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We consider the aggregation equation ρ t − ∇ • (ρ∇K * ρ) = 0 in R n , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of R n and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric densityρ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for whichρ is the steady state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction-diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε → 0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j = 1, . . . , K at unknown spot locations xj ∈ Ω for j = 1, . . . , K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/ log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates Sj and xj for j = 1, . . . , K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green's function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths Sj , for j = 1, . . . , K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.

Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to self-assembly of nanoparticles. In the presence of long-range attraction and short-range repulsion, such systems can exhibit bound states. We use linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the well-posedness of the ring. In addition, weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry. Many of these patterns have been observed in nature, although a general theory has been lacking, in particular how small changes to the interaction potential can lead to large changes in the self-organized state.

An optimization problem for the fundamental eigenvalue λ 0 of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of radius ε 1, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue λ 0 is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains N small traps. For small hole radii ε, a two-term asymptotic expansion for λ 0 is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations x i , for i = 1,. .. , N, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize λ 0. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes λ 0. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize λ 0. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer-Meinhardt reactiondiffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg-Landau theory of superconductivity.

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Large systems of particles interacting pairwise in d-dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, codimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a non-local linear stability analysis for particles uniformly distributed on a d − 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in selfassembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

Abstract. In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We review and expand upon recent developments of this class of problems as well as present new results. As in previous work, we leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence of the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the "designer potential" problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model. IntroductionMathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive. While some models include anisotropy in this communication (e.g. an organism's eyes may have a limited field of vision) simplified isotropic interactions have been shown to capture many important swarming behaviors including milling [20,10]. More recently it has been shown [17,38,37] that the competition between the desire to avoid collisions and the desire to remain in a cohesive swarm can sometimes result in simple radially symmetric patterns such as rings, annuli and uniform circular patches and other times result in exceedingly complex patterns. Moreover how modelers select the strength and form of the repulsion near the origin