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 more rapid decline of non-adapted genotypes and, meanwhile, increase the selective advantage of adaptive genotypes, which can have opposing effects on the probability of rescue. More generally, it is unclear if the probability of rescue increases or decreases with stronger selection, with important implications for herbicide/drug resistance evolution. By investigating various scenarios of rescue, we show that, in general, rescue becomes more likely with stronger selection when: 1) the rate of evolution has a stronger response to selection and 2) the rate of population decline is more rapid. A stronger evolutionary response to selection occurs when adaptive alleles are more dominant or the level of inbreeding is higher, or the genetic variance is larger and the environmental shift is more rapid for rescue via the evolution of quantitative traits. The intuition is that a stronger evolutionary response to selection increases the marginal benefit of increased selection, while a more rapid decline of the population lowers the marginal cost of a reduction in mean absolute fitness.