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

##### August 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.

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

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

June 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 impacts, argue ...

#### Loss of tropical forests reduces rainfall

September 5, 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.

#### 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 ...

#### Vegetation as seen by Suomi NPP

June 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 a partnership between ...

#### Deserts 'greening' from rising CO2

July 3, 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.

#### Saving Earth's water from toxic waste

August 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

#### Researchers plumb the secrets of tissue paper

August 24, 2016

Canada's tissue manufacturers are now much closer to producing the perfect paper, thanks to new UBC research.

#### One of the most significant Etruscan discoveries in decades names female goddess Uni

August 24, 2016

Archaeologists translating a very rare inscription on an ancient Etruscan temple stone have discovered the name Uni—an important female goddess.

#### Is divorce seasonal? Study shows biannual spike in divorce filings

August 21, 2016

To everything there is a season—even divorce, new research from University of Washington sociologists concludes.

#### More than a few good men: Study finds counterintuitive outcomes of gender imbalance

August 24, 2016

Contrary to traditional expectations of unbalanced sex ratios, places with more men than women do not typically experience higher rates of family and social instability, according to a University of Utah study. The study, ...

#### Urban sociologists call for expanding concepts of 'livable cities'

August 24, 2016

A commentary in the current issue of the journal Nature, co-written by Hillary Angelo, UC Santa Cruz assistant professor of sociology, argues that while big cities appear to be islands of sustainable living, issues of social ...

#### An inflexible diet led to the disappearance of the cave bear

August 23, 2016

Senckenberg scientists have studied the feeding habits of the extinct cave bear. Based on the isotope composition in the collagen of the bears' bones, they were able to show that the large mammals subsisted on a purely vegan ...