Stable moving patterns in the 1-D and 2-D Gray-Scott Reaction-Diffusion System  

I presented some of the material from my 2009 paper, and a few of the more interesting videos from the xmorphia exhibit to the Mathematics department at Rutgers University as part of their Fall 2010 series in Mathematical Physics.

Here are the slides in two formats, Powerpoint and Adobe PDF:

Stable moving patterns (Rutgers talk) -- slides (PPS) (13.1 MB)

Stable moving patterns (Rutgers talk) -- slides (PDF) (12.6 MB)

Both contain extensive graphics and links to the videos I prepared for the talk (which are on YouTube). Some software (e.g. PDF plug-in or a PPS viewer) might not be able to jump to the links directly, but you can cut and paste the URL text (example: "youtube.com/v/kXDTqqgrYCg") into your web browser's address bar. (The same links are also in the slide text below)

In addition to the videos linked from the slides, the participants also saw the videos from the following pages in my main Gray-Scott web exhibit:

k=0.055, F=0.098
k=0.055, F=0.098
(showed briefly while looking for a better example)

k=0.055, F=0.102
k=0.055, F=0.102
(showed briefly while looking for a better example)

k=0.057, F=0.098
k=0.057, F=0.098
(persistent stripes that form a linked network of polygonal cells, with gradual evolution into fewer and larger cells reminiscent of soap bubbles)

k=0.057, F=0.014
k=0.057, F=0.014
(showed briefly while looking for a better example)

k=0.055, F=0.014
k=0.055, F=0.014
(low-U high-V spots that move a bit, then "split" into two spots which then move and split again, etc.)


What follows is the text of each slide with links to the relevant images and videos. This is intended for readers with a slower connection, or who prefer to go to the YouTube videos directly from the browser without the need to download the slides.


Stable moving patterns in the 1-D and 2-D

Gray-Scott Reaction-Diffusion System

figure: two spots far
figure: two spots far
  figure: U-shaped pattern
figure: U-shaped pattern
  figure: 4 spots zigzag
figure: 4 spots zigzag

figure: three spots 1-D
figure: three spots 1-D

figure: five spots
figure: five spots
  figure: ''landspeeder''
figure: "landspeeder"

Robert Munafo

Rutgers Mathematical Physics Seminar Series

2010 Dec 9, 2 PM — Hill 705



slide 1


Contents (outline of this talk)

slide 2


Brief History: Initial Motivation

figure: 1994 Williams Caltech website
figure: 1994 Williams Caltech website

Roy Williams "Xmorphia" web exhibit (Caltech, 1994)

figure: PDE formulas
figure: PDE formulas

Pearson "Complex patterns in a simple system" (Science 261 1993) (illust. next slide)

Lee et al. "Experimental observation of self-replicating spots in a reaction-diffusion system" (Nature 369 1994)

(showing part of fig. 1 from Lee et al., and their form of the PDE formulas)

Ferrocyanide-iodate-sulphite reaction in gel reactor

( K4Fe(CN)6.3H2O , NaIO3 , Na2SO3 , H2SO4 , NaOH , bromothymol blue )



slide 3


Brief Intro: Parameter Space

Parameter space and four pattern examples, from Pearson (ibid. 1993)

(showing fig. 3 and part of fig. 2 from Pearson (ibid. 1993))

Coloring imitates bromothymol blue pH indicator as in Lee et al. "Pattern formation by interacting chemical fronts" (Science 261 1993): blue = low U; yellow = intermediate; red = high U

(showing fig. 3 from Miyazawa et al.)

