Stability Analysis of Flock and Mill Rings for Second Order Models in Swarming

Abstract

We study the linear stability of flock and mill ring solutions of two individual based models for biological swarming. The individuals interact via a nonlocal interaction potential that is repulsive in the short range and attractive in the long range. We relate the instability of the flock rings with the instability of the ring solution of the first order model. We observe that repulsive-attractive interactions lead to new configurations for the flock rings such as clustering and fattening formation. Finally, we numerically explore mill patterns arising from this kind of interactions together with the asymptotic speed of the system.

Ring states in swarmalator systems

“…Perhaps using the more realistic Winfree model [4], which has richer phase dynamics, would lead to more interesting swarmalator phenomena when K is positive. Here, we develop the stability theory for ring states of the swarmalator model defined in the main text, using techniques similar to those developed in [42,69,75,76]. It is convenient to use complex notation to describe the ring phase wave state.…”

Kinetic Models for Topological Nearest-Neighbor Interactions

“…. , x N are drawn randomly and independently with probability ρ (1) N . Using the combinatorial approach of [10] to evaluate this probability, we obtain…”

“…Proposition 6 (Case λ(N ) = 1). Assume that (f (1) N ) N ∈N and (ρ (1) N ) N ∈N converge toward smooth functions f and ρ respectively. If the convergence of (f…”

Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

“…Many nonlocal interaction equations, such as aggregation equations [10,11,16,17,32,41,43] and related models for biological swarming [18,31,44,46,53,54], support stationary solutions where the entire particle density concentrates on a co-dimension one manifold [2,3,4,9,35,36,42,52,55,56,57]. The nonlocal aggregation equation in R d , (1.1) ρ t + ∇ · (ρu) = 0, u(x, t) = R d g 1 2 |x − y| 2 (x − y)ρ(y, t) dy, serves as a canonical example of this type of behavior.…”

“…We then apply the theory to some simplified nonlocal equations. tions [10, 11, 16, 17, 32, 41, 43] and related models for biological swarming [18,31,44,46,53,54], support stationary solutions where the entire particle density concentrates on a co-dimension one manifold [2,3,4,9,35,36,42,52,55,56,57]. The nonlocal aggregation equation in R d ,…”