  We describe the behaviour at threshold of S-wave poles of the scattering
matrix within a four-parameter model for non-exotic meson-meson scattering.
This model accommodates all non-exotic mesons, hence also the light scalar
mesons, as resonances and bound states characterised by complex singularities
of the scattering amplitude as a function of the total invariant mass.
  The majority of the full S-matrix mesonic poles stem from an underlying
confinement spectrum. However, the light scalar mesons K0*(830), a0(980),
f0(400--1200), and f0(980) do not, but instead originate in 3P0-barrier
semi-bound states. We show that the behaviour of the corresponding poles is
identical at threshold.