Parameter space visualizations for several reaction- diffusion models, from fig. 3, Miyazawa et al. "Blending of animal colour patterns by hybridization" (Nature Comm. 1(6) 2010)


slide 4


Brief History: 2009 Website Project

figure 1: parameter map image from my website
figure 1: parameter map image from my website

Pearson: 34 sites; 0.045 Williams: 45 sites; 0.032 Munafo: 150+ sites; 0.03

False color images: purple = lowest U, red = highest U, pastels = dU/dt > 0


slide 5


New pattern types (2009)

negative soap-bubbles image
negative soap-bubbles image

k=0.057, F=0.09

positive soap-bubbles image
positive soap-bubbles image

k=0.059, F=0.09

exotic diversity image
exotic diversity image

k=0.061, F=0.062

self-sustaining spirals image
self-sustaining spirals image

k=0.045, F=0.014

(The above 4 images have arrows pointing to a copy of the parameter map image from the previous slide, and showing the locations of the k and F values for each image)

(Xmorphia parameter map image again)

Munafo (2009) www.mrob.com/pub/comp/xmorphia


slide 6


Inherent Instability


Turing-F600-k620.mp4watch (YouTube)youtube.com/v/kXDTqqgrYCg

k=0.062, F=0.06

Periodic boundary conditions; size 3.35w × 2.33h

Each second is 1102 dtu

Initial pattern of low-level random noise (0.4559<U<0.4562; 0.2674<V<0.2676)

Final values: 0.35<U<0.90; 0.00<V<0.36 (approx.)

(Contrast-enhanced images; lighter = higher U)

figure: parameter space with Turing and exotic regions
figure: parameter space with Turing and exotic regions

Gray-Scott parameter values for Turing instability (green; width to scale) and stable moving patterns (blue; width exaggerated). The video is at X


slide 7


How can we trust these results?

- Eliminate suspicion of numerical error

- Quantify the sensitivity of these pattern phenomena to precision, randomness, reaction parameters and other environmental conditions


slide 8


Supplement to Slide 8

My diagram illustrating the effects of "precision", "noise" and "randomness" on the robustness of pattern-forming phenomena

  ^ Q | R L | | ^ | : | : | : | : | M MO:M OM P ---E--------:-----------> | N

This is an abstract (qualitative conceptual) representation of two different types of phenomena that affect pattern-forming systems, viewed as "dimensions" or "parameters" of the laws of physics.

Q represents the amount of quantization that results from the (finite, not infinitesimal) size of atoms. This determines the relative strength of brownian motion and other similar phenomena.

N represents the amount of aberration resulting from numerical methods — roundoff error, the choice of a certain finite grid size to represent a continuous system, etc.

The other letters represent the location of certain pattern-forming systems that have been studied in published work:

P : Pearson (ibid. 1993)

M : Munafo (my various Gray-Scott simulation experiments from 1994 to 2009)

O : Other published Gray-Scott simulations over the period 1994-2009

E : Exact systems (mathematical analysis and proofs)

R : The real world

L : Stochastic simulations, such as certain papers by Prof. Joel Lebowitz of the Rutgers mathematics department.

The vertical dotted line represents one author (Munteanu et al. "Pattern formation in noisy self-replicating spots" Int'l. Jrl. of Bifurcation and Chaos 12(16) 2006) who added various amounts of "Gaussian noise" to a simulation and observed a substantial qualitative change in pattern type (from stripes to spots) as one goes from zero noise up to a level one might expect in real-world experiments, in Gray-Scott simulations with k=0.0655, F=0.05 and all other details as in Pearson (ibid. 1993). I have reproduced these results in my simulations.

Note that these "systems" exist in different types of "reality" : the physical reality of the world around us, the "virtual reality" inside a computer, and the "abstract" existence of mathematical truth. For this and other reasons, there are many more than just two "dimensions" at play and so the chart above is just an illustration of the concept.

I sought to address the general issue of numerical error ("M" systems as compared to "E") and ignoring statistical mechanics ("E" systems as compared to "R").


Verification Examples

figure: two spots (close)
figure: two spots (close)

figure: U-shaped pattern
figure: U-shaped pattern

Two spots maintain the same distance while the pair rotates toward a 45o alignment

U-shaped pattern moves at about 1 dlu per 62,000 dtu (dimensionless units of length, time)

measured value = true value + simulation error + measurement error

simulation error = f(stability, precision, grid spacing, ...)

