  In this paper we consider two continuous-mass population models as analogues
of logistic branching random walks, one is supported on a finite trait space
and the other one is supported on an infinite trait space. For the first model
with nearest-neighbor competition and migration, we justify a well-described
evolutionary path to the short-term equilibrium on a slow migration time scale.
For the second one with an additional evolutionary mechanism-mutation, a jump
process-trait substitution tree model is established under a combination of
rare mutation and slow migration limits. The transition rule of the tree highly
depends on the relabelled trait sequence determined by the fitness landscape.
The novelty of our model is that each trait, which may nearly die out on the
migration time scale, has a chance to recover and further to be stabilized on
the mutation time scale because of a change in the fitness landscape due to a
newly entering mutant.
