A Diffusion Approximation Based on Renewal Processes with Applications to Strongly Biased Run–Tumble Motion

Abstract

Document VersionAbstract We consider organisms which use a renewal strategy such as runtumble when moving in space, for example to perform chemotaxis in chemical gradients. We derive a diffusion approximation for the motion, applying a central limit theorem due to Anscombe for renewal-reward processes; this theorem has not previously been applied in this context. Our results extend previous work, which has established the mean drift but not the diffusivity. For a classical model of tumble rates applied to chemotaxis, we find that the resulting chemotactic drift saturates to the swimming velocity of the organism when the chemical gradients grow increasingly steep. The dispersal becomes anisotropic in steep gradients, with larger dispersal across the gradient than along the gradient. In contrast to one-dimensional settings, strong bias increases dispersal. We next include Brownian rotation in the model and find that, in limit of high chemotactic sensitivity, the chemotactic drift is 64 % of the swimming Center for Ocean Life, National Institute of Aquatic Resources, Technical University of Denmark, Jaegersborg Allé 1, DK-2920 Charlottenlund, Denmark. E-mail: uht@aqua.dtu.dk 2 Uffe Høgsbro Thygesen velocity, independent of the magnitude of the Brownian rotation. We finally derive characteristic time scales of the motion that can be used to assess if the diffusion limit is justified in a given situation. The proposed technique for obtaining diffusion approximations is conceptually and computationally simple, and applicable also when statistics of the motion is obtained empirically or through Monte Carlo simulation of the motion.