Mathematics might seem like a strange tool to use to describe the biological world. Nature can seem pretty random. It turns out, though, that the plants and animals and bacteria and the other forms of life that surround us display extraordinary mathematical relationships.

In the cases where we can quantify a biological process, mathematical equations can often help describe that phenomenon and allow us to make conjectures about it as it changes. Several of us at the Santa Fe Institute are studying these surprising mathematical regularities.

My own research in ecology asks how many species coexist in a particular ecosystem, and what rules govern that coexistence. It's an important question. We worry a lot about biodiversity these days, and we are aware that human activities tend to inhibit biodiversity. If we want to know just how detrimental such losses of diversity are to the natural world on which we depend, and even predict the effects of future losses, it is helpful to understand what drives and constrains coexistence. For that we need mathematics.

The classic explanation is that each species in an ecosystem plays a unique role; biologists say a species occupies a distinct "niche." But some high diversity ecosystems seem to turn this conventional wisdom on its head.

In tropical forests, for example, we sometimes see hundreds of tree species in a small area, many of which are making their living in much the same way. In other words, many species seem to play very similar roles within an ecosystem. It's not impossible to reconcile this with the notion of niches, but it's clear that species' differences play a more subtle role than our current understanding allows.

Coexistence touches on another big mystery for ecologists: Patterns of species biodiversity are strikingly similar across different types of ecosystems. When we ask, how does total biodiversity change in relation to habitat area, or how many rare and how many abundant species do we expect, we find that for many different systems, the same numerical patterns crop up again and again. Why should this be? What underlying processes cause the coexistence patterns in a rainforest to look, numerically, like the coexistence patterns in a coral reef?

These kinds of quantitative generalities bring us to theory -- which allows us to think about ecosystems more abstractly -- and, in conjunction with data, might help elucidate some general, and useful, concepts. What predictions for patterns of diversity in space and time can we make, and do they work? Why does relatively simple, universal behavior emerge from complex ecological communities? How did these systems evolve to be this way, and are they robust in the face of upsets, such as climate change or habitat destruction?

We have a long way to go before we can answer these questions. We are just at the stage where we can begin to describe what needs to be quantified. But if we can integrate the lessons of ecological research, I think we humans can learn to better coexist with our natural world.

For me, these universal patterns raise a further question: What if our insights about diversity prove to be useful beyond ecology? What if diversity follows similar patterns wherever it is found -- in ecosystems and in human systems? We're beginning to see that it might.

With my SFI colleagues and a student participating in the Institute's Research Experiences for Undergraduates (REU) program, I'm looking at data from the business world -- treating it as an ecology of human institutions, if you will. Take the restaurant industry as an example. It's an industry with a great deal of diversity, from family-run establishments to large chains, from fast food to fine dining. There's also "species" coexistence: Why do we have both a Wendy's and a McDonald's on the same block, for example? And there are quantitative features similar to those in ecology; spatial patterns of diversity in industry share many of the hallmarks of the patterns we find in biological systems.

My research in ecology started with a mystery primarily of interest to scientists. But there is more to these questions than puzzles for scientists. Understanding the rules by which ecosystems are constructed and maintained will be crucial in helping manage the impacts of global climate change. I believe the same kind of ecological thinking can be useful in entirely different fields; bringing ideas from ecology to the study of human institutions has the potential to help us become more productive, allow us to live together more harmoniously, and maybe better able to withstand societal shifts and changes. We have to be careful, of course, about applying our models and intuitions in the wrong way. But in the signature SFI style we're beginning to see the potential for a very fruitful cross-fertilization of fields.

James O'Dwyer is incorporating elements of math, physics and biology to better understand the interactions of living things in their environments. His appointment as a Santa Fe Institute Omidyar Fellow was preceded by a postdoctoral fellowship in ecology at the University of Oregon. He holds a master's degree in mathematics and a Ph.D. in theoretical physics from Cambridge University as well as a master's in physics from Durham University.

ABOUT THE SERIES

The Santa Fe Institute is a private, nonprofit, independent research and education center founded in 1984 where top researchers from around the world gather to study and understand the theoretical foundations and patterns underlying the complex systems that are most critical to society — economies, ecosystems, conflict, disease, human social institutions and the global condition. This column is part of a series written by researchers at the Santa Fe Institute and published in The New Mexican.