Doctoral Dissertations

Date of Award

5-1997

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Thomas G. Hallam

Committee Members

James A. Drake, Sergey Gavrilets, Louis J. Gross, Mark Kot

Abstract

The basis for this project, retrospective risk assessment, entails determining from the output of a Daphnia population model, the level of a stressor that has been affecting the population. In this study, the stressor is a toxic chemical that causes sublethal effects. Our goal is to find an indicator that accurately estimates the chemical concentration level to which a model Daphnia population has been exposed.

Initial investigations for indicators focus on the dynamical behavior of the Daphnia population model. The dynamical systems approaches of frequency analysis, phase portrait reconstruction, and rotation number are used to examine the time-series data generated by the model for trends that could be indicators of chemical concentration. However, for each of these approaches as chemical concentration increases no discernible pattern is observed.

This led to considering simple descriptive statistics of the time-series data as possible indicators. The mean values of the time-series data plotted against chemical concentration yielded one- to-one graphs that can be used to estimate chemical concentration. However, the maximum and minimum values of the time-series data produce a range of variation around each mean value which gives a spectrum of predicted chemical concentrations. This spectrum can be so large that this indicator is rendered useless.

Because time-series data did not provide any useful indicators, the population level parameter juvenile period length is then considered as a possible indicator. The juvenile period length data obtained from the model has a one-to-one correspondence with chemical concentration and no range of variation making it an appropriate indicator of chemical concentration.

Lastly, simpler models that are analytically tractable but retain important features of the original Daphnia population model are analyzed. To examine the effect of juvenile period length, a delay differential equation model is formulated. Analysis reveals that juvenile period length alone cannot reproduce the dynamical behavior of the original Daphnia population model. A system coupling an ordinary differential equation and a difference equation is then formulated to investigate the effect of the periodic birth cycle of the original Daphnia population model. This system produces stable periodic solutions reminiscent to those of the original Daphnia population model.

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