  Linearly-sloped or `ramp' potentials belong to a class of core-softened
models which possess a liquid-liquid critical point (LLCP) in addition to the
usual liquid-gas critical point. Furthermore they exhibit thermodynamic
anomalies in the density and compressibility, the nature of which may be akin
to those occurring in water. Previous simulation studies of ramp potentials
have focused on just one functional form, for which the LLCP is
thermodynamically stable. In this work we construct a series of ramp
potentials, which interpolate between this previously studied form and a
ramp-based approximation to the Lennard-Jones (LJ) potential. By means of Monte
Carlo simulation, we locate the LLCP, the first order high density liquid
(HDL)-low density liquid (LDL) coexistence line, and the line of density maxima
for a selection of potentials in the series. We observe that as the LJ limit is
approached, the LLCP becomes metastable with respect to freezing into a
hexagonal close packed crystalline solid. The qualitative nature of the phase
behaviour in this regime shows a remarkable resemblance to that seen in
simulation studies of accurate water models. Specifically, the density of the
liquid phase exceeds that of the solid; the gradient of the metastable LDL-HDL
line is negative in the pressure (p)-temperature (T) plane; while the line of
density maxima in the p-T plane has a shape similar to that seen in water and
extends well into the {\em stable} liquid region of the phase diagram. As such,
our results lend weight to the `second critical point' hypothesis as an
explanation for the anomalous behaviour of water.
