Superdiffusion and encounter rates in diluted, low dimensional worlds

Rate limitation due to encounters is fundamental to many ecologicalinteractions. Since encounter rate governs reaction rates, and thus, dynamics of systems, it deserves systematic study. In classicalpopulation biology, ecological dynamics rely on the assumption ofperfectly mixed interacting entities (e.g., individuals, populations,etc.) in a spaceless world. The so-called mean field assumptionassumes that encounter rates are driven exclusively by changes in thedensity of the interacting entities and not on how they aredistributed or move in space. Therefore, the mean field assumptiondoes not give any insight into relevant spatiotemporal statisticalproperties produced by the trajectories of moving entities throughspace. In the present study, we develop spatially explicitsimulations of random walking particles (i.e., Lévy walkers) toevaluate encounter rate constraints beyond the mean fieldassumption. We show that encounter rate fluctuations are driven notonly by physical aspects such as the size or the velocity of theinteracting particles, but also by different motion patterns. Inparticular, superdiffusion phenomena might be relevant at lowdensities and/or low spatial dimensionality. Finally, we discusspotential adaptive responses of living organisms that may allowindividuals to control how they diffuse through space and/or thespatial dimensions employed in the exploration process.