Boston University / Center for BioDynamics / Research / Pattern Formation Research


Pattern Formation Research

Contents

  1. Control of spatiotemporal waves and patterns
  2. Pattern formation in bacterial colonies
  3. Pattern formation in the Gray-Scott model
  4. Annhilation reaction
  5. Patterns in economics and social behavior


1. Control of spatiotemporal waves and patterns

The appearance of spiral waves in excitable and oscillatory media is often an undesirable effect, leading to unpredictable consequences for many applications. For instance, a single spiral wave in cardiac tissue has been identified as a likely cause of monomorphic ventricular tachycardia. Accordingly, there is a need to develop effective methods to control and annihilate spiral waves. Recently, Grigory Osipov, Boris Shulgin and Jim Collins have proposed mechanisms for controlling the movement and suppression of spiral waves in discrete excitable media. They showed that the controlled drift and subsequent annihilation of a spiral wave can be achieved through the combination of two factors: the introduction of small spatial inhomogeneities in the medium and the interaction of the wave with the boundaries of the medium. For this scheme, the inhomogeneities can be introduced in the spatial distribution of the relaxation parameters or in the coupling coefficient of the slow variables of the partial cells making up the medium.

Related Publications
 

G.V. Osipov, B.V. Shulgin and J.J. Collins. (1998) "Controlled movement and suppression of spiral waves in excitable media", Phys. Rev. E, 58: 6955-6958.

Osipov and Collins have also proposed a general mechanism for suppressing non-steady state motions -- propagating pulses, spiral waves, spiral-waves chaos -- in excitable media. This approach is based on two points: (1) excitable media are multistable, and (2) traveling waves in excitable media can be separated into fast and slow motions, which can be considered independently. Osipov and Collins showed that weak impulses can be used to change the values of the slow variable at the front and back of a traveling wave, which leads to wavefront and waveback velocities that are different from each other. This effect can destabilize the traveling wave, resulting in a transition to the rest state.

Related Publications
 

G.V. Osipov and J.J. Collins. "Using weak impulses to suppress traveling waves in excitable media", Phys. Rev. E (In Press).

2. Pattern formation in bacterial colonies

Medvedev, Kaper and Kopell analyzed the dynamics associated with some bacterial colonies (e.g. Proteus mirabilis) whose colony front undergoes periodic transitions between expansion and stopping; during the latter, there is consolidation of the colony behind the front. The work uses nonlinear parabolic partial differential equations with density- dependent diffusivity. It concentrates on the switching mechanisms, especially the switch from consolidation to expansion. The paper shows that the transition can be understood from the dynamics of the diffusivity, and that the transition moment occurs when some wave in diffusivity dynamics occuring inside the colony catches up to the colony boundary.

Related Publications
 

G. Medvedev, T. Kaper and N. Kopell, "A reaction-diffusion system with periodic front dynamics", Submitted. Also CBD Technical Report # 12.

3. Pattern formation in the Gray-Scott model

The irreversible 1-D Gray Scott model from chemistry consists of a pair of reaction-diffusion equations with cubic autocatalysis. This model contains a wide variety of stationary and time-dependent singular solutions, including self-replicating pulse solutions, as discovered by John Pearson and coworkers. T. Kaper and D. Morgan, along with collaborators Arjen Doelman (University of Amsterdam), Wiktor Eckhaus (Utrecht University), Robert A. Gardner (University of Massachusetts, Amherst), and Paul Zegeling (Utrecht University), briefly survey existence and stability results for the these solutions, beginning with the stationary solutions.

Using methods of geometric singular perturbation theory, they show that there exist various localized and large-amplitude single-pulse stationary solutions, for which the pulse height increases and the pulse width decreases as the activator diffusivity vanishes. These one-pulse solutions correspond to singularly perturbed homoclinic orbits with slow and fast segments in the fourth-order ODE system that governs stationary solutions. They also construct fast-slow multi-circuit homoclinic orbits and multiple pulse periodic orbits, which correspond to stationary, localized multi-pulse homoclinics and spatially-periodic multi-pulse patterns in the PDE. The stationary periodic patterns are observed to be the attractors in the pulse-replication regime in the full PDE.

Related Publications
 

A. Doelman, T.J. Kaper, and P. Zegeling, "Pattern formation in the one-dimensional Gray-Scott model", Nonlinearity, 10, 523-563.

The same group has done a complete stability analysis of the stationary one-pulse homoclinic and spatially-periodic patterns. They explicitly compute the Hopf bifurcation curves in the critical scaling through which these patterns gain stability. These bifurcation curves are ordered according to the length of the period, and as the activator decay rate decreases, the one-pulse homoclinic is the first to gain stability and then the periodic solutions with shorter and shorter periods do so. This stability analysis is carried out by converting the full fourth-order eigenvalue problem to a second-order nonlocal eigenvalue problem, derived by imposing matching the jump discontinuities in the first derivative of the first eigenfunction component across the pulses. All of the formal matching results for the single-pulse homoclinic solutions are corroborated by an Evans functions analysis. In addition, they show using topological shooting that these solutions successively disappear in a sequence of bifurcation curves as B is further decreased, and these disappearance curves agree well with the numerically observed splitting boundaries in the scaling regime where the self-replication occurs.

Related Publications
 

A. Doelman, R. A. Gardner and T.J. Kaper, "Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotics approach", Physica D, (1998) 122, 1-36.

