Explicit Flock Solutions for Quasi-Morse potentials

Abstract

We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, that are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness is proven for three space dimension, whilst existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.

26 pages

References in corpus (6)

• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• A model for rolling swarms of locusts
• Dimensionality of Local Minimizers of the Interaction Energy
• Equilibria of biological aggregations with nonlocal repulsive-attractive interactions
• Nonlinear stability of flock solutions in second-order swarming models
• A new interaction potential for swarming models

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