Nonlocal macroscopic limits of kinetic equations in biological and robotic systems

Abstract

This thesis addresses the derivation, analysis and numerical analysis of nonlocal partial differential equations from individual movement. More specifically, my main results concern how fractional diffusion and swarming behaviour arise as a limit of microscopic kinetic models for interacting particles. Applications of these results to biology, swarm robotic systems and networks are discussed. In the following I summarize the content of this thesis, which is motivated by using the analysis of PDEs to shed light on macroscopic behaviour in complex systems. Nonlocal diffusion from microscopic movement We assumed the motion of the individuals, in the case of chemotaxis, follows a velocity jump process, characterized by long runs according to an approximate Lévy distribution, interrupted by instantaneous reorientations. From the resulting kinetic equation obtained from this microscopic movement, and using a perturbation argument, we derived nonlocal Patlak-Keller-Segel equations in the appropriate limit. The resulting system involves fractional Laplacians that describe the nonlocal movement of the individuals. Subsequent work studied more complex search strategies. Motivated by recent experimental results from T. Harris et al. [105] in the case of T cells migrating through chronically-infected brain tissues, we considered that the runs are further interrupted by long pauses according to a Lévy distribution. No directional motion is present in this case. We obtained two coupled kinetic equations for the moving and resting populations. Solving the equation for the resting population introduces a nonlocal delay in time, consistent with those observed in experiments. The simple structure of this equation allows analytic insights not directly visible from the microscopic model. In particular, we found an explicit fundamental solution in R n , allowing us to study hitting times, showing the effect of the delay and the advantages of a Lévy search strategy over Brownian motion. Interacting particles with Lévy strategies and alignment We considerd Lévy robotic systems which combine superdiffusive random movement with emergent collective behaviour from local communication and alignment. We de rived a fractional PDE from the movement strategies of the individual robots, intro ducing long range interactions and alignment into the analysis. The resulting kinetic model is studied at short and long time scales. Applications we study include targeting efficiency and optimal search strategies. We showed that the system allows efficient parameter studies for a search problem, ad dressing basic questions such as the optimal number of robots needed to cover an area. Validation against concrete robotic simulations with e-puck robots are also included in collaboration with computer scientists from the Edinburgh Centre for Robotics. Superdiffusion in complex systems: metaplex networks In complex, non convex geometries, the fractional Laplacian, based on the Euclidean distance, is not the physically relevant operator to describe superdiffusive behaviour. Based on an approach used in complex systems to study superdiffusion in networks, where particles are allowed to hop to non-nearest neighbours, we propose to study diffusion using a network of subdomains, corresponding to the nodes of a graph. We introduced a general framework for diffusion in networks with internal structure in the nodes: metaplex networks. We illustrate its use in a toy model and in real-world networks. The results shed light on the rich and substantially different nature of the dynamics of metaplexes and the interplay of their internal and external structure.