We consider a reaction-diffusion system for spatial spread of pest resistance to host plant resistance genes which is based on the Lotka-Volterra predator-prey equations, with logistic growth of the resource level and a diffusion term added to account for spatial spread of the pest. The model is phenotype specific, in which a pest subpopulation's fitness comes down to a balance between its resource assimilation rate and its respiration rate. We derive an expression for the rate of spatial spread of the resistant pest types from an initial point source, and discuss its relevance for adaptive pest resistance management strategies. Using results for an analogous single-species reaction-diffusion model in heterogeneous media, we consider the likely impact of pest-susceptible plant refugia on the speed of the travelling wave of resistant pests, and simultaneously the expected trade-off, in terms of crop yield decrease, when refugia are included. We also explore the possibility that resistance breaking by the pest population is not an inevitable phenomenon, particularly when refugia of the appropriate size are used.