CFL (Courant-Friedrichs-Lewy 1928) stability criterion (for the Laplacian term):

              C Δx2t

(A higher value means greater stability. Constant C depends on e.g. k and F for the Gray-Scott system)

slide 9


Verification Procedure

screen shot of the same test running in five different-sized grids
screen shot of the same test running in five different-sized grids

Δx Δt Δx2t pixels(typ.)
"std" 0.00699 0.5 9.78×10-5 128×128
"s1.4" 0.00495 0.177 1.38×10-4 180×180
"s2" 0.00350 0.0625 1.96×10-4 256×256
"s2.8" 0.00247 0.0221 2.77×10-4 360×360
"s4" 0.00175 0.00781 3.91×10-4 512×512

(amount of calculation increases by 4√2 each time: ratio of 1024 to 1 between "std" and "s4")

Expected Results:

(two-spot pattern): Movement is 100% spurious: measurements should tend towards zero

(U-shaped pattern): If real, velocity should clearly converge on a nonzero value


slide 10


Verification Examples — Results

figure: two spots (close)
figure: two spots (close)

Suspect rotating 2-spot phenomenon

cross-qtr. relative model max dU/dt interval velocity   std 8.97e-7 5.6e5 1.0 s1.4 4.57e-7 1.12e6 0.50 s2 2.31e-7 2.18e6 0.26 s2.8 1.158e-7 4.34e6 0.129 s4 5.80e-8 8.7e6 0.064

figure: U-shaped pattern
figure: U-shaped pattern

Velocity of U-shaped pattern

model dist pixels time velocity (meas.err)   std 0.55418 79.2 35076 1/63294(45) s1.4 0.59863 121.1 37342 1/62379(64) s2 0.57042 163.1 35456 1/62159(27) s2.8 0.56947 230.3 35258 1/61913(44) s4 0.56553 323.5 35009 1/61905(12)

Same tests using 4th order Runge-Kutta

model dist pixels time velocity (meas.err)   std 0.57937 82.9 36559 1/63101 s1.4 0.56053 113.4 35021 1/62479 same s2 0.56351 161.2 35008 1/62124 as s2.8 0.56517 228.6 35003 1/61934 above s4 0.56574 323.6 35002 1/61870

Limits on parameter k (when F=0.06) for stability of U-shaped pattern

model minimum maximum   std 0.0608833 0.0609829 s1.4 0.0608796 0.0609831 s2 0.0608777 0.0609831 s2.8 0.0608767 0.0609830 s4 0.0608762 did not test   measurement error in all values is +-1 in the last digit

slide 11


Immunity to Noise


U-noise-immunity.mp4watch (YouTube)youtube.com/v/_sir7yMLvIo

k=0.0609, F=0.06

Periodic boundary conditions; size 3.35w × 2.33h

1 second ≈ 1100 dtu

4 U-shaped patterns traveling "up"

Systematic noise perturbation applied once per 73.5 dtu

Amplitude of each noise event starts at 0.001 and doubles every 11,000 dtu (10 seconds in this movie)

At noise level 0.064, three patterns are destroyed; noise amplitude is then diminished to initial level

(False-color: yellow = high U; pastel = positive dU/dt )

slide 12


Symmetry-based Instability Tests


Daedalus-stability.mp4watch (YouTube)youtube.com/v/fWfsMVEeP5k

k=0.0609, F=0.06

Periodic boundary conditions; size 2.36w × 1.65h

Manually constructed initial pattern based on parts of naturally-evolved systems

1 second = 265 dtu

Two sets of three noise events; noise event amplitudes are 10-5, 0.001 and 0.1; pattern allowed to recover after each event

Coloring (left image) same as before

Coloring (images below): white = positive dU/dt, black = negative dU/dt, shades of gray when magnitude of dU/dt is less than about 5×10-13

figure: landspeeder (full color)
figure: landspeeder (full color)

figure: landspeeder (derivative only)
figure: landspeeder (derivative only)

