The definition of isolation distances between genetically-modified (GM) and conventional crops that keep outcrossing below a certain threshold is very different from one European country to the other. For instance, the isolation distances between GM and non-GM maize crops currently ranges between 15 m and 800 m across the EU. Such differences reflect a lack of scientific knowledge of the processes driving pollen transport, i.e. transport by the wind and by pollination insects. While the former can be modeled quite accurately with high-resolution atmospheric-dispersion models, modeling pollen transport by insects is more difficult.

A number of recent studies have shown that pollination insects like honey bees can travel large distances (> 1km) when their ressources are scarce. This means that the seemingly random displacements of pollination insects is more akin to a Levy random walk (RW) than to a Brownian RW. Levy RW's consist of frequently occurring short displacements with more occasional longer displacements. These longer displacements are in turn punctuated by even rarer and longer displacements, and so on. The probability of seeing such long displacements with a Brownian RW is close to zero as the tails of the Gaussian displacement kernel decay exponentially. By assuming that pollination insects perform a Levy RW, we can explicitly take into account the probability of transporting GM pollen over large distances and hence better evaluate the risk of outcrossing.

At a macroscopic level, a cloud of particles that follow a Levy RW can be represented by a fractional-order diffusion equation. In such an equation, the diffusion term is represented by a non-local integro-differential operator that results in a faster diffusion rate than classical second-order operators. This faster-than-normal diffusion process is called superdiffusion and is currently widely used to model the anomalous transport processes observed in complex systems.

In this work, we propose a mechanistic model for pollen dispersal that is based on bee movement through space.The model involves 2 coupled equations: one for motile pollen that is transported by bees and the other for the stationary pollen that has been deposited by the bees. Since bees are assumed to follow a Levy RW, the equation for motile pollen involves a fractional-order diffusion term. The model is calibrated and validated by using published data on bees displacement patterns and experimental data from an apple orchard with a row of 200 transgenic source trees carrying the GUS marker gene. Fruits from neighboring non-GM trees were gathered at various distances to estimate the fraction of outcrossing.

By solving numerically the model equations, we show how the pollen dispersal is modified when assuming that pollination insects follow a Levy RW. The tails of the stationary pollen distribution decay more slowly and hence spread over a larger area. The isolation distances required to keep outcrossing below a certain threshold are then substantially increased. Our results suggest that classical models based on the Brownian motion assumption might seriously underestimate the risk associated with GM pollen outcrossing in conventional crops.