In the ocean, propagules with a planktonic stage are typically dispersed some distance downstream of the parent generation, introducing an asymmetry to the dispersal. Ocean-dwelling species have also been shown to exhibit chaotic population dynamics. Therefore, we must better understand chaotic population dynamics under the influence of asymmetrical dispersal. Here, we examine a density-dependent population in a current, where the current has both a mean and stochastic component. In our finite domain, the current moves offspring in the downstream direction. This system displays a rich variety of dynamics from chaotic to steady-state, depending on the mean distance the offspring are moved downstream, the diffusive spread of the offspring, and the domain size. We find that asymmetric dispersal can act as a stabilizing or destabilizing mechanism, depending on the size of the mean dispersal distance relative to the other system parameters. As the strength of the current increases, the system can experience period-halving bifurcation cascades. Thus, we show that stability of chaotic aquatic populations is directly tied to the strength of the ocean current in their environment, and our model predicts increased prevalence of chaos with decreasing dispersal distance. Climate change is likely to alter the dispersal patterns of many species, and so our results have implications for conservation and management of said species. We discuss the management implications, particularly of exploited species.