A pattern with an instability that this test does not help reveal

slide 13


Logarithmic Timebase, Long Duration

Eight solitons in a 2-D system, k=0.067, F=0.046 :
t=78,125
t=78,125
;
t=1,250,000
t=1,250,000
;
t=2×107
t=2×107
;
t=3.2×108
t=3.2×108

figure 2: pairs of solitons in 2-D and 1-D
figure 2: pairs of solitons in 2-D and 1-D

(larger, upper part of figure) Pair of solitons in 2-D system, k=0.068, F=0.042
(smaller, lower part of figure) Pair of solitons in 1-D system, k=0.0615, F=0.04


slide 14


Discovery - Great Diversity


Original-F620-k610-fr159.mp4watch (YouTube)youtube.com/v/wFtXwFfrwWk

k=0.061, F=0.062

Periodic boundary conditions; size 3.35w × 2.33h

1 second ≈ 1190 dtu

Initial pattern of a few randomly placed spots of relatively high U on a "blue" background (secondary homogeneous state, approx. U=0.420, V=0.293)

(False-color: yellow = high U; pastel = positive dU/dt )

figure 1: normal soliton
figure 1: normal soliton

Ordinary spot soliton (k=0.067, F=0.062)

figure 2: negative soliton
figure 2: negative soliton

"negative soliton" (k=0.061, F=0.062)

In figures 1 and 2: White curve = U; Black curve = V; dotted line shows cross-section taken

("iota" image from figure 2 in Pearson (ibid. 1993))
"negative solitons" in Pearson (ibid. 1993)

"Pattern ι is time independent and was observed for only a single parameter value."

(parameters unpublished, probably k=0.06, F=0.05)


slide 15


Discovery - Moving U Pattern


U-discovery-F620-k609-fr521.mp4watch (YouTube)youtube.com/v/xGMuuPYhLiQ

Original coloringyoutube.com/v/ypYFUGiR51c

k=0.0609, F=0.062

Periodic boundary conditions; size 3.35w × 2.33h

1 second ≈ 3900 dtu

Initial pattern and coloring same as before

figure: parameter space with &i& and X
figure: parameter space with ι and X
This video is at X; Pearson's type ι is shown


slide 16


Long Duration Test

Different Behaviors at Multiple Time Scales


Exponential-time-lapse.mp4watch (YouTube)youtube.com/v/-k98XOu7pC8

k=0.0609, F=0.062

Periodic boundary conditions; size 3.35w × 2.33h

Initial pattern of spots (randomly chosen U<1, V>0) on solid red (U=1, V=0) background; coloring same as before

Video uses accelerating time-lapse: simulation speed doubles every 6.7 seconds


slide 17


Complex Interactions


complex-interactions-1.mp4watch (YouTube)youtube.com/v/hgTBOf7gg8E

k=0.0609, F=0.06

Periodic boundary conditions; size 3.35w × 2.33h

Each second is 10,000 dtu

Manually constructed initial pattern based on parts of naturally-evolved systems

Coloring same as before


slide 18


Slow Movement, Rotation


slow movers and rotaters.mp4watch (YouTube)youtube.com/v/PB3lPMhwIo0

k=0.0609, F=0.06

Periodic boundary conditions; size 3.35w × 2.33h

Each second is 100,000 dtu

Manually constructed initial pattern based on parts of naturally-evolved systems

Coloring same as before

figure 1: rotating ''target with 9 spots'' undergoing tests
figure 1: rotating "target with 9 spots" undergoing tests

Stability analysis of another slow-rotating pattern

figure 2: 4-spot pattern undergoing tests
figure 2: 4-spot pattern undergoing tests
Stability testing of 4-negaton pattern; stable form shown at right


slide 19


A Gray-Scott Pattern Bestiary

figure: a large collection of patterns
figure: a large collection of patterns

k=0.0609, F=0.06 Contrast-enhanced grayscale, lighter = higher U

Most patterns were manually constructed from parts of naturally occurring forms

Note: Many of these are not yet thoroughly tested

    - move in a straight line, if is has (only) bilateral symmetry

    - rotate, if it has (only) rotational symmetry

    - move on a curved path, if it has no symmetry at all


slide 20


Relation to Other Work:

