Ecological traps are threats to organisms, and exist in a range of biological systems. A subset of ecological trap theory is the "ethological trap," whereby behaviors canalized by past natural selection become traps when environments change rapidly. Invasive predators are major threats to imperiled species and their ability to exploit canalized behaviors of naive prey is particularly important for the establishment of the predator and the decline of the native prey. Our study uses ecological theory to demonstrate that invasive predator controls require shifts in management priorities. Total predation rate (i.e., total response) is the product of both the functional response and numerical response of predators to prey. Functional responses are the changes in the rate of prey consumption by individual predators, relative to prey abundance. Numerical responses are the aggregative rates of prey consumption by all predators relative to prey density, which change with predator density via reproduction or migration, in response to changes in prey density. Traditional invasive predator management methods focus on reducing predator populations, and thus manage for numerical responses. These management efforts fail to manage for functional responses, and may not eliminate impacts of highly efficient individual predators. We explore this problem by modeling the impacts of functional and numerical responses of invasive foxes depredating imperiled Australian turtle nests. Foxes exhibit exceptionally efficient functional responses. A single fox can destroy >95% of turtle nests in a nesting area, which eliminates juvenile recruitment. In this case, the ethological trap is the "Arribada" nesting strategy, an emergent behavior whereby most turtles in a population nest simultaneously in the same nesting grounds. Our models show that Arribada nesting events do not oversaturate foxes, and small numbers of foxes depredate all of the nests in a given Arribada. Widely scattering nests may reduce fox predation rates, but the long generation times of turtles combined with their rapid recent decline suggests that evolutionary responses in nesting strategy may be unlikely. Our study demonstrates that reducing populations of highly efficient invasive predators is insufficient for preserving native prey species. Instead, management must reduce individual predator efficiency, independent of reducing predator population size.