We model the conditions for pest eradication in a reaction–diffusion system made of a prey and a gener-alist predator through spatial impulsive control within a bounded domain. The motivating example is the control of the invasive horse chestnut leafminer moth through the yearly destruction of leaves in autumn, in which both the pest and its parasitoids overwinter. The model is made of two integro-partial differential equations, the integral portion describing the within-year immigration from the whole domain. The problem of pest eradication is strongly related to some appropriate eigenvalue problems. Basic properties of the principal eigenvalues of these problems are derived by using of Krein–Rutman's theorem and of comparison results for parabolic equations with non-local terms. Spatial control of the pest can be achieved, if one of these principal eigenvalues is large enough, at an exponential rate. This is true without and with parasitoids, the latter case being of course more rapid. We discuss the possible implementation of these results to the leafminer invasion problem and discuss complementary methods.