Habitat sinks can attract dispersing animals if high mortality or breeding failure are difficult to detect (e.g., when due to human hunting or pollution). Using a simple deterministic model, we explore the dynamics of such source-sink systems considering three scenarios: an avoided sink, no habitat preference, and an attractive sink. In the second two scenarios, there is a threshold proportion of sink habitat above which the whole population decreases to extinction, but this extinction threshold varies with habitat preference and the relative qualities of the two habitat types. Hence, it would be necessary to know the habitat preferences of any species in a source-sink system to interpret data on population increases and declines. In the attractive sink scenario, small changes in the proportion of sink habitat may have disproportionate effects on the population persistence. Also, small changes in growth rates at the source and the sink severely affect the threshold and the time of extinction. For some combinations of demographic parameters and proportion of habitat sink, the decline affects the source first; thus, during some time, it will be hidden to population monitoring at the sink, where numbers can even increase. The extinction threshold is also very sensitive to the initial population sizes relative to carrying capacity. Attractive sinks represent a novel aspect of source-sink dynamics with important conservation and management implications.