  Motivated by modeling the dynamics of a population living in a flowing medium
where the environmental factors are random in space, we have studied an
asymmetric variant of the one-dimensional contact process, where the quenched
random reproduction rates are systematically greater in one direction than in
the opposite one. The spatial disorder turns out to be a relevant perturbation
but, according to results of Monte Carlo simulations, the behavior of the model
at the extinction transition is different from the (infinite randomness)
critical behavior of the disordered, symmetric contact process. Depending on
the strength $a$ of the asymmetry, the critical population drifts either with a
finite velocity or with an asymptotically vanishing velocity as $x(t)\sim
t^{\mu(a)}$, where $\mu(a)<1$. Dynamical quantities are non-self-averaging at
the extinction transition; the survival probability, for instance, shows
multiscaling, i.e. it is characterized by a broad spectrum of effective
exponents. For a sufficiently weak asymmetry, a Griffiths phase appears below
the extinction transition, where the survival probability decays as a
non-universal power of the time while, above the transition, another extended
phase emerges, where the front of the population advances anomalously with a
diffusion exponent continuously varying with the control parameter.
