A Lévy-flight diffusion model to predict transgenic pollen dispersal

Abstract

The containment of genetically modified (GM) pollen is an issue of significant concern for many countries. For crops that are bee-pollinated, model predictions of outcrossing rates depend on the movement hypothesis used for the pollinators. Previous work studying pollen spread by honeybees, the most important pollinator worldwide, was based on the assumption that honeybee movement can be well approximated by Brownian motion. A number of recent studies, however, suggest that pollinating insects such as bees perform Lévy flights in their search for food. Such flight patterns yield much larger rates of spread, and so the Brownian motion assumption might significantly underestimate the risk associated with GM pollen outcrossing in conventional crops. In this work, we propose a mechanistic model for pollen dispersal in which the bees perform truncated Lévy flights. This assumption leads to a fractional-order diffusion model for pollen that can be tuned to model motion ranging from pure Brownian to pure Lévy. We parametrize our new model by taking the same pollen dispersal dataset used in Brownian motion modelling studies. By numerically solving the model equations, we show that the isolation distances required to keep outcrossing levels below a certain threshold are substantially increased by comparison with the original predictions, suggesting that isolation distances may need to be much larger than originally thought.

Epidemics on plants: Modeling long-range dispersal on spatially embedded networks

“…For instance, most of the transport of pollen between plants is carried out by wind or insect pollinators. This biological process is vital for the survival of the species, but it is also important for understanding transgenic pollen dispersal [13]. Of similar importance is the spreading of evolutionary novelties across populations.…”

“…The incorporation of long-range dispersal effects, either as diffusive processes or by including long-range jumps, has been the topic of many researches [22,23,24,25,26,27,28,29]. Recently, Vallaeys et al [13] have stressed that the diffusive process "often seriously underestimates dispersal distances", and on the other hand, pure Lévy movements "often overestimates dispersal distances". Thus, methods that account for an equilibrated balance between diffusive and long-range dispersal are still needed for modeling epidemic processes in plants [13].…”

“…Recently, Vallaeys et al [13] have stressed that the diffusive process "often seriously underestimates dispersal distances", and on the other hand, pure Lévy movements "often overestimates dispersal distances". Thus, methods that account for an equilibrated balance between diffusive and long-range dispersal are still needed for modeling epidemic processes in plants [13]. Another important challenge in modeling plant diseases is the necessity to consider the spatial characteristics of the plots or fields in which the plants are embedded.…”

Analysis of temporal patterns in animal movement networks

“…The black kite and the wolf trajectories appeared different from both Brownian and Lévy motions thus suggesting the possibility to study these movements using more complex behavioural models. Indeed, Brownian motion often underestimates long range movements while pure Lévy walk often overestimates them (Vallaeys, Tyson, Lane, Deleersnijder, & Hanert, 2017). More realistic motions might also be tested in the future (e.g.…”

The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

“…A more recent example is that of Chen et al [83], who showed from experiments that active transport within living cells described by time-dependent Brownian walks can self-organize into (truncated) Lévy walks. Other examples demonstrating this concept can be found elsewhere in the literature, e.g., swarming bacteria [84], pollen dispersal [85], etc. Despite this, from an ecological viewpoint, such as insect trapping, the motivation behind time-dependent diffusion, and how this is linked to the type of mechanisms involved, to date, has not been clearly understood [52].…”