(343a) Verified Probability Bounds Analysis around Bifurcations in An Ecosystem Model

Nonlinear systems of ordinary differential equations (ODEs) arise in many models in engineering and the sciences. Such systems usually must be solved numerically, and may admit a variety of possible qualitative and quantitative behaviors. The difficulty in obtaining the numerical solution of such systems increases if initial conditions or parameters are uncertain. If a range of possible initial conditions includes a separatrix, or if a range of uncertain parameter values includes a bifurcation, then it is possible that multiple, distinctively different long-term behaviors may result. In such situations, rigorously capturing all possible behaviors of a nonlinear ODE system can become quite difficult.

In this presentation, we will demonstrate an approach for verified probability bounds analysis for situations in which the intervals of uncertainty may enclose a bifurcation or separatrix. That is, given probability bounds for the uncertainties, in the form of probability boxes (p-boxes), we seek to determine verified bounds on the probability distributions of the state variables at specified times of interest. This makes it possible to determine rigorous bounds on the probability that some specified outcome for a state variable will be achieved. This is a core problem in ecosystem modeling for risk assessment and management.

To do this, we build on the general approach described by Lin and Stadtherr [1] for the verified solution of nonlinear ODE models with uncertain parameters and initial states. Given bounds on the uncertain quantities, this technique (VSPODE) computes rigorous bounds on the trajectories of the state variables, based on use of an interval Taylor series method to represent dependence on time and on use of Taylor models to represent dependence on the uncertain quantities. For propagation of the imprecise probabilities describing the uncertainties, we extend an approach recently described by