A recurrent idea in science is that the loss of stability of an equilibrium position through diffusion can lead to the creation of patterns. The idea goes back to Turing in his famous 1952 paper “On the chemical basis of morphogenesis,” which proposes a model for morphogenesis through chemical reaction-diffusion. The same idea can explain the shapes on Earth. We practically never observe a large flat landscape of sand on Earth. Why? Indeed, on a flat area there would always be a few irregularities. When the wind starts blowing, sand grains are transported and deposited behind the irregularities, thus making them grow. After a while, we observe a pattern of dunes. The “diffusion” in this case comes from the wind. The same occurs when the wind induces a regular pattern of waves on the surface of the water. The original equilibrium, consisting of a flat surface area, loses its stability when the wind associated with the diffusion starts to blow. This loss of stability is associated with the appearance of another equilibrium, consisting of a pattern of regular dunes for the sand or of regular waves for the water.
Spot vegetation around the ruins of Plazuelas in Mexico.
Similar patterns can be observed for vegetation and, what is remarkable, they can also be explained through a reaction-diffusion model. I met Antonello Provenzale in Rome during the workshop “Models and Methods for Mathematics of Planet Earth” at the Istituto Nazionale di Alta Matematica (INdAM); he provided me references on the subject, including some of his own articles. The vegetation patterns occur in semi-arid regions, where there is not enough water for a full vegetation cover. What are the diffusing substances in the model? They are the vegetation on the one hand and the surface water and ground water on the other. The roots of the plants can use some of the water in the soil covered by no vegetation. Also, the vegetation protects the soil from evaporation, and this feed-back mechanism allows the persistence of the vegetation patches. The models predict four types of vegetation patterns, depending on the parameters, which can be the quantity of water and the slope of the surface:
- Spots are observed in the most arid conditions;
- When there is more water, we can observe labyrinths;
- The last pattern before full vegetation is isolated gaps;
- Finally, stripes are observed when the surface is slanted.
These four types are observed in vegetation patterns occurring in nature. And, what is more remarkable, they are the same patterns that are observed in animal coatings and well explained through a reaction-diffusion model!
The model explaining vegetation patterns exhibits hysteresis. What does it mean? If the quantity of water is above a threshold allowing for a patched vegetation pattern but below the quantity of water necessary for a full vegetation pattern, then we have two stable equilibria: one with no vegetation, and one with a vegetation pattern. If the quantity of water decreases below the threshold, the vegetation pattern disappears and the land becomes a desert. But it is not sufficient that the water comes back above the threshold to restore the vegetation. Indeed, the feedback mechanisms are no more present and cannot help to the development of new vegetation patches on a desert: a higher level of water is needed to restore some vegetation. The lesson is that it is much easier to prevent desertification than to restore vegetation once it has disappeared.
Christiane Rousseau
Patterns are fascinating, and they can be found in many places on Earth. Last month, I visited the Galápagos Islands, where I noticed a hexagonal pattern on a mountain slope on Isabella Island. The pattern was formed by the Palos Santos trees, which grow in a more or less hexagonal pattern. They were dormant at the time of my visit (it was the dry season), but the pattern was visible from a distance. (It would be more obvious from the air.)
It was not clear what causes this particular pattern. The explanation given by the naturalist was that it optimized water intake by the trees. Hexagonal patterns optimize the ratio of the surface area to the length of the boundary, so among all regular patterns they would maximize the ratio of the area covered by the tree canopy to the length of its drip line. Could this be the explanation? Or is it another instance of a Turing pattern? If so, what is the diffusion process?
Hans Kaper