Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

Abstract

Conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles. We address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities. Currently, no existence theory for such potentials is available. We develop and compare two complementary solution methods, a continuous pseudoinverse method and a discrete stochastic lattice approach, and formally show a connection between the two. Interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive-repulsive potential are determined. For a swarming Morse potential, characteristic slow-fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches. The discrete approach is found to be remarkably robust to modifications in movement rules, related to the potential function. The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations.

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

“…The form of the kernel (.) clearly plays a central role in model predictions, but our kernels (O2) and (O3) are widely used in applications to swarming; for detailed discussions of predictions from other kernels, see for example Carrillo et al (2014), Hackett-Jones et al (2012). Also, although diffusion is omitted in most models for swarming, its inclusion has been considered by some authors (e.g.…”

A review of swarm robotics tasks

“…For example, aggregation obtained with artificial physics methods [14,15,16,17,18,19] generally does not lend itself to a precise identification of discrete aggregates. In [38], where robots are able to sense the population density in their neighborhood, the aggregation dynamics is obtained with robots increasing and decreasing the randomness of their movement based on the local density: while robots in low-density areas move in a highly random fashion, in high-density areas the stochastic component of robot movement is lower, leading to robots "settling down" in regions with a high number of neighbors.…”

“…Usually, the force is attractive if the distance between two agents is greater than a target value, and is repulsive for smaller distance values. 8 Although artificial physics has been successfully used to formally characterize the aggregation dynamics of swarms of autonomous agents [14,15,16,17,18], its practical implementation in artificial systems with real robots imposes some requirements on robot sensing capabilities which may not be cost-effective. Robots with local sensing abilities are typically characterized by a limited range of visibility, which influences the capability to perceive other robots in the environment; determination of the relative orientation of neighbors may be affected by a high error, such as when infrared technology is used; mechanical constraints usually determine a saturation effect in robot actuators, effectively limiting the amplitude of the control inputs which regulate robot motion.…”

“…A non-local advection type is considered, and steady state solutions for the equation are derived for different kernel functions (called interaction potentials in [15]). The solutions show the formation of various aggregation patterns starting from a uniform agent distribution.…”

“…In [15], the spatial patterns formed by agents are studied via a conservation law without a diffusion term: the advection equation. A non-local advection type is considered, and steady state solutions for the equation are derived for different kernel functions (called interaction potentials in [15]).…”