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Motivated by modeling the dynamics of a population living in a flowing mediumwhere the environmental factors are random in space, we have studied anasymmetric variant of the one-dimensional contact process, where the quenchedrandom reproduction rates are systematically greater in one direction than inthe opposite one. The spatial disorder turns out to be a relevant perturbationbut, according to results of Monte Carlo simulations, the behavior of the modelat the extinction transition is different from the (infinite randomness)critical behavior of the disordered, symmetric contact process. Depending onthe strength $a$ of the asymmetry, the critical population drifts either with afinite velocity or with an asymptotically vanishing velocity as $x(t)\simt^{\mu(a)}$, where $\mu(a)<1$. Dynamical quantities are non-self-averaging atthe extinction transition; the survival probability, for instance, showsmultiscaling, i.e. it is characterized by a broad spectrum of effectiveexponents. For a sufficiently weak asymmetry, a Griffiths phase appears belowthe extinction transition, where the survival probability decays as anon-universal power of the time while, above the transition, another extendedphase emerges, where the front of the population advances anomalously with adiffusion exponent continuously varying with the control parameter.

Extinction transition in stochastic population dynamics in a random, convective environment