  Variability on external conditions has important consequences for the
dynamics and the organization of biological systems. In many cases, the
characteristic timescale of environmental changes as well as their correlations
play a fundamental role in the way living systems adapt and respond to it. A
proper mathematical approach to understand population dynamics, thus, requires
approaches more refined than, e.g., simple white-noise approximations. To shed
further light onto this problem, in this paper we propose a unifying framework
based on different analytical and numerical tools available to deal with
"colored" environmental noise. In particular, we employ a "unified colored
noise approximation" to map the original problem into an effective one with
white noise, and then we apply a standard path integral approach to gain
analytical understanding. For the sake of specificity, we present our approach
using as a guideline a variation of the contact process--which can also be seen
as a birth-death process of the Malthus-Verhulst class--where the propagation
or birth rate varies stochastically in time. Our approach allows us to tackle
in a systematic manner some of the relevant questions concerning population
dynamics under environmental variability, such as determining the stationary
population density, establishing the conditions under which a population may
become extinct, and estimating extinction times. We focus on the emerging phase
diagram and its possible phase transitions, underlying how these are affected
by the presence of environmental noise time-correlations.
