  The role of the selection pressure and mutation amplitude on the behavior of
a single-species population evolving on a two-dimensional lattice, in a
periodically changing environment, is studied both analytically and
numerically. The mean-field level of description allows to highlight the
delicate interplay between the different time-scale processes in the resulting
complex dynamics of the system. We clarify the influence of the amplitude and
period of the environmental changes on the critical value of the selection
pressure corresponding to a phase-transition "extinct-alive" of the population.
However, the intrinsic stochasticity and the dynamically-built in correlations
among the individuals, as well as the role of the mutation-induced variety in
population's evolution are not appropriately accounted for. A more refined
level of description, which is an individual-based one, has to be considered.
The inherent fluctuations do not destroy the phase transition "extinct-alive",
and the mutation amplitude is strongly influencing the value of the critical
selection pressure. The phase diagram in the plane of the population's
parameters -- selection and mutation is discussed as a function of the
environmental variation characteristics. The differences between a smooth
variation of the environment and an abrupt, catastrophic change are also
addressesd.
