"Chaos und Voraussagbarkeit in Ökologischen Systemen"

(Chaos and Predictability in Ecological Systems)

Dr. M. Davison, Physiologisches Institut

Was ist der Inhalt meines/unseres Beitrags (abstract)?

Environmental systems tend to be complicated, with many variables that may interact in a nonlinear way. As a result, realistic mathematical models of such systems usually involve nonlinear equations. Nonlinear equations can display counterintuitive behaviour and are, under appropriate conditions, capable of producing chaotic dynamics which display a sensitive dependence on initial conditions.

These counterintuitive behaviours makes it difficult, for example, to design sustainable resource extraction policies, particularly when it is impossible to determine the effects of these policies experimentally. The recent collapse of cod stocks on the Atlantic coast of Canada and the economic and social disruption that this has entailed show the importance of designing such policies correctly.

If the qualitative behaviour of the systems are counterintuitive and the equations themselves have chaotic solutions, can any predictions of the effect of a given environmental policy be made? Fortunately, the answer is "yes”. Tools from the mathematical discipline of ergodic theory allow accurate predictions of the evolution of probability densities associated with nonlinear systems to be made. In many cases, these probability densities converge to an invariant probability density. Once such an invariant density has been found a great deal of information about the dynamics of the system can be obtained. Classical statistical measures of the dynamics can be computed. Sophisticated "multifractal” data analyses may be performed. Such multifractal analyses allow the novel "Landsberg-Rényi Disorder” of the system to be obtained. Furthermore, assessment of the effect on these statistical results of a change in the mathematical model describing the system is often simplified by the use of these ergodic tools.

To illustrate some of these ideas we present a detailed analysis of what is arguably the simplest interesting nonlinear model of population dynamics, the discrete logistic map. This is a model of the change with time of the population of a single species. The model is a modification of a simple model of geometric growth to account for the finite carrying capacity of a region. It is an interesting model from a mathematical perspective because, depending on the value of the growth rate, it displays all possible types of behaviour: populations which die out, populations that explode, populations that oscillate periodically, and populations whose time variation is chaotic. We study this system using the techniques described above and show the power of these techniques to find structure in apparently unstructured data.

From this model we can see that, even if the dynamics produced are chaotic, underlying statistical regularities exist and can be calculated. So in principle, useful predictions may be obtained about ecological systems, even in the face of chaos.

Was sind meine/unsere Absichten betreffend Forschung (und allenfalls Lehre) im Bereich des Beitragsthemas für die nächsten 2-5 Jahre?

Our first step in the mathematical analysis of an ecological system is to build a mathematical model of that system. Then we ask, "Can this model explain our observations?” But what do we mean by

"explain?” Does simple reproduction of observed behaviour, within suitable bounds due to error and noise, constitute explanation? No! We're interested in much more than reproduction of the data we already have, or even of accurate prediction of what the system we examine will do in the future, assuming no changes in parameter. For example, suppose we've succeeded in constructing a model that mimics the population of cod stocks on the Grand Banks off Newfoundland and can even predict what next year's population will be on the basis of the previous years. That's good, but what we're really interested in is the effect of a change in fishing quotas on the population of the cod stocks. Thus we want our model to be a good predictor of what happens when we change the underlying conditions. Many models can mimic what we see, particularly since we see only short and noisy time series. But only a restricted subset of the mimicking models will give us adequate insights into the behaviour of the system once altered. For this reason it is important to develop statistical tools that are as good as possible at rejecting models that do not mimic the data. Of course, that is not sufficient, as even then there will be multiple mimicking models -- at some point the biological insight of, in our example, the fisheries ecologist, must enter. Nonetheless, good rejection criteria are of the utmost importance as they allow the "model space” of possible models to be pruned as much as possible.

For this reason we are actively investigating "Landsberg-Rényi Disorder” measures, and some derived "Complexity” measures for their power in discriminating between mathematical models. These disorder measures are entropy-based but independent of system size, rather like fractal dimensions, with which, indeed, they share deep links. The statistical behaviour of their address-sample estimators is well understood, and their calculation from datasets is routine once certain parameters related to how the data is binned are chosen.

At the moment we are using the Landsberg-Rényi disorder to study the discrete logistic map x(n+1) = mx(n)[(1-x(n)], which demonstrates all possible statistical behaviours: fixed point, periodicity, or chaos, depending on the value of the growth parameter m. The statistical behaviour of this map is a very delicate function of m. We study the ability of the disorder measures to discriminate between these cases -- the optimal measure to use and the optimal binning with which to calculate this measure, as a function of the size of the available dataset. We investigate the effect of measurement error or "noise” on this optimization process. As the mathematical questions involved here are both subtle and difficult, we expect to be engaged in this phase of the study for approximately a year. Once we have created these optimized statistical tools, and checked them against other known models, we will apply them to the study of real ecological, epidemiological, and climatological datasets and to use them as rejection criterion for models of these datasets.

Welches sind die Bezüge zwischen dem Thema/Inhalt des Beitrags und einem oder mehreren der drei Teilschwerpunkte des GUS?

The broader purpose of our research into nonlinear dynamics is to gain insight into the often counterintuitive behaviour of complicated ecological systems. Such insight is needed if sustainable and environmentally responsible programs of resource management and extraction are to be designed. Thus there is a connection with the first, Umweltverantwortliches Handeln, of the GUS Teilschwerpunkte.

The program outlined above arises from the mathematical theory of discrete maps, and as such is well suited to the analyis of not only ecological data but also epidemiological data. Recent years have seen many outbreaks of "new” and terrifying epidemics such as AIDS and Ebola. Speculative theories relating these outbreaks to massive environmental change have been proposed. Our tools will be as useful in the study of models of this nature as they will be in the study of natural resource extraction models, making a link with the third Teilschwerpunkt, Umwelt und Gesundheit.

Another area using complicated, nonlinear mathematical models containing many interacting variables is climatology. Global warming is an area of particular current interest. The models in this area are not discrete maps but ordinary, partial, and integro-differential equations. Nonetheless, Landsberg-Rényi disorders may be calculated for the output of these models and for real datasets, and could, potentially, be used as rejection criteria for such models. These climatological models need to have predictive power, not only under current conditions but also under altered conditions, if they are to be of use for construction of adequate programs of climate change amelioration. Thus there is a link between our research and the second Teilschwerpunkt, Klimaforschung und Immissonsökologie.


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