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, which 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 are proven for three space dimensions, while 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.
Highlights
Self-organization, complex pattern formation, and rich dynamic structures are common features of collective motion of individuals
We first turn to the existence theory of flock profiles in three space dimensions, as in this case the Bessel functions in the potential as well as in all subsequent computations reduce to trigonometric functions
We analyzed the solvability of convolution equations that describe particular solutions in aggregation or self-propelled interacting particle models equipped with radially symmetric interaction potentials
Summary
Self-organization, complex pattern formation, and rich dynamic structures are common features of collective motion of individuals. The self-propulsion term αvi − β|vi|2vi with the Rayleigh-type dissipation can be generalized to the form f(|vi|)vi for some function f : [0, ∞) → R, such that f(0) > 0 and f(υ) becomes negative when υ is large enough In both models, the potential W is assumed to be repulsive at short range (U(r) decreases for small r > 0) and attractive at long range (U(r) increases for r large enough). Flock profiles play an important role on the dynamics of (1.2) since they form a stable family of attracting solutions as shown in [10] for general potentials under suitable conditions.
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