"Negaton" Clusters and Targets

(Showing parts of fig. 1 and fig. 5 from Schenk et al. (1998))

From Schenk et al. "Interaction of self-organized quasiparticles..." (Physical Review E 57(6) 1998), fig. 1 and 5 The pattern marked * (subfigure "b") is stable in Gray-Scott at {k=0.0609, F=0.06}, the pattern marked × (subfigure "e") is not.

figure: target pattern
figure: target pattern

Target pattern in Gray-Scott, k=0.0609, F=0.06, with U and V levels at cross-section through center

(Showing part of fig. 4.37 from Schenk PhD dissertation)

Target pattern from C. P. Schenk (PhD dissertation, WWU Münster, 1999) p. 116 fig. 4.37


slide 21


Relation to Other Work: Halos

figure: two ''negatons'' at stable weakly-bound distance
figure: two "negatons" at stable weakly-bound distance

Negatons with halos (Gray-Scott system, k=0.0609, F=0.06, lighter = higher U; exaggerated contrast)

(showing fig. 11 from Barrio et al. 2009)

Spots with halos in numerical simulation by Barrio et al. "Modeling the skin patterns of fishes" (Physical Review E 79 031908 2009) fig. 11

(showing fig. 3d from Stollenwerk web page)

Spots with halos in gas-discharge experiment by Lars Stollenwerk ("Pattern formation in AC gas discharge systems", website of the Institute of Applied Physics, WWU Münster, 2008) fig. 3d


Halos also appear in models by Schenk, Purwins, et al (ibid., 1998 and 1999, shown on previous slide) in work related to gas discharge experiments


slide 22


One-Dimensional Gray-Scott Model

(showing fig. 9 from Mazin et al. 1996)

Single 1-D negaton inside a growing region of solid "blue state" at k=0.06, F=0.05, from Mazin et al. "Pattern formation in the bistable Gray-Scott model" (Math. and Comp. in Simulation 40 1996) fig. 9

figure 1: spiral and fast pulse
figure 1: spiral and fast pulse

2-D spirals and 1-D pulse at k=0.047, F=0.014

figure 2: two slow-moving patterns
figure 2: two slow-moving patterns

Representative 2-D and 1-D patterns at k=0.0609, F=0.06


(Note: Non-existence results of Doelman, Kaper and Zegeling (1997) and of Muratov and Osipov (2000) are not applicable because they concern models with a much higher ratio DU/DV )


slide 23


Open Questions

— As parameter k is increased, leading end of double-stripe (shown) moves faster, but trailing end moves slower and the object lengthens; in the other direction (decreasing k) the reverse is true

— When these two speeds are closely matched, the U shape (shown) neither grows nor shrinks — why?

— Universal presence of other pattern types suggests this; parameter space maps should make it easy to find; nearby Turing effect is possibly relevant

— 1-D systems seem particularly well suited to this task

— Shape of negaton "halos" is easy to solve

— Most existing work applies conditions or limits that exclude the commonly studied DU/DV=2 systems

slide 24


Discussion

screen image of my Gray-Scott program in July 1994
screen image of my Gray-Scott program in July 1994

18 MFLOPs/sec. Note ASCII text rendering of pattern

my Gray-Scott video screensaver, September 2010
my Gray-Scott video screensaver, September 2010

(depicts a Gray-Scott image using a brightness map of my face to control parameter k)

Robert Munafo       mrob.com/sci

mrob.com/pub/comp/xmorphia

(email: click "contact" below)

(the background of this slide is a detailed parameter map image generated in 2004, showing k from 0.0582 to 0.0682 and F from 0.031 to 0.051, in a color scheme that depicts low U values as blue and high U values as yellow.)


slide 25



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