The Economic Threshold With a Stochastic Pest Population: An Application to the European Red Mite

This paper applies option value theory to determine the timing and value of pesticides in controlling crop damage. Entomologists and economists have both sought to determine '~ction" or 'economic" thresholds which would trigger the application of a pesticide, or an alternative pest control policy. These concepts are reviewed, and the differences arising in each discipline are noted. Two mathematical models are developed to more precisely define these concepts. The first model is deterministic, and builds on the work ofHeadley (1972) and Hall and Norgaard (1973). The second model is stochastic, and assumes that the pest population can be described by a 'tliffusion process. " This formulation falls naturally into a class of models that have been used successfully in the field of finance to determine the optimal time to exercise a stock option. Within the pest control context, it is optimal to spray as soon as the stochastically evolving pest population hits a 'stopping frontier. " With a fixed harvest date this frontier is a positively­ sloped, convex curve in time-pest space. This implies that the closer one gets to harvest, the larger the pest population must be to trigger a pesticide application. Both models are applied to the European Red Mite, a foliar pest of apples. Geometric Brownian motion is used to model red mite density. In the stochastic model, the pesticide application is delayed until the pest population reaches a larger size. This stopping value increases with the volatility of the pest density process because of the increased likelihood that predators or other random factors might 'haturally" reduce the pest population, thus avoiding the cost ofpesticide application. To our knowledge, option value theory has not been applied to the problem of pest control. The theory, however, is well suited to the problem of optimal timing in a stochastic environment and is certainly ­ consistent with the integrated pest management (IPM) philosophy. Optimal timing may reduce the overall amount of chemicals applied to crops and soils. THE ECONOMIC THRESHOLD WITH A STOCHASTIC PEST POPULATION AN APPLICATION TO THE EUROPEAN RED MITE L Introduction and Overview The concepts of economic thresholds (ET) and economic injury level (EIL) are still "the backbone ofprogressive concepts in insect control, namely, integrated control, insect pest management" (poston, Pedigo, and Welch (1983)). These concepts were introduced by entomologists to provide practical decision rules for using pesticides in a context of complex interactions between plants and pests. The most widely accepted definition of economic threshold was first given by Stem et al. (1959) as the "density at which control measures should be initiated to prevent an increasing pest population from reaching the economic injury leve1. " The ET is thus an operating rule intimately tied to the EIL, which is "the lowest population density that will cause economic damage", where the later was taken to be the amount of injury which justifies the cost ofartificial control measures. Although quite useful, these definitions needed some clarification regarding the amount of injury that should trigger some action to control a pest population. -