(200a) Modeling Impacts of Contaminants in An Aquatic Community: Bounding Effects of Uncertainty

Numerical ecosystem models are important for their use in studying environmental phenomena and potential effects of anthropogenic disturbances, allowing for experiments using a number of possible conditions without causing harm to the environment. As an example, food chains and webs may be modeled using systems of ordinary differential equations (ODEs), in which the state variables represent the population of different trophic levels within an ecosystem. The effects of outside disturbances are generally not precisely known, but can often be bounded by ranges of values or probability distributions. For example, introduction of a pollutant may cause the intrinsic death rates in an ecosystem to change, but the exact values of those parameters are uncertain. Therefore, in numerical simulations, it is desirable to rigorously capture al possible results (population trajectories) for a range of parameter values. It is also not uncommon in nonlinear population models to encounter bifurcations, or qualitative changes in the long-term behavior of a system given a change in a parameter. It is useful to provide verified computations across such a parameter, and specifically, to propagate any probabilistic knowledge over time.

In this presentation, we apply a method for the verified solution of nonlinear ODE models, thus computing rigorous bounds on the population of a species over a given time period, based on the ranges of uncertain values. The method is based on the general approach described by Lin and Stadtherr [1], which uses an interval Taylor series to represent dependence on time, and Taylor models to represent dependence on uncertain parameters and/or initial conditions. We also demonstrate an approach for the propagation of uncertain probability distributions in one or more model parameter and/or initial condition through a population model using a method, based on Taylor models and probability boxes (p-boxes) [2], that propagates these distributions through the dynamic model. As a result, we obtain a p-box describing the probability distribution for each species population at any given time of interest. As opposed to the traditional Monte Carlo simulation approach, which may not accurately bound all possible results of nonlinear models under uncertainty, this Taylor model method for verified probability bounds analysis fully captures all possible system behaviors.

The methods presented are applied to a mathematical model of an aquatic community consisting of the benthic consumer Dreissena polymorpha (zebra mussel), the pelagic consumer Daphnia magna (water flea), and a common food source, the phytoplankton Chlamydomonas reinhardtii. The parameters in the model are experimentally determined include additional modifying parameters for the effects of chemical contamination [3]. The results are compared to those for the same model obtained by Monte Carlo simulations.

[1] Lin, Y., Stadtherr, M.A. Validated Solutions of Initial Value Problems for Parametric ODEs. Applied Numerical Mathematics, 57: pp. 1145--1162, 2007.

[2] Enszer, J.A, Stadtherr, M.A. Rigorous Propagation of Imprecise Probabilities in Process Models. In Proceedings of the 7th International Conference on Foundations of Computer-Aided Process Design, Breckenridge, CO, June 7-12, 2009.

[3] Kulacki, K.J., Costello, D.M., Enszer, J.A., and Lamberti, G.A. Predicting the Toxicity of Novel Chemicals to Benthic and Pelagic Organisms Using Experimentation and Mathematical Modeling. Presented at the Society of Environmental Toxicology and Chemistry, Ohio Valley Chapter, 25th Annual Meeting, Bloomington, IN, October 3, 2008.