### Probability Bounds Analysis In Modeling Nonlinear Ecosystem Dynamics

Joshua A. Enszer and Mark A. Stadtherr. Chemical and Biomolecular Engineering, University of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, IN 46556

One major motivation for studying ecological systems with numerical models is the potential use of such models in studying environmental impacts from outside influences, such as the introduction of a pollutant, or other anthropogenic disturbances. This allows for experimentation under a wide variety of possible conditions without causing harm to real-world ecosystems. Food chains and webs in ecology may be modeled using systems of ordinary differential equations (ODEs), in which the state variables represent the population of different species within the ecosystem, and the equations capture the interactions between the different species. The effects of outside disturbances on ecosystem characteristics are generally not precisely known, but can often be bounded by a range of values. 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. This means that the ecosystem model includes parameters that are not precisely known, but which can be bounded or described with some probability distribution. Thus, in the numerical simulation of ecosystem dynamics, it is desirable to be able to rigorously capture all possible results (population trajectories) over a range of parameter values. If a probability distribution is available for the uncertain parameters, then it is also desirable to be able to compute bounds on the resulting probability distribution for the population values at any particular time.

In this presentation, we demonstrate 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 uses 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. Assuming an uncertain probability distribution for each parameter and/or initial condition of interest, we use a method, based on Taylor models and probability boxes (p-boxes) and recently described by Enszer et al. [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 tested on a series of small food chains or webs representable by nonlinear ODE systems, including the tritrophic Rosenzweig-MacArthur model. The results are compared to those 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., Lin, Y., Ferson, S., Corliss, G.F., Stadtherr, M.A. Propagating Uncertainties in Modeling Nonlinear Dynamic Systems. In Proceedings of the 3rd International Workshop on Reliable Engineering Computing, Georgia Institute of Technology, Savannah, GA: pp. 89--105, 2008.