model, with displacement-dependent stiffness
constants and a Morse on-site potential, is analyzed numerically; depending on
the stiffness parameter, it is shown to have either (i) a second-order
transition with "nu_perpendicular" = - beta = 1, "nu_parallel" = gamma/2 = 2
(characteristic of short range attractive part of the Morse potential) or (ii)
a first-order transition with finite melting entropy, discontinuous fraction of
bound pairs, divergent correlation lengths, and critical exponents
"nu_perpendicular" = - beta = 1/2, "nu_parallel" = gamma/2 = 1.
