Refining self-propelled particle models for collective behaviour

Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion

Robustness of Cucker-Smale flocking model

Consider a system of autonomous interacting agents moving in space, adjusting each own velocity as a weighted mean of the relative velocities of the other agents. In order to test the robustness of the model, we assume that each pair of agents, at each time step, can fail to connect with certain probability, the failure rate. This is a modification of the (deterministic) Flocking model introduced by Cucker and Smale in Emergent behavior in flocks, IEEE Trans. on Autom. Control, 2007, 52 (May) pp. 852-862. We prove that, if this random failures are independent in time and space, and have linear or sub-linear distance dependent rate of decay, the characteristic behavior of flocking exhibited by the original deterministic model, also holds true under random failures, for all failure rates.Comment: 9 pages, 3 figure

Particle based gPC methods for mean-field models of swarming with uncertainty

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.Comment: Communications in Computational Physics, to appea