  We consider the soft-core Widom-Rowlinson model for particles with spins and
holes, on a Cayley tree of degree d, depending on repulsion strength beta
between particles of different signs and on an activity parameter lambda for
particles. We analyse Gibbsian properties of the time-evolved intermediate
Gibbs measure of the static model, under a spin-flip time evolution, in a
regime of large repulsion strength beta. We first show that there is a
dynamical transition, in which the measure becomes non-Gibbsian at large times,
independently of the particle activity, for any d greater or equal 2. In our
second and main result, we also show that for large beta and at large times,
the measure of the set of bad configurations (discontinuity points) changes
from zero to one as the particle activity increases, assuming that d is greater
or equal than 4. Our proof relies on a general zero-one law for bad
configurations on the tree, and the introduction of a set of uniformly bad
configurations given in terms of subtree percolation, which we show to become
typical at high particle activity.
