Boundaries of sustainability in simple and elaborate models of agricultural pest control with a pesticide and a non-toxic refuge.

In two models of pest control using a pesticidal crop along with a non-pesticidal refuge to prevent the development of resistance, we numerically compute the bifurcations that bound the region in parameter space where control is sustainable indefinitely. An exact formula for one of the bifurcation surfaces in one of the models is also found. One model is conceptual and as simple as possible. The other is realistic and very detailed. Despite the great differences in the models, we find the same distinctive bifurcation structure. We focus on the parameters that determine: (i) the restriction of pest exchange between the crop and the refuge, which we call 'screening' the refuge, and (ii) the recessiveness of the resistance trait. The screened refuge technique is seen to work in the models up to quite high values of fitness of resistant heterozygotes, that is, even when resistance is not strongly recessive.

Analysis of sustainable pest control using a pesticide and a screened refuge.

We describe and analyze a 'screened refuge' technique for indefinitely sustaining control of insect pests using transgenic pesticidal crops or an applied pesticide, even when resistance is not recessive. The screen is a physical barrier that restricts pest movement. In a deterministic discrete-time model of the use of this technique, we obtain asymptotic analytical formulas for the two important equilibria of the system in terms of the refuge size and the pest fitnesses, mutation rates, and mobility out of and into the refuge. One of the equilibria is stable and is the point at which the pest population is controlled. The other is a saddle whose stable manifold bounds the basin of attraction of the former: its location provides a measure of the tolerance of the control mechanism to perturbations in the resistant allele density.

A situation in which a local nontoxic refuge promotes pest resistance to toxic crops.

In order to delay the development of pest resistance to genetically engineered insecticidal crop varieties, it is current practice to grow "refugees" of non-toxic plants close to insecticidal crops. We model such a toxic/nontoxic crop complex as an open system with a small stream of toxin-susceptible immigrants. We find that, for intermediate values of the dominance of a pest gene for resistance to the toxin, the local refuge can spoil the benefit that is provided by the immigrant stream. We provide formulas for some important boundaries in parameter space.