Asymptotic Dynamics of Attractive-Repulsive Swarms

Abstract

We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady-state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more δ-functions.

Finite-time blow-up of L∞-weak solutions of an aggregation equation

“…Another open area is blow-up for initial data which is not radially symmetric. Blow-up for a large class of initial data was proven in one spatial dimension by comparison with a Burgers-like dynamics in [4]; it was also studied, both analytically and computationally, in [17]. However, nothing is known for higher dimensions.…”

“…Applying this differential inequality, along with an appropriate continuation theorem on solutions of (1.1), it follows that whenever the initial data u 0 satisfies u 0 (x) ≥ D C at some point x the solution exhibits finite time blow up in the L ∞ norm. Analytic and computational aspects of blow-up of (1.1) in one dimension were also studied in [17]. As the space dimension increases, ∆(e −|x| ) becomes progressively less singular; in dimensions two and higher ∆(e −|x| ) does not contain a Dirac delta function, its singular part is of the form 1 |x| .…”

Ring patterns and their bifurcations in a nonlocal model of biological swarms

“…Mathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2].…”

Partial differential equation models in the socio-economic sciences

“…A particularly interesting instance is the coupling of aggregation and repulsion, where the first one is usually on a large range (to stay in the overall group) and the second acts on a long range (to keep some space needed for each agent). A variety of novel developments has been motivated by such models [24][25][26][27], in particular the large-time behaviour and analysis of stationary patterns have received considerable attention [28][29][30]. In this issue, Carrillo et al [31] analyse such steady states in the case when both the repulsive as well as the aggregative force are modelled by integral operators.…”