We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes, and generally display similar dynamics. The same is not true of the corresponding PDE models. Indeed, while both PDE models have multiple stationary states, for the globally scaled PDE such (space-homogeneous) equilibria are unstable for certain parameter regimes, with the instability leading to traveling wave solutions, while they are always stable for the locally scaled one, which never produces traveling waves. This observation is based on a careful numerical study of the model, supported by further analysis.
The phenomenon of collective navigation has received considerable interest in recent years. A common line of thinking, backed by theoretical studies, is that collective navigation can improve navigation efficiency through the 'many-wrongs' principle, whereby individual error is reduced by comparing the headings of neighbours. When navigation takes place in a flowing environment, each individual's trajectory is influenced by drift. Accordingly a potential discrepancy emerges between an individual's intended heading and its actual heading. In this study we develop a theoretical model to explore whether collective navigation benefits are altered according to the form of heading information transmitted between neighbours. Navigation based on each individual's intended heading is found to confer robust advantages across a wide spectrum of flows, via both a marked improvement in migration times and a capacity for a group to overcome flows unnavigable by solitary individuals. Navigation based on individual's actual headings is far less effective, only offering an improvement under highly favourable currents. For many currents, sharing actual heading information leads to journey times that exceed those of individual navigators, negating the suggested benefits proposed by the 'many-wrongs' principle.
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