Is a larger refuge always better? Dispersal and dose in pesticide resistance evolution

The evolution of resistance against pesticides is an important problem of modern agriculture. The high-dose/refuge strategy, which divides the landscape into treated and non-treated (refuge) patches, has proven effective at delaying resistance evolution. However, theoretical understanding is still incomplete, especially for combinations of limited dispersal and partially recessive resistance. We reformulate a two-patch model based on the Comins model and derive a simple quadratic approximation to analyze the effects of limited dispersal, refuge size and dominance for high efficacy treatments on the rate of evolution. When a small but substantial number of heterozygotes can survive in the treated patch, a larger refuge always reduces the rate of resistance evolution. However, when dominance is small enough, the evolutionary dynamics in the refuge population, which is indirectly driven by migrants from the treated patch, mainly describes the resistance evolution in the landscape. In this case, for small refuges, increasing the refuge size will increase the rate of resistance evolution. Our analysis distils major driving forces from the model, and can provide a framework for understanding directional selection in source-sink environments.