Impulsive culling of a structured population on two patches

We consider a model for a creature inhabiting two patches between which migration may occur. The creature is assumed to have a life cycle with two stages, namely juvenile and adult, giving rise to a delay differential system. The creature could represent an insect crop pest whilst the patches could represent neighbouring farms. Given that it is common to control crop pests by adult impulsive culling, we impose an adult impulsive culling regime on each patch. We find conditions on the regimes such that the pest will be eradicated on both patches simultaneously. The regime on one patch is assumed to be independent of the regime on the other patch to reflect the possibility that the patches represent farms with different owners where each owner has autonomy in their pest control decisions. In the special case where the birth functions on both patches are of an Allee type, we calculate explicit finite upper bounds for the number of culls needed on each patch to guarantee eradication. Simulations corroborate our theoretical results.