A. Doelman, R. A. Gardner and T.J. Kaper, "A stability index analysis of 1-D patterns of the Gray-Scott model", Submitted.

Dave Morgan has shown that the Gray-Scott model possess a large family of spatially periodic patterns. They are born in a Turing/ Ginzburg-Landau bifurcation, and he and collaborators have analytically continued them from the regime in which their periods are O(1) to the regime in which their periods are asymptotically large, as measured with respect to the activator diffusivity. Moreover, it is shown that, inside each existence interval, an Eckhaus subband of these periodic states are stable. Using results of G. Schneider, the full nonlinear stability is also shown. Finally, these subbands are continued numerically into the regime in which the inhibitor feed rate is asymptotically small, and an entire Busse balloon of stable periodic states is uncovered. These periodic states attract a wide variety of initial data, including large-amplitude fronts propagating into a linearly stable homogeneous state, small-amplitude fronts moving into a linearly unstable state, and oscillatory data.

Related Publications
 

D. Morgan, A. Doelman, and T.J. Kaper, "Spatially periodic patterns in the 1D Gray-Scott model", (1998) In preparation, to appear as a CBD Technical Report.

Finally, this group has extended the existence and stability results to a much larger regime in the parameter space. Most importantly, they find classes of slowly modulated two pulse solutions, whose dynamics are essential to understanding the dynamical onset of the splitting bifurcations. These slowly modulated two pulse solutions consist of left and right moving pulses, symmetrically displaced about the origin, whose (slow) velocity decreases slowly in time. Explicit ODEs for the slowly-varying wave speeds are derived for the various regimes in parameter space. The existence of these solutions is demonstrated both via classical matched asymptotics and via the same geometric singular perturbation theory methods used above, and the study of their linear stability is accomplished via an extension of the NLEP theory developed in a previous publication. The evolution of these slowly-modulated two pulse solutions leads to new insight into the rich dynamics exhibited during the actual splitting (or "self-replication"). Moreover, these results are finding application to a broad class of other coupled reaction-diffusion equations, as we are presently showing.

Related Publications
 

A. Doelman, W. Eckhaus and T.J. Kaper, "Slowly-modulated two pulse solutions and the onset of splitting bifurcations in the 1-D Gray-Scott model", (1998) In preparation, to appear as a CBD Technical Report.

A. Doelman, R. A. Gardner and T.J. Kaper, "Stability of large-amplitude pulse solutions in coupled reaction-diffusion equations", (1998) In preparation.

4. Annihilation reaction

S. Redner and collaborators have determined the mechanism that leads to spatial organization in the two-species annihilation reaction A+B--->0, and have used this insight to understand how the interplay between external input, basic aspects of the particle transport mechanism, and the microscopic reaction itself, the overall reaction kinetics and the growth of spatial heterogeneities.

Two-species annihilation
Spatial organization of reactants A (red) and B (blue) in two-species annihilation.

Related Publications
 

"Spatial Structure in Diffusion-Limited Two-Species Annihilation", F. Leyvraz and S. Redner, Phys. Rev. A 46, 3132, (1992).

"Kinetics and Spatial Organization in Competitive Reactions", S. Redner and F. Leyvraz, in Fractals and Disordered Systems, Vol. II, eds. A. Bunde and S. Havlin (Springer-Verlag 1993).

"Kinetics of A+B--->0 with Driven Diffusive Motion", I. Ispolatov, P. L. Krapivsky, and S. Redner, Phys. Rev. E 52, 2540, (1995).

"Heterogeneous Catalysis on a Disordered Surface", L. Frachebourg, P. L. Krapivsky, and S. Redner, Phys. Rev. Lett. 75, 2891, (1995).

5. Patterns in economics and social behavior

The dynamics of continuously growing vicious civilizations (domains), which engage in "war" whenever two domains meet, has been investigated. In a war, the smaller domain disappears, while the larger domain shrinks in size by a fraction f of the casualties of the loser. In the fair war regime of 1/2 < f < 1, steady egalitarian competition between comparable size domains arises. For unfair wars, 0 < f < 1/2, superpowers ultimately dominate via a power-law coarsening with non-universal f-dependent exponents.

Related Publications
  "On War: The Dynamics of Vicious Civilizations", I. Ispolatov, P. L. Krapivsky, and S. Redner, Phys. Rev. E 54, 1274, (1996).

A model for the evolution of the wealth distribution in an economically interacting population has been introduced, in which a specified amount of assets are exchanged between two interacting individuals. For "random" exchange, either individual is equally likely to gain in a trade, while in "greedy" exchange, the richer gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like wealth distribution. Multiplicative processes have also been investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises, in agreement with individual wealth distributions in developed economies.

Related Publications
  "Wealth Distribution Asset Exchange Models", S. Ispolatov, P. L. Krapivsky, and S. Redner, Eur. Phys. J. B 2, 267, (1998).

Numerical data for the distribution of scientific citations have been examined for: (i) papers published in 1981 in journals catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank is consistent with a power-law dependence for leading rank papers, with exponent close to -1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay N(x)~ x^-3. Work is now being pursued to test the universality of this result for related popularity measures, such as the distribution of book sales or the distribtion of movie ticket sales.

Related Publications
  "How Popular is Your Paper? An Empirical Study of the Citation Distribution", S. Redner, Eur. Phys. J. B 4, 131, (1998).


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