The Discovery Decline Phenomenon (DDP) requires that firm-level discoveries of a non-renewable resource depend, non-linearly, upon cumulative industry wide exploration. In this paper, an optimal rule for the exploration effort that adheres to the DDP is derived. Data from the United Kingdom Continental Shelf (UKCS) are applied to the model where it is found that cumulative exploration is a highly significant determinant of firm-level exploration, and that prices and taxes also determine exploration effort. The 'Hubbert peak' in discoveries in the UKCS is estimated to be 1989 thereby improving upon previous estimates but also implying that the DDP in this region is already quite advanced.