Errata and Minor Additions, 2009-SMP  

This page lists errata and minor additions being considered by the author for the paper, Stable localized moving patterns in the 2-D Gray-Scott model. Go here to view the paper, illustrations, and other related material.

Major Issues

The work has been challenged on the basis of the following issues, which the author is working to address:


Section 2, paragraph 3 ("The initial state was...") : "perhaps due to the color of ferrocyanide" should be changed to "due to the color of the pH indicator bromothymol blue" and the reference changed (to Lee et. al., Science 261 1993).

Section 4, paragraph 6 ("Many stripelike patterns...") : "They are stable in the central linear section" needs to be clarified. These linear features are stable along the dimension perpendicular to their length, but are unstable in the other dimension. Any deviation from a straight line will increase, first slowly and then with increasing speed, forming meanders like those of a river in a floodplain.

Section 8, final paragraph : "category 4" should be "class 4" because that's what Wolfram calls it in his 1984 paper.

References : These probably need to be in order first cited, rather than the present chronological.

References, Wolfram 1984 : In title, "University" should be "Universality" (Wolfram also makes this error in the paper — Section I, paragraph 2, sentence 6. Perhaps my spell check methods are 25 years out of date :-)

References, Munteanu 2006 : The second author should have an acute accent over the e in his last name: Andreea Munteanu and Ricard V. Solé.

Figure 1, caption : The background values of u and v are (0.4201, 0.2878).

Possible Additions

Section 7, final paragraph : Add a reference to the following paper:

A.W. Liehr, A.S. Moskalenko, Yu.A. Astrov, M. Bode, and H.-G. Purwins, Rotating bound states of dissipative solitons in systems of reaction-diffusion type, European Physical Journal B 37 (2004) 199-204.

noting their experimental observation of two spots in a "bound state".

Section 7 : Add a reference to the following paper:

Rafael A. Barrio, Ruth E. Baker, Benjamin Vaughan, Jr., Karla Tribuzy, and Marcelo R. de Carvalho, Modeling the skin pattern of fishes, Physical Review E 79 (2009) 031908.

This paper discusses the modeling of skin coloration patterns in Potamotrygon motoro, a freshwater stingray. Here I wish to note the halos seen around the spots that appear in the two-layer coupled system in their figure 11. These spots do not display a preference for certain inter-spot distances (the "bound states" of Liehr et. al.), and I suspect it is because of the strong coupling. Similar halos are also seen in the figures 14-16.

Given the above additions and the already somewhat lengthy section 7, I am also considering removing some details and condensing the rest, to maintain the present length of less than one full column.

Section 8, near paragraph 5 : In one dimension, standard analysis (by Laplace transform) readily shows that the "halos" of a single spot in isolation have spacing and relative amplitude levels that agree with what is seen in simulation. More involved analysis would be required to prove stability and movement of multi-spot patterns.

The pattern shown here is stable and moves to the right at 1 lu per 1.19×1010 tu:

F=0.06, k=0.0609
F=0.06, k=0.0609

The top strip shows the pattern, colored as in my two-dimensional figures. The white and black curves show levels of u and v respectively, with the total height of the figure representing a scale of 0.0 to 1.0. This simulation was performed with Δx = 1/286 and Δt = 1/16.