Hence, vegetation in semi-arid environments (or regions with low rainfall) self-organizes into patterns or “bands.” The pattern formation occurs where stripes of vegetation run parallel to the contours of a hill, and are interlaid with stripes of bare ground. Banded vegetation is common where there is low rainfall. In a paper published last month in the SIAM Journal on Applied Mathematics, author Jonathan A. Sherratt uses a mathematical model to determine the levels of precipitation within which such pattern formation occurs.
“Vegetation patterns are a common feature in semi-arid environments, occurring in Africa, Australia and North America,” explains Sherratt. “Field studies of these ecosystems are extremely difficult because of their remoteness and physical harshness; moreover there are no laboratory replicates. Therefore mathematical modeling has the potential to be an extremely valuable tool, enabling prediction of how pattern vegetation will respond to changes in external conditions.”
Several mathematical models have attempted to address banded vegetation in semi-arid environments, of which the oldest and most established is a system of partial diﬀerential equations, called the Klausmeier model.
The Klausmeier model is based on a water redistribution hypothesis, which assumes that rain falling on bare ground infiltrates only slightly; most of it runs downhill in the direction of the next vegetation band. It is here that rain water seeps into the soil and promotes growth of new foliage. This implies that moisture levels are higher on the uphill edge of the bands. Hence, as plants compete for water, bands move uphill with each generation. This uphill migration of bands occurs as new vegetation grows upslope of the bands and old vegetation dies on the downslope edge.
In this paper, the author uses the Klausmeier model, which is a system of reaction-diffusion-advection equations, to determine the critical rainfall level needed for pattern formation based on a variety of ecological parameters, such as rainfall, evaporation, plant uptake, downhill flow, and plant loss. He also investigates the uphill migration speeds of the bands. “My research focuses on the way in which patterns change as annual rainfall varies. In particular, I predict an abrupt shift in pattern formation as rainfall is decreased, which dramatically affects ecosystems,” says Sherratt. “The mathematical analysis enables me to derive a formula for the minimum level of annual rainfall for which banded vegetation is viable; below this, there is a transition to complete desert.”
The model has value in making resource decisions and addressing environmental concerns. “Since many semi-arid regions with banded vegetation are used for grazing and/or timber, this prediction has significant implications for land management,” Sherratt says. “Another issue for which mathematical modeling can be of value is the resilience of patterned vegetation to environmental change. This type of conclusion raises the possibility of using mathematical models as an early warning system that catastrophic changes in the ecosystem are imminent, enabling appropriate action (such as reduced grazing).”
The simplicity of the model allows the author to make detailed predictions, but more realistic models are required to further this work. “All mathematical models are a compromise between the complexity needed to adequately reflect real-world phenomena, and the simplicity that enables the application of mathematical methods. My paper concerns a relatively simple model for vegetation patterning, and I have been able to exploit this simplicity to obtain detailed mathematical predictions,” explains Sherratt. “A number of other researchers have proposed more realistic (and more complex) models, and corresponding study of these models is an important area for future work. The mathematical challenges are considerable, but the rewards would be great, with the potential to predict things such as critical levels of annual rainfall with a high degree of quantitative accuracy.”
With 2013 being the year of “Mathematics of Planet Earth (MPE),” mathematics departments and societies across the world are highlighting the role of the mathematical sciences in the scientific effort to understand and deal with the multifaceted challenges facing our planet and our civilization. “The wider field of mathematical modeling of ecosystem-level phenomena has the potential to make a major and quite unique contribution to our understanding of our planet,” says Sherratt.
Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semi-arid Environments V: The Transition from Patterns to Desert
Jonathan A. Sherratt
SIAM Journal on Applied Mathematics, 73 (4), 1347–1367 (Online publish date: July 3, 2013).
The paper will be available for free access at the link above from September 4 – December 4, 2013.
About the author:
Jonathan A. Sherratt is a professor in the Department of Mathematics at Heriot-Watt University, and at Maxwell Institute for Mathematical Sciences in Edinburgh, United Kingdom.
The Society for Industrial and Applied Mathematics (SIAM), headquartered in Philadelphia, Pennsylvania, is an international society of over 14,000 individual members, including applied and computational mathematicians and computer scientists, as well as other scientists and engineers. Members from 85 countries are researchers, educators, students, and practitioners in industry, government, laboratories, and academia. The Society, which also includes nearly 500 academic and corporate institutional members, serves and advances the disciplines of applied mathematics and computational science by publishing a variety of books and prestigious peer-reviewed research journals, by conducting conferences, and by hosting activity groups in various areas of mathematics. SIAM provides many opportunities for students including regional sections and student chapters.
Karthika Muthukumaraswamy | Source: EurekAlert!
Further information: www.siam.org
More articles from Ecology, The Environment and Conservation:
Neue Akteure im Ökosystem der Arktis
12.12.2013 | Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar- und Meeresforschung
New rearing method may help control of the western bean cutworm
09.12.2013 | Entomological Society of America
A unique solar panel design made with a new ceramic material points the way to potentially providing sustainable power cheaper, more efficiently, and requiring less manufacturing time.
It also reaches a four-decade-old goal of discovering a bulk photovoltaic material that can harness energy from visible and infrared light, not just ultraviolet light.
Scaling up this new design from its tablet-size prototype to a full-size solar panel would be a large step toward making solar power affordable compared with ...
Atlantische Flohkrebse pflanzen sich jetzt auch in arktischen Gewässern fort
Biologen des Alfred-Wegener-Institutes, Helmholtz-Zentrum für Polar- und Meeresforschung (AWI), haben zum ersten Mal nachgewiesen, dass sich in den arktischen Gewässern westlich Spitzbergens auch Flohkrebse aus dem wärmeren Atlantik fortpflanzen.
Diese überraschende Entdeckung deute auf einen möglichen Wandel der arktischen Zooplankton-Gemeinschaft hin, berichten die Wissenschaftler und Wissenschaftlerinnen in der Fachzeitschrift Marine Ecology ...
The molecular architecture of three key proteins and their complexes reveals how plants fine-tune their immune response to pathogens
Plants rarely get sick in their natural environment. When the threat of infection arises, a quick decision is made about the necessary countermeasures. The course is set by a protein which forms complexes with its partner proteins for this purpose.
Jane Parker from the Max Planck Institute for Plant Breeding ...
Researchers studying speciation of butterfly orchids on the Azores have been startled to discover that the answer to a long-debated question "Do the islands support one species or two species?" is actually "three species".
Hochstetter's Butterfly-orchid, newly recognized following application of a battery of scientific techniques and reveling in a complex taxonomic history worthy of Sherlock Holmes, is arguably Europe's rarest orchid species. Under threat in its mountain-top retreat, the orchid urgently requires conservation recognition.
A lavishly illustrated publication, titled "Systematic revision of Platanthera in ...
Researchers from Brown University and the University of Hawaii have found some mineralogical surprises in the Moon's largest impact crater.
Data from the Moon Mineralogy Mapper that flew aboard India's Chandrayaan-1 lunar orbiter shows a diverse mineralogy in the subsurface of the giant South Pole Aitken basin.
The differing mineral signatures could be reflective of the minerals dredged up at the time of the giant impact 4 billion years ago, ...
12.12.2013 | Life Sciences
12.12.2013 | Earth Sciences
12.12.2013 | Studies and Analyses
11.12.2013 | Event News
10.12.2013 | Event News
05.12.2013 | Event News