When does the probability of evolutionary rescue increase with the strength of selection?

Populations may be rescued from extinction via sufficiently rapid adaptive evolution. Evolution is faster with stronger selection but this may come with a demographic cost, creating opposing effects on evolutionary rescue. The outcome of this trade-off determines the optimal strategy for avoiding herbicide/drug resistance evolution. Here we examine the effect of stronger selection, and the associated demographic change, across four models of evolutionary rescue. We find that stronger selection cannot facilitate rescue in two quite different population genetic models unless it increases the absolute fitness of the rescue homozygote. Similarly, in a quantitative genetic model of rescue, where stronger selection leaves maximum fitness unchanged, stronger selection cannot facilitate rescue despite elevating the rate of evolution. We also explore a quantitative genetic model of rescue in a gradually changing environment, generalizing the finding that an intermediate selection strength maximizes survival at steady-state to a wider class of fitness functions.