Mathematics professors at Aalborg University use spatial statistics to help biologists understand biodiversity in the world’s rainforests. Using mathematical models, they can see patterns and correlations that are otherwise invisible.

By using statistical models and advanced mathematics, the two mathematics professors Rasmus Waagepetersen and Jesper Møller from the Department of Mathematical Sciences at AAU help biology researchers understand why the trees of the rainforest grow where they do. By studying the spatial distribution of trees over time, mathematicians have come up with new clues for understanding how life in the rainforest develops.

## Thousands of trees

In order to understand the biodiversity of the world’s rainforests, biologists select areas of 1 x 0.5 km in various places on the planet. They register the positions of the different trees and plants within these areas in order to gain an overview of how many different species there are and where and how they grow.

Unlike forests in temperate regions like Denmark, where we typically see a dozen different kinds of trees, the rainforest houses a huge variety of species. "There are often several hundred different species in an area of 50 hectares of rainforest, and several hundred thousand trees in total”, Rasmus Waagepetersen says.

The data gathered provide biologists with an indication of how many different species exist in a given area, but they do not necessarily make them wiser as to why the trees are located where they are, or what determines the great diversity of trees in the rainforest.

What biologists are interested in finding out is why there are so many different species. "You would think that a few species would eventually outcompete the others. Yet that is not the case. The rainforest has been around for millions of years, and it has maintained a very high level of biodiversity. Biologists would love to know why this is the case and to understand how biodiversity works”, says Rasmus Waagepetersen.

## Order in chaos

If you look at the distribution of trees in the rainforest as dots on a map, it seems completely random at first. In some places, certain species are located in dense clusters; elsewhere, they seem to grow at random, scattered amongst each other.

In order to understand how this diversity arises and the underlying reasons why the trees grow where they do, biologists must lean on advanced statistics.

They will typically have a number of hypotheses about the factors that come into play. These may be different soil conditions, the altitude of the terrain, proximity to water, spreading of seeds via various animals, etc., yet these still cannot explain everything.

- “With the models we have developed, we can consider all the different conditions that biologists know about, and see if it makes sense in terms of where the trees are located. However, this also means that we can see whether there are other factors that biologists may have overlooked, or do not know about. Lots of unknown factors may come into play”, Rasmus Waagepetersen explains.

## Explanatory variables

The two researchers’ statistical models do not relate specifically to biological causalities. However, they can be used to study whether the biologists’ data support their theories about how things are correlated.

- “We cannot necessarily see what contributes to biodiversity being the way it is. That’s not how mathematics works. But we can see if a pattern suggests other significant factors that influence where and how trees grow”, says Jesper Møller and continues: “We call these 'explanatory variables' – i.e. external factors that may or may not affect the outcome”, he explains.

“With the statistical models and methods, we have developed, we can see if an explanatory variable is missing – if there is an apparent correlation in the randomness that must have a cause.”

## Understanding the world through mathematics

"The work to map the foundation of the rainforest’s biodiversity is based on over 35 years of fundamental research in mathematics”, professor Jesper Møller explains.

- “This is not something that has been developed specifically to map the biodiversity of rainforests. It is just one of many areas where we use mathematical and statistical models to understand how the world works and why things happen the way they do”, he says.

By using point processes, where events or other subjects of study are marked as points on a map in two or three dimensions, in principle it is possible to decipher patterns in anything from traffic accidents on the road networks to meteor impacts on Earth, or how our brain cells are organised.

- “Basically, it’s about using advanced statistics and mathematics to better understand insanely intricate processes that otherwise look completely incoherent and chaotic”, says Jesper Møller.