Finding the Tipping Point

The following was written by Karna Gowda, a graduate student at Northwestern University in the Department of Engineering Sciences and Applied Mathematics, and SAMSI visiting fellow for the Ecology Program this year.

Group shot of Tipping Point working group

Pictured (L-R): Christopher Strickland, SAMSI postdoc, Jake Norton, NCSU graduate student, Karna Gowda (on the screen), graduate student at Northwestern University and Lou Gross, University of Tennessee-Knoxville. Not Pictured but frequent participators are: Philip Dixon, Iowa State and Michael Just, NC State

Imagine balancing a pencil on your finger. If you tap one end a little bit, it will return to balance. Tap it more forcefully, and it will fall. This simple example illustrates one notion of a “tipping point”: the tipping point here is the smallest force you can apply that causes the pencil to fall.

Tipping points are thresholds. When crossed, significant change follows. A classic example occurs in lake ecosystems, where small changes in water nutrient composition can result in sudden qualitative changes in the water’s clarity due to growth of microorganisms. A change like this can disrupt the balance of the ecosystem. Sunlight levels drop more rapidly as the water’s clarity drops, which reduces growth of food resources for many aquatic species, including fish.

There are numerous other examples of potential tipping points in ecosystems, many of which may result in a loss of biodiversity and land degradation. This has led researchers to study how data and mathematical models can be used to predict tipping points, so that catastrophe can be averted. As part of the SAMSI Program on Mathematical and Statistical Ecology (2014-2015), we in the “Tipping Points with Forestry Applications” working group broadly focus on this topic. How can data from forest inventories, bee population surveys, and long-term grassland ecosystem studies be used to understand and potentially predict tipping points?

A current focus of our group is on the time series analysis methodology developed by Sugihara et al. (Science 2012). The methodology is based on the idea of attractor reconstruction, wherein partial information about a highly complex (nonlinear and potentially chaotic) dynamical system is used to construct a representative picture of broad system behavior. This picture is used to assess whether different elements are causally related (i.e. coupled or driven) within this dynamical system. Sugihara et al. use time series of sardine and anchovy populations, along with sea surface temperature data, to infer that the populations are driven by temperature and are not clearly affected by one another.

So far, the working group has explored how this methodology can be used to understand the relationships between populations and their drivers when ecosystem data are sparse. Attractor reconstruction techniques require ample well-resolved data, and often these data simply do not exist. A proposed work-around entails using multiple time series collected from different points in space in lieu of only a single time series. Though the data at each point in space may be short, putting together the data over space may provide the attractor reconstruction technique with enough information to draw a clear conclusion.

Understanding the relationships between ecosystem populations and their drivers is a crucial aspect of tipping point prediction, and is an inherently interdisciplinary task. The SAMSI Program on Mathematical and Statistical Ecology has brought together researchers from a number of backgrounds to tackle problems such as this. Collaboration between statisticians, ecologists and mathematicians within the Tipping Points working group so far has been illuminating and fruitful, and we look forward to sharing our insights as they develop!