Biological invasions often damage island ecosystems. One such damaging consequence of biological invasions is hyperpredation. Hyperpredation is the increase in predation pressure from a generalist predator following the introduction of an alternative prey, typically a consequence of apparent competition between the two prey. Models for this have been devised that demonstrate this effect. However, hyperpredation may not always occur or may not always occur at the same strength. Here, we investigate how different mechanisms affect the magnitude of hyperpredation: (i) saturation of the predator's functional response, (ii) predator interference and (iii) non-predatory competition among predators. We find that all three mechanisms generally reduce hyperpredation. Predator saturation can actually overturn hyperpredation into hypopredation, an increase in native prey, as a result of apparent predation between the two prey. This occurs when the alternative prey is ‘poisoned prey’, i.e. prey that have a handling time cost greater than the nutritional benefit for the predator. Consuming ‘poisoned prey’ can result in an increase or decrease in predator density. Conversely, we also identify scenarios in which interference and competition may increase hyperpredation. Based on these insights, we conclude that the invasion of established ecosystems by non-native prey can lead to more diverse consequences than previously thought. Potential control measures should take these effects into account.