02.09.2013

The greater the plant density in a given area, the greater the amount of rainwater that seeps into the ground. This is due to a higher presence of dense roots and organic matter in the soil. Since water is a limited resource in many dry ecosystems, such as semi-arid environments and semi-deserts, there is a benefit to vegetation to adapt by forming closer networks with little space between plants.

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.

Desert steppes in Yol Valley in Mongolia. Photo Credit: Christineg (Source: Dreamstime)

“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.

Source Article:

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.

About SIAM

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 | EurekAlert!

Further information:

http://www.siam.org

**Further reports about:**
> Applied and Environmental Microbiology
> SIAM
> Sherratt
> annual rainfall
> mathematical model
> mathematics
> semi-arid environment

Successful calculation of human and natural influence on cloud formation

04.11.2016 | Goethe-Universität Frankfurt am Main

Invasive Insects Cost the World Billions Per Year

04.10.2016 | University of Adelaide

A multi-institutional research collaboration has created a novel approach for fabricating three-dimensional micro-optics through the shape-defined formation of porous silicon (PSi), with broad impacts in integrated optoelectronics, imaging, and photovoltaics.

Working with colleagues at Stanford and The Dow Chemical Company, researchers at the University of Illinois at Urbana-Champaign fabricated 3-D birefringent...

In experiments with magnetic atoms conducted at extremely low temperatures, scientists have demonstrated a unique phase of matter: The atoms form a new type of quantum liquid or quantum droplet state. These so called quantum droplets may preserve their form in absence of external confinement because of quantum effects. The joint team of experimental physicists from Innsbruck and theoretical physicists from Hannover report on their findings in the journal Physical Review X.

“Our Quantum droplets are in the gas phase but they still drop like a rock,” explains experimental physicist Francesca Ferlaino when talking about the...

The Max Planck Institute for Physics (MPP) is opening up a new research field. A workshop from November 21 - 22, 2016 will mark the start of activities for an innovative axion experiment. Axions are still only purely hypothetical particles. Their detection could solve two fundamental problems in particle physics: What dark matter consists of and why it has not yet been possible to directly observe a CP violation for the strong interaction.

The “MADMAX” project is the MPP’s commitment to axion research. Axions are so far only a theoretical prediction and are difficult to detect: on the one hand,...

Broadband rotational spectroscopy unravels structural reshaping of isolated molecules in the gas phase to accommodate water

In two recent publications in the Journal of Chemical Physics and in the Journal of Physical Chemistry Letters, researchers around Melanie Schnell from the Max...

The efficiency of power electronic systems is not solely dependent on electrical efficiency but also on weight, for example, in mobile systems. When the weight of relevant components and devices in airplanes, for instance, is reduced, fuel savings can be achieved and correspondingly greenhouse gas emissions decreased. New materials and components based on gallium nitride (GaN) can help to reduce weight and increase the efficiency. With these new materials, power electronic switches can be operated at higher switching frequency, resulting in higher power density and lower material costs.

Researchers at the Fraunhofer Institute for Solar Energy Systems ISE together with partners have investigated how these materials can be used to make power...

Anzeige

Anzeige

Event News

ICTM Conference 2017: Production technology for turbomachine manufacturing of the future

16.11.2016 | Event News

Innovation Day Laser Technology – Laser Additive Manufacturing

01.11.2016 | Event News

#IC2S2: When Social Science meets Computer Science - GESIS will host the IC2S2 conference 2017

14.10.2016 | Event News

Latest News

UTSA study describes new minimally invasive device to treat cancer and other illnesses

02.12.2016 | Medical Engineering

Plasma-zapping process could yield trans fat-free soybean oil product

02.12.2016 | Agricultural and Forestry Science

What do Netflix, Google and planetary systems have in common?

02.12.2016 | Physics and Astronomy

VideoLinks

NASA | A Year in the Life of Earth's CO2

NASA Computer Model Provides a New Portrait of Carbon Dioxide

Black Holes Come to the Big Screen

The new movie "Interstellar" explores a longstanding fascination, but UA astrophysicists are using cutting-edge technology to go one better.

NASA's Swift Mission Observes Mega Flares from a Mini Star

NASA's Swift satellite detected the strongest, hottest, and longest-lasting sequence of stellar flares ever seen from a nearby red dwarf star.

NASA | Global Hawks Soar into Storms

NASA's airborne Hurricane and Severe Storm Sentinel or HS3 mission, will revisit the Atlantic Ocean for the third year in a row.

Baffin Island - Disappearing ice caps

Giff Miller, geologist and paleoclima-tologist, is walking the margins of melting glaciers on Baffin Island, Nunavut, Canada.

The Infrasound Network and how it works

The CTBTO uses infrasound stations to monitor the Earth mainly for atmospheric explosions.

B2B-VideoLinks

Drying and curing of paints on glass and ceramics

Bright and brilliant paints on glass and ceramics require safe solutions for drying and curing.

JULABO World of Temperature

Explore the World of Temperature with JULABO - Superior Temperature Technology for a Better Life.

Acoustic Wave Separation: How It Works

In this animated video, see how Acoustic Wave Separation technology works in full detail.

Infrared Heat for printed electronics

Drying and sintering of printed electronics by specialty light sources from Heraeus

All about Data Logger, how to use

Wolfgang Rudolph explains: all information worth knowing about the data logger and the practical test by means of a drone