Tuning spatial distributions of selection pressure to suppress emergence of resistance

Control measures such as insecticides or antimicrobials are used to contain biological agents such as pathogenic bacteria and vectors of human and plant diseases, respectively. Following control measure application, a resistant subpopulation may eventually rise to such frequency that the control measure will be rendered ineffective: The timescale over which this occurs is the 'effective lifetime' of the control measure. Prolonging this timescale relaxes urgency at which novel control measures needs to be developed. Spatial heterogeneity in control measure application can influence the rate at which resistance to the control measure evolves; in the agricultural context, this fact is exploited by distributing insecticides in mosaics across cropping regions in order to slow the rate of resistance evolution. Contemporary and historical modeling practices, which aim to inform agricultural practices, often employ assumptions which squeeze out the impact of the spatio-temporal heterogeneity endemic to nature and aim to identify how the fitness cost of a resistant population can be leveraged to prevent the rise of resistance. In this paper, we present a minimal model of continuous dispersal and spatio-temporal heterogeneity in selection pressure distribution. We assume the resistant sub-population to have negligible fitness cost to resistance. We find that even in that case the spatial distribution of selection pressure may be tuned in order to minimize the initial rate at which resistance evolves, thus increasing the effective lifetime of a pesticide. We demonstrate that the existence of an optimal distribution is a consequence of two processes: the selection against a susceptible strain when the selection pressure is active, and the subsequent re-invasion of the susceptible strain into the space once the selection pressure has been alleviated. In summary, we find that optimal distribution of a control measure can slow resistance evolution - even in absence of fitness costs to resistance - by balancing selection and re-invasion dynamics.