Stepwise versus gradual deterioration: Consequences for evolutionary rescue

Environmental change depresses adaptation and threatens the population’s persistence. Here, we investigate when asexual populations avoid extinction through evolutionary rescue under two distinct modes of environmental variation: a gradual, though discrete, shift of the optimum or a stepwise protocol involving discrete jumps separated by static intervals. We use individual-based simulations, in which individuals have an explicit genetic basis and fitness values are obtained from a three layer genotype-phenotype-fitness map that couples a NK genotype-to-phenotype map with Fisher’s geometric model. We quantify extinction risk P ext , the predictability of endpoints P 2 , the effective number of endpoints N eff = 1 / P 2 , and the fraction G of genotypes visited along successful trajectories by varying the stress amplitude δ , the pace of change (window τ and total duration T shift ), and degree of epistasis K of the genotype-to-phenotype map. Overall, P ext increases monotonically with δ , but slower change or a finer partition of the same total shift (larger τ or longer T shift ) consistently shifts the rescue–extinction transition to higher δ . Increasing the ruggedness of the landscape is beneficial because it lowers P ext when environmental change is fast or abrupt, as larger K effectively amplifies single-step phenotypic effects and enables rescue with fewer mutations. However, under very slow change, ruggedness can fragment long adaptive routes, increasing the chance of getting trapped in suboptimal domains of the fitness landscape, and thus increasing extinction risk. Successful lineages traverse only a small fraction of genotype space (strong contingency), while endpoints remain moderately repeatable.