When does stronger selection increase the probability of evolutionary rescue?

After an environmental change, populations may be rescued from extinction through adaptive evolution. Stronger environmental changes often cause a faster demographic decline but also more rapid adaptive evolution, with opposing effects on rescue. More generally, it is unclear if the probability of rescue increases with stronger selection, with implications for herbicide/drug resistance evolution. Here we show that, across many scenarios of rescue, stronger selection promotes rescue when: (1) the rate of evolution has a stronger response to selection and (2) the rate of population decline is more rapid. Condition (1) occurs when adaptive alleles are more dominant or the level of inbreeding is higher or, in quantitative genetic models, the genetic variance is larger and the environmental shift is more rapid. Intuitively, a stronger evolutionary response to selection increases the marginal benefit of increased selection, while a faster demographic decline lowers the marginal cost of a reduction in mean fitness.