Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation
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Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation P. Aceves-Sanchez1 · P. Degond2 · E. E. Keaveny2 · A. Manhart3 S. Merino-Aceituno4,5 · D. Peurichard6
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Received: 24 April 2020 / Accepted: 8 September 2020 / Published online: 25 September 2020 © The Author(s) 2020
Abstract We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of nonlinear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive. Keywords Self-propelled particles · Hydrodynamic limit · Pattern formation · Stability analysis · Gradient flow · Non-local interactions Mathematics Subject Classification 35Q70 · 82C05 · 82C22 · 82C70 · 92B25 · 92C35 · 76S05
1 Introduction This work is devoted to deriving and analysing a model of collectively moving selfpropelled particles that interact with a complex, heterogeneous environment. The field
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11538020-00805-z) contains supplementary material, which is available to authorized users.
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A. Manhart [email protected]
Extended author information available on the last page of the article
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of collective dynamics studies what happens when a large number of agents, which can be animals, people, micro-organisms, crystals, etc., interact with each other. A particular focus is the emergence of large-scale order or patterns. Famous examples include global alignment in crystals (de Gennes and Prost 1993), lane formation for people (Feliciani and Nishinari 2016), waves and aggregation in bacteria (Shimkets 1990; Ben-Jacob et al. 2000), milling in schools of fish (Shaw 1978) or swarming in birds (Cavagna et al. 2010). All these examples have in common that local, small-scale interaction rules between individuals lead to global, large-scale patterns. These patterns are typically hard or impossible to predict from the local interaction rules; hence, their understanding requires the use of either extensive simulations or mathematical analysis. Combining Collective Dynamics and Environmental Effects In many systems, one also needs to take into account the environment to be able to explain observed patterns in collective phenom
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