Superdiffusion and encounter rates in diluted,low dimensional worlds

Rate limitation due to encounters is fundamental to many ecological interactions. Since encounter rate governs reaction rates, and thus, dynamics of systems, it deserves systematic study. In classical population biology, ecological dynamics rely on the assumption of perfectly mixed interacting entities (e.g., individuals, populations, etc.) in a spaceless world. The so-called mean field assumption assumes that encounter rates are driven exclusively by changes in the density of the interacting entities and not on how they are distributed or move in space. Therefore, the mean field assumption does not give any insight into relevant spatiotemporal statistical properties produced by the trajectories of moving entities through space. In the present study, we develop spatially explicit simulations of random walking particles (i.e., Lévy walkers) to evaluate encounter rate constraints beyond the mean field assumption. We show that encounter rate fluctuations are driven not only by physical aspects such as the size or the velocity of the interacting particles, but also by different motion patterns. In particular, superdiffusion phenomena might be relevant at low densities and/or low spatial dimensionality. Finally, we discuss potential adaptive responses of living organisms that may allow individuals to control how they diffuse through space and/or the spatial dimensions employed in the exploration process.