Ecology is a science moving away from its natural history roots. Mathematical models of population growth have played an important role in increasing the rigor of the discipline. The models which describe the dynamics of population growth have increased in sophistication and complexity. For a practicing ecologist, who is testing ecological theory in the laboratory or in the field, the large number of diverse models makes the process of finding and applying models difficult. The goal of this thesis is to find, build, and apply the most appropriate tools for selecting population models. This will be done by developing tools which provide a correct assessment of the uncertainty in the internal structure of the model and the quality of the data used to parameterize and test the model. Rather than making qualitative assessments of model quality these tools will make quantitative statements. Metrics for model quality will lead to a more rigorous model selection process.

For a model there are two basic types of uncertainty that need to be handled: reducible and inherent uncertainty in the system. Reducible uncertainty can be made smaller through effort, for example, using more precise measurement instruments. Inherent variation by its nature cannot be reduced through effort; this type of uncertainty is known as stochastic variation. An example of stochastic variation is the amount of rain fall in the month of March at a particular location. The amount of rain changes from year to year and is best described not as a single number or formula but as a probability distribution.

The mathematics of handling stochastic variation in models has a rich and long history. It has been applied to many ecological models (Lewontin and Cohen, 1969; May, 1973). The development of mathematical methods for propagating reducible uncertainty is more recent. Interval analysis, which defines an arithmetic on intervals of the real line, is a technique for propagating uncertainty in mathematical expressions (Moore, 1966, 1979; Hayes, 2003). An interval is a range with its endpoints defined by a pair of numbers with the first number being less than or equal to the second. The upper and lower values of an interval bound the measurement. No assumptions about the distribution of a measurement between the endpoints need to be made when encoding a measurement as an interval. An interval can be used to represent the uncertainty in not knowing the exact values of an input parameter to a model. The implications of applying arithmetic on intervals to ecological models is currently not known. Applying interval arithmetic to ecological models should provide a tool for selecting models based on complexity.

Consider a three component model with each component having two links with a total of three links. Each component describes a resource pool like energy or biomass. Each link is a differential equation which describes how the amount of biomass or energy changes the rate of flow between the two connected components. Biological evidence shows that rather than a single link (one-way flow) there is feedback between the connected components. Now there is a total of six links in the system. Additional biological research shows that rather than there being three components there are total of six components. One can see how a model like this can quickly grow in complexity to a point where the model becomes a tangled web of equations. If the uncertainty in the inputs to the systems are represented as intervals and correctly propagated through the bounds on the output should reveal if any conclusions can be made. If the bounds are too wide to draw any useful information out of the system then there are two potential solutions. The first is to reduce the width of the bounds on the input parameters until conclusions can be drawn from the model. The second is to construct a simpler model to use with fewer components and links to the components. While the simpler model might be biologically less realistic, it is tractable and can give real results.

Applying the methods described above should lead to selection of better ecological models. The result over time is that the pool of available models to use will become more likely to give results. This will make the process of selecting and testing models easier. For the theoretician a better pool of models will lead to more focused work, and, hopefully, to advances in understanding the mechanisms of population growth. The expected result of the selection process is that the overly complex models will be discarded and a set of simple models will be maintained.

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