A general differential equation model of the temporal evolution of a pest population density when subjected to control measures using a pesticide, is investigated. The model is based on logistic growth combined with population dispersal described by diffusion, and pesticide action is characterised by its LD90 toxicity measure and a consumption limited by a saturation pesticide density. Solutions are found for a number of assumed scenarios concerning the initial pest distribution and pesticide application strategy. Criteria are established for pesticide toxicity and application density to ensure eradication, and the efficiency of the strategies investigated are compared with regard to total pesticide consumption. It is shown that rather general conclusions can be reached, such as that it is inevitable that a pesticide residue is left after full eradication has been accomplished.