Math and Polar Bears

“The book of nature is written in the language of mathematics” - Galileo

Mathematics surrounds us in our everyday lives, from simple calculations of household budgets to complicated models for local weather predictions. It’s generally recognized that math is a key component in jobs like engineering, but did you know that ecologists use math to test theories and make predictions?

I pursued my PhD at the University of Toronto specifically to learn how a field ecologist, like me, can combine field data on polar bears (e.g. how many seals they eat, how much they move) with mathematical models to make predictions about polar bears in the face of climate change.

Global warming has caused a decline in the sea ice habitat polar bears depend on to catch seals, and scientists predict further sea ice loss. To understand how polar bears will be impacted we need to know their survival and reproductive rates under these low sea ice conditions. However, we don’t have field measurements, since they are never-before-seen conditions, and we can’t assume the rates won’t change because we know bears need ice to eat and thus survive. So, what can we do?

Enter mathematics!

The cool thing about math is that once you figure out a relationship, you can use it to explore different scenarios and find unknown values. Think about the classical Pythagorean theorem: If we know the length of two sides of a right triangle, we can determine the length of the third. Similarly, if we determine the relationship between sea ice and polar bear survival/reproductive rates, then we can predict how those rates will change under never-before-seen sea ice conditions. But how do we find the relationship between sea ice and polar bear survival/reproduction?

The key is energy!

Everyone has heard of counting calories. By tracking the amount of incoming and outgoing energy (e.g., recording that pizza you ate and that hour on the treadmill), we can determine why we are gaining or losing weight. The same idea applies to polar bears! The relationship between energy intake and energy use determines if an individual bear will survive and/or reproduce. By understanding how much energy a bear needs to power biological processes, such as walking or milk production for cubs, we can begin to understand how these processes might be jeopardized under food stress.

My work involves creating mathematical equations for each biological process a polar bear must perform. Using the data we already have from polar bears in zoos and from wild Western Hudson Bay polar bears, I can determine the relationship between a bear’s age, sex, and size and their resulting energy use. Knowing these relationships, we can then give our model bears different amounts of incoming energy and explore how the survival and reproduction for the model bears reacts to those different energy inputs.

Using data that is available now to develop and verify mathematical relationships allows me to assess how and when polar bears reproduction and survival will be impacted by sea ice decline and associated food shortages. By combining Arctic sea ice predictions with mathematical models of polar bear energetics, my colleagues and I made predictions for the persistence of polar bear subpopulations under different carbon emission scenarios. 

With today’s rapidly changing environment, techniques that allow us to reliably predict under never-before-seen conditions are essential for understanding how our actions today are shaping our tomorrow.  

Dr. Stephanie Penk is currently a postdoctoral researcher at the University of Montana. Her focus is on the development and correct use of mathematical models in ecology.