  There exist a number of results proving that for certain classes of
interacting particle systems in population genetics, mutual invadability of
types implies coexistence. In this paper we prove a sort of converse statement
for a class of one-dimensional cancellative systems that are used to model
balancing selection. We say that a model exhibits strong interface tightness if
started from a configuration where to the left of the origin all sites are of
one type and to the right of the origin all sites are of the other type, the
configuration as seen from the interface has an invariant law in which the
number of sites where both types meet has finite expectation. We prove that
this implies noncoexistence, i.e., all invariant laws of the process are
concentrated on the constant configurations. The proof is based on special
relations between dual and interface models that hold for a large class of
one-dimensional cancellative systems and that are proved here for the first
time.
