Nonlinear stability of flock solutions in second-order swarming models

In this paper we consider interacting particle systems which are frequently used to model collective behavior in animal swarms and other applications. We study the stability of orientationally aligned formations called flock solutions, one of the typical patterns emerging from such dynamics. We provide an analysis showing that the nonlinear stability of flocks in second-order models entirely depends on the linear stability of the first-order aggregation equation. Flocks are shown to be nonlinearly stable as a family of states under reasonable assumptions on the interaction potential. Furthermore, we numerically verify that commonly used potentials satisfy these hypotheses and investigate the nonlinear stability of flocks by an extensive case-study of uniform perturbations.Comment: 22 pages, 1 figure, 1 tabl

Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernels are constructed by inverting Fredholm integral operators or by employing certain integral identities. These solutions are expected to be the global energy stable equilibria and to characterize the generic behaviors of stationary solutions for more general interactions

On global minimizers of repulsive-attractive power-law interaction energies

We consider the minimisation of power-law repulsive-attractive interaction energies which occur in many biological and physical situations. We show existence of global minimizers in the discrete setting and get bounds for their supports independently of the number of Dirac Deltas in certain range of exponents. These global discrete minimizers correspond to the stable spatial profiles of flock patterns in swarming models. Global minimizers of the continuum problem are obtained by compactness. We also illustrate our results through numerical simulations.Comment: 14 pages, 2 figure

Collective behavior of animals: swarming and complex patterns

En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo que nos referimos usando la voz inglesa swarming. Estos modelos se basan en EDOs (ecuaciones diferenciales ordinarias) y muestran un comportamiento asintótico complejo y rico en patrones, que mostramos numéricamente. Además, comentamos cómo se conectan estos modelos de partículas con las ecuaciones en derivadas parciales para describir la evolución de densidades de individuos de forma continua. Las cuestiones matemáticas relacionadas con la estabilidad de de estos modelos de EDP's (ecuaciones en derivadas parciales) despiertan gran interés en la investigación en biología matemáticaIn this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology researc

Single to double mill small noise transition via semi-Lagrangian finite volume methods

We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse this fact accurately, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in terms of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space

Existence of Compactly Supported Global Minimisers for the Interaction Energy

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun

Explicit flock solutions for Quasi-Morse potentials

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.Comment: 26 page