Predator-prey theory suggests that generalist predators are linked to demographic stability of prey whereas specialists are destabilizing. We overview the demographic consequences of different predation strategies and hypothesize that subsistence hunting occurs opportunistically, persecution hunters behave like specialist predators, and recreational hunters behave like generalist predators. Under this hypothesis, persecution hunting would have destabilizing effects, whereas the effects of subsistence and recreational hunting would be neutral or stabilizing. We found poor empirical support for this hypothesis, but there was scarce empirical data. Recreational hunters mainly hunted opportunistically and hunting as managed persecution followed a type III functional response, i.e. with low hunting intensity at low game abundances and a switch to an increased intensity at some level of abundance. We suggest that recreational hunters have limited destabilizing effects on game populations and that hunting may be an ineffective way of complete the removal of invasive species. We urge for further studies quantifying the responses of hunters to game abundances, in particular studies evaluating the responses of subsistence hunters and illegal persecution.