Using math models to make predictions: How vegetation competes for rainfall in dry regions

Aug 30, 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 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 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 , 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 di?erential 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 seeps into the soil and promotes growth of new foliage. This implies that 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.

Explore further: Deserts 'greening' from rising CO2

More information: Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semi-arid Environments V: The Transition from Patterns to Desert, epubs.siam.org/doi/abs/10.1137/120899510

Related Stories

Deserts 'greening' from rising CO2

Jul 03, 2013

Increased levels of carbon dioxide (CO2) have helped boost green foliage across the world's arid regions over the past 30 years through a process called CO2 fertilisation, according to CSIRO research.

Vegetation as seen by Suomi NPP

Jun 21, 2013

(Phys.org) —Images crafted from a year's worth of data collected by the Suomi NPP satellite provide a vivid depiction of worldwide vegetation. Suomi NPP, short for National Polar-orbiting Partnership, is ...

Elevated carbon dioxide making arid regions greener

May 31, 2013

Scientists have long suspected that a flourishing of green foliage around the globe, observed since the early 1980s in satellite data, springs at least in part from the increasing concentration of carbon dioxide in Earth's ...

Rising CO2 levels likely to change vegetation locally more so than globally: study

Jun 28, 2012

(Phys.org) -- In all the talk about global warming as a result of human created CO2 emissions, it seems other impacts of higher levels of carbon dioxide on the environment tend to get overlooked. One of those ...

Loss of tropical forests reduces rainfall

Sep 05, 2012

Deforestation can have a significant effect on tropical rainfall, new research confirms. The findings have potentially devastating impacts for people living in and near the Amazon and Congo forests.

Saving Earth's water from toxic waste

Aug 20, 2013

Scientists have devised a better way to protect groundwater from acids, heavy metals and toxic chemicals, helping to secure the Earth's main freshwater supply.

Recommended for you

Egypt archaeologists find ancient writer's tomb

Apr 19, 2014

Egypt's minister of antiquities says a team of Spanish archaeologists has discovered two tombs in the southern part of the country, one of them belonging to a writer and containing a trove of artifacts including reed pens ...

Researchers create methylation maps of Neanderthals and Denisovans, compare them to modern humans

Apr 18, 2014

(Phys.org) —A team of Israeli, Spanish and German researchers has for the first time created a map of gene expression in Neanderthals and Denisovans and has compared them with modern humans. In their paper ...

3 Qs: Economist makes the case for new quasi-experiments as a way of studying environmental issues

Apr 18, 2014

How can scholars get traction on environmental problems, particularly those relating to pollution? In an essay appearing in this week's issue of the journal Science, MIT economist Michael Greenstone, along ...

New specimens of Yanornis indicate a digestive system of living birds

Apr 18, 2014

In a recent paper describing ten new specimens of Yanornis martini identified by the director of the Shandong Tianyu Museum of Natural History Mr. Xiaoting Zheng, an international team of scientists lead ...

Study finds law dramatically curbing need for speed

Apr 18, 2014

Almost seven years have passed since Ontario's street-racing legislation hit the books and, according to one Western researcher, it has succeeded in putting the brakes on the number of convictions and, more importantly, injuries ...

Changing dinosaur tracks spurs novel approach

Apr 17, 2014

Paleontologists are using a range of old and new techniques to map the Broome Sandstone dinosaur trackways.

More news stories

Researchers create methylation maps of Neanderthals and Denisovans, compare them to modern humans

(Phys.org) —A team of Israeli, Spanish and German researchers has for the first time created a map of gene expression in Neanderthals and Denisovans and has compared them with modern humans. In their paper ...

Last Week's Best—Quantum mechanics breakthrough, 3-D printed human heart, and paraplegia therapy

(Phys.org) —Hello readers—we'd like to try something new here at Phys.org and Medical Xpress: offer a weekly summary every Monday highlighting what we feel are the most important stories of the past ...

Earliest ancestor of land herbivores discovered

New research from the University of Toronto Mississauga demonstrates how carnivores transitioned into herbivores for the first time on land.

Neanderthals and Cro-magnons did not coincide on the Iberian Peninsula

The meeting between a Neanderthal and one of the first humans, which we used to picture in our minds, did not happen on the Iberian Peninsula. That is the conclusion reached by an international team of researchers ...

Clippers and coiners in 16th-century England

In 2017 a new £1 coin will appear in our pockets with a design extremely difficult to forge. In the mid-16th century, Elizabeth I's government came up with a series of measures to deter "divers evil persons" ...

Growing app industry has developers racing to keep up

Smartphone application developers say they are challenged by the glut of apps as well as the need to update their software to keep up with evolving phone technology, making creative pricing strategies essential to finding ...

Monoprice takes on Amazon in trade of cheap electronics

You'd only have to drive by the empty shells of Circuit City stores, and soon RadioShacks, to see why a company in Rancho Cucamonga, Calif., represents a nightmare for the retail electronics industry.