Anomalous Transport with Intrinsic Biophysical Properties: Levy Walks and Networks

Abstract

This journal format thesis is an exploration of the various aspects pertaining to anomalous random walks; so-called for the differing trajectories (or other aspects) of the motion which set it apart from Brownian motion. Each question posed in this exploration is addressed in a different paper. Overall, the thesis addresses questions regarding the velocity of the walkers, the time scales over which anomalous behaviour occurs, and how the space across which the walkers move affects the final results. The questions have relevance in fields spanning intracellular transport, epidemiology, and robotics, to name only a few. A key feature of the thesis is the close link to empirical enquiries in the endeavour to construct a more realistic description of random walks. The aim is for the derived results to be more easily applicable to experimental investigations by considering intrinsic properties assigned to the random walkers. The first paper addresses the velocity of a group of random walkers with anomalous behaviour, via the analysis of the front propagation velocity. When the population of random walkers is mortal (can grow or decline), the 'death' rate of the walkers is crucially important and tempers the front velocity. This is true regardless of the considered birth dynamics, so long as the population is sustained. This slowing down with increased mortality counteracts the persistence often seen in anomalous transport. The second paper addresses the issue of initial conditions for mortal walkers, and how sensitive the long-term properties of the system are to assumptions made regarding how the random walkers started moving. The effects of initial conditions vary with the motility patterns of the random walkers, though in all cases a plateau in motility is reached when the time scales approach the life expectancy of the walkers. We find that the front velocity is independent of our assumptions regarding initial conditions. We then shift the focus of the thesis from continuous to discrete space. In the third paper this is motivated by considering random walkers moving across a network. The front velocity is no longer a meaningful quantity, whereupon we instead investigate the times spent in each node. We present empirical evidence for the existence of nodes which dominate the movement across the network by effectively 'trapping' walkers. Those random walkers which leave the trapping nodes will keep returning. The fourth paper investigates how the competition between two nodes on a network affects the long-term distribution of the walker population if both nodes are trapping. We find that the trapping effects of a node can be tempered such that a smaller population is found in such 'traps', in accordance with principles like finite size restrictions. Finally, the fifth paper studies how trapping (or tempering thereof) can arise spontaneously if an individual walker's motion depends on the surrounding walkers. We find that both these effects can arise by introducing a mean-field population dependence of the future walker motility, resulting in self-organised tempering and aggregation in a trapping node. The effects of papers three and four can thus arise via self-organising criticality in moving from a non-trapping to a trapping state, or vice versa.

Date of Award	31 Dec 2018
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Sergei Fedotov (Supervisor) & Oliver Dorn (Supervisor)