Theodore Kolokolnikov

Swarm dynamics and equilibria for a nonlocal aggregation model

We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR 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 density ¯ ρ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which ¯ ρ is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

PREDICTING PATTERN FORMATION IN PARTICLE INTERACTIONS

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, co-dimension 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 nonlocal 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 self-assembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps

An optimization problem for the fundamental eigenvalue λ0 of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of radius e � 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 λ0 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 e, a two-term asymptotic expansion for λ0 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 xi ,f ori =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 λ0 .F or ac lass of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes λ0. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize λ0. 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.

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary. The reciprocal o...

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

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.

The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation

The stability of two different types of stripe solutions that occur for two different forms of the Gierer--Meinhardt (GM) activator-inhibitor model is analyzed in a rectangular domain. For the basic GM model with exponent set

A minimal model of predator-swarm interactions.

(p,q,r,s)

Phytoplankton depth profiles and their transitions near the critical sinking velocity.

, representing the powers of certain nonlinear terms in the reaction kinetics, a homoclinic stripe is constructed whereby the activator concentration localizes along the midline of the rectangular domain. In the semistrong regime, characterized by a global variation of the inhibitor concentration across the domain, instability bands with respect to transverse zigzag instabilities and spot-generating breakup instabilities of the homoclinic stripe are determined analytically. In the weak interaction regime, where both the inhibitor and activator concentrations are localized, the spectrum of the linearization of the homoclinic stripe is studied numerically with respect to both breakup and zigzag instabilities. For certain exponent sets near the existence threshold of this hom...

On ring-like solutions for the Gray-Scott model: existence, instability and self-replicating rings

We propose a minimal model of predator–swarm interactions which captures many of the essential dynamics observed in nature. Different outcomes are observed depending on the predator strength. For a ‘weak’ predator, the swarm is able to escape the predator completely. As the strength is increased, the predator is able to catch up with the swarm as a whole, but the individual prey is able to escape by ‘confusing’ the predator: the prey forms a ring with the predator at the centre. For higher predator strength, complex chasing dynamics are observed which can become chaotic. For even higher strength, the predator is able to successfully capture the prey. Our model is simple enough to be amenable to a full mathematical analysis, which is used to predict the shape of the swarm as well as the resulting predator–prey dynamics as a function of model parameters. We show that, as the predator strength is increased, there is a transition (owing to a Hopf bifurcation) from confusion state to chasing dynamics, and we compute the threshold analytically. Our analysis indicates that the swarming behaviour is not helpful in avoiding the predator, suggesting that there are other reasons why the species may swarm. The complex shape of the swarm in our model during the chasing dynamics is similar to the shape of a flock of sheep avoiding a shepherd.

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems

We consider a simple phytoplankton model introduced by Shigesada and Okubo which incorporates the sinking and self-shading effect of the phytoplankton. The amount of light the phytoplankton receives is assumed to be controlled by the density of the phytoplankton population above the given depth. We show the existence of non-homogeneous solutions for any water depth and study their profiles and stability. Depending on the sinking rate of the phytoplankton, light intensity and water depth, the plankton can concentrate either near the surface, at the bottom of the water column, or both, resulting in a “double-peak” profile. As the buoyancy passes a certain critical threshold, a sudden change in the phytoplankton profile occurs. We quantify this transition using asymptotic techniques. In all cases we show that the profile is locally stable. This generalizes the results of Shigesada and Okubo where infinite depth was considered.