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Mathematics and the environment

Florida Everglades

Credit: Tetra Images Photography/Shutterstock

Some people claim that the discoveries made by scientists contribute to the destruction of the natural environment. Professor Louis Gross at the University of Tennessee shows that the case can equally be made for the opposite. He is a mathematical ecologist, applying advanced mathematics to the problems of managing the natural environment to maximise the benefits to the whole natural system. The pressures of human life have an effect on the rest of nature and by understanding how the relationships work, everyone and everything might get some of what they want.

It turns out that these problems are not trivial mathematically. The flow pattern of a river might have a linear relationship with the rainfall in a particular place, but what happens when the river bursts its banks? Or if it rains after a period of drought? And how do you know what the rainfall is going to be anyway?

Not only are many natural processes essentially stochastic they also require nonlinear algebra to describe them. Getting meaningful results is a huge mathematical and computational exercise. This is why Gross, like many scientists from other, more conventional fields, has turned his attention to the mathematics of the natural world – it has some of the most interesting mathematical problems. Mathematical biology has achieved a high profile through cell biology and genomics, but at the scale of the whole ecosystem it is still in its emerging stage and the field has many opportunities to do new things.

Professor Alan Hastings, from the Department of Environmental Science and Policy at the University of California, Davis, has known Gross since graduate school where they had the same supervisor. He preceded Gross as President of the Society for Mathematical Biology. He says: ‘He is very outgoing and enthusiastic. He has done a tremendous amount of organisational work and is very active in doing things which have a world wide impact.

‘His major contributions have been in formulating and leading in the development of computer models for understanding the dynamics of ecosystems. He has been very involved in issues around training in mathematical biology. He is certainly one of the earlier people in this field.’

Lou Gross was born in Philadelphia. His father was an electrical engineer and his mother a bookkeeper. His interest in science began on a planetarium visit in the second grade. This grew into a fascination with radio-astronomy which in turn drove him to study hard at mathematics. This did not make him into a nerd – he was also a very keen boy scout and sports runner.

He went to Drexel University in Philadelphia, which offered a cooperative education programme allowing students to work for six months and then study for six months. For six months a year he was paid to work at the National Radio Astronomy Observatory in Charlottesville, and for the rest of the year he studied mathematics.

He was not just involved in lowly work but was encouraged to work on new ideas. He spent a lot of time developing computer programmes to control the interferometers that were being developed at the time. He had access to an IBM 360 mainframe with all of 360k of main memory, which taught him a lot about writing efficient programmes.

He says: ‘I was not taken with computing at the time, I just saw it as a tool. As a mathematician you always look for the most appropriate way to analyse your problem. I was looking at convolution integrals for interferometry and how to relate that to the data that was coming in, and using a computer was a way to do that.’

Louis Gross
Gross: 'Outgoing and enthusiastic.'

He graduated in mathematics, with a minor in physics. But after three years of working in radio astronomy, he decided that it was time to change tack. He says: ‘I came to the realisation that the really good people not only had good mathematical ability, but they also had good hands-on skills and could fix equipment. They also had huge physical intuition and I realised I did not have that physical intuition, which is why I decided to get into biological science. I always had an interest in the outdoors, because I was very active in the scouting movement, which had a strong emphasis on natural history.’

Gross had already become interested in mathematical biology before he entered the graduate programme at Cornell. It was an emerging field at the time, although some aspects had been around for many years, such as population genetics. He had taken a course at Drexel on the mathematics of human biology with a focus on demography and had become enthralled by the subject. He liked the stochastic mathematics and the real world possibilities of applying mathematics to biology.

He joined a small applied mathematics group at Cornell and did some work on neural network models of the human brain. He did not think that was going to go anywhere very soon, but at the same time he started taking a course in ecology from Simon Levin. He also met his other great mentor Brian Chabot, who was a plant physiology ecologist.

He says: ‘I realised that there had not been very much work done on photosynthesis. It was an opportunity for me to do something that I had not done before, which is laboratory work. As part of my dissertation I wrote a paper which was published on the dynamics of photosynthesis. I looked at how carbon uptake changes in response to changes in light. There are still many opportunities in biology that are completely new.’

After gaining his PhD he was certain about what he wanted to do – and made his first, and so far only, career move. He says: ‘I immediately accepted a position at the University of Tennessee because of my colleague Thomas Hallam, who had been hired here specifically to build a programme in mathematical biology with an emphasis on ecology. There were very few other places that were so forward thinking and had recognised the importance of the subject. I was appointed to the mathematics department, but I also held a position in ecology, although that was not a department at the time. I continued my work on photosynthesis here, before being drawn into other things. Even now, the strong programmes in the field do not have large groups working on them, and yet that is enough. Computational biology has grown as a field, but this tends to mean things below the organism scale – like genomics.’

Gross accepts that the growth of mathematics in fields like genomics and cell biology has gained a higher public profile than his work in recent years, because people feel that it has a direct impact on human health. His work has concentrated on a much larger scale, but he believes it can also have a direct impact on human life.

He says: ‘My major research project over the past two decades has been on Everglades restoration. This is an example of a large piece of real estate in which the driving force for the natural systems and social systems is the water. Where the water is makes a big difference to where the people are and where they are not, as well as the organisms and the responses of the whole system. It is a tremendously interlinked system where the human systems have an impact on the natural systems.

‘Unfortunately there are a very large number of stakeholders who have different views as to how the system might be appropriately managed. We use computational models rather than mathematical models, because it’s extremely difficult to write down any set of equations that are complicated enough to capture the essence of the system and still be able to say anything useful. These models can determine the implications of different human actions. What we say is, if you manage the system this way, how does that impact the system of the Everglades and we have come up with a hydrological model look at the implications of different plans on the biota.

‘One of the things that we have learned from theory in ecology is the scales in which you are interested and the scales in which they respond are tremendously important. In the Everglades, the spatial scales are rather broad, so what is happening in one part of the system may be very much decoupled from other parts of the system, unless something like the water is coupling the system. Unfortunately, the water is coupling the system so you cannot take a simple average in the way that you would with a linear process and expect it to work out. That is one of the things that we have tried to deal with in the Everglades models. What happens in one part of space should be discernable from what is happening in another part of space.

We have always been asked for one number that says one management plan is better than another management plan and we have always refused to do it. We have come down to producing maps that display how natural systems work with time series. We developed a methodology called relative assessment with two maps that represent different management schemes and show the differences between them. You can then do a time series of these maps and people can pick out the nonlinearities and see how certain populations have crashed in response to a change in the water.’

One project Gross’s group has been examining is saving the Florida panther, which is related to the cougar, and was on the verge of extinction with a population below 30 in 1995. The plan was to import female cougars from Texas who had once interbred with Florida Panthers; now the population has grown to 100. The group is working on population augmentation model to help plan such projects.

‘These models were not available at the time and we are now working on them. It was an example of a relatively simple ecological idea, but the mathematics had not been worked out. It was basically a problem in optimal control – a long standing field in mathematics – which has differential equations with a control parameter that you can modify to maximise or minimise something under certain constraints. This is a relatively new field in biological systems, but it has applications in other areas.

‘What we are trying to do with the Everglades is work out what to do, where to do it and when and how to assess what you have done. This is highly computational and involves a lot of intensive parallel computing, because when you do something in one place it affects what happens in another place. You have a huge control space with sometimes millions of control variables.’

He says a lot of ecological models can also be used in other fields, such as infection control. In many cases, he says, there are ‘rules of thumb’ that people have used for many years to make decisions on managing natural systems. The mathematical models can sometimes come up with alternative strategies, which can be shown to be more effective under certain circumstances. Gross has discovered that many issues in managing the environment, from controlling the activity of hunters to keep bears away from human settlements to controlling forest fires, are actually very interesting mathematical problems, often involving stochastic process. In many ways there are parallels with the ways that advanced mathematics has become an important part of the creation of investment strategies in the financial markets.

Gross has been at the University of Tennessee in Knoxville since 1979 and has no inclination to do the common academic shuffle between institutions to get resources, greater salary or prestige. He was brought in by a university management which, in many ways, was ahead of the game in encouraging interdisciplinary research activities and as a result he has been given a long leash. Of course his department has not been expensive to run, with many millions of dollars being brought in through research grants as the field of environmental management has acquired a higher political profile.

He says: ‘What has been good about staying here is that we have been able to build one of the best programmes in the world in mathematical biology. We have done this in a collaborative way. The Institute for Environmental Modelling has allowed us to work with a wide variety of people from different specialities. There are few other universities where it would be so easy to set up an interdisciplinary research institute.

‘Modern biology is very interdisciplinary and, in order to train the next generation of biologists, we will need to inculcate them with a lot of other specialities. One of the challenges is how to modify the undergraduate curricula so that they see other subjects. One way to do this is with electives, the other is to have truly interdisciplinary courses.’

So how was he able to persuade his university to pioneer this approach? ‘Well, it didn’t cost them very much. We got some stationery and we were up and running,’ says Gross. Since then his institute has brought in some $15m in external funding. One well-known outputs from his research is the SADA software package which is used for planning site remediation all over the world. Its development is funded by the Nuclear Regulatory Commission.

Another reason for his decision not to move is the ‘academic two body problem’. His wife is also an academic in poetry and although he has been courted by many other institutions he has not been able to tackle the double hire issue. He is also very happy where he is.

In the future he is hopeful that the growing profile of mathematical biology at the cell level will eventually spread to his work at a macroscopic level. He says that one of the problems is that many of the academic meetings have become quite specialised. Mathematical papers are presented at meetings about a particular area of biology and people in that field are unwilling at the moment to go to meetings where all they talk about is mathematical biology, but across many fields.

Gross has served as President of the Society for Mathematical Biology and he hopes that in the future this will emerge as more of a field in its own right. He says: ‘There are so many things that you can learn from how people in other areas of biology are using mathematical methods. There are many routes to how the field is advanced. Some of these developments will occur in mathematics departments and others will happen in biological science research. For example, there is a growing field of agent-based modelling, which has so far seen few applications outside ecology. They can be equally applied to many other fields and the computational tools are becoming available and people without a huge amount of coding experience can do something new.’

Much of Gross’s work has been around parallel processing simply because in ecology you have to do it because of the synchrony of biological systems. Gross believes that the same approach can be used in human biology by studying ‘whole human’ responses.

Gross has developed a number of interests outside of his university work. Most notably he is the sound engineer for a local theatre in Knoxville. He has published articles in journals about designing theatre sound systems.

This stems from a deep interest in what might be called folk music, but his tastes are very broad. Another reason to stay put in Tennessee?

Lou Gross CV




1974 BS in Mathematics with Honours, Drexel University

1979 PhD in Applied Mathematics, Cornell University.

Professional appointments

1979-1985: Assistant Professor, Department of Mathematics and Graduate Program in Ecology, University of Tennessee, Knoxville.

1985-1992: Associate Professor, Department of Mathematics, University of Tennessee, Knoxville,

1992-1997: Professor, Department of Mathematics, University of Tennessee, Knoxville,

1997-present: Professor, Departments of Ecology and Evolutionary Biology and Mathematics University of Tennessee, Knoxville,

1998-present: Director, The Institute for Environmental Modelling, University of Tennessee, Knoxville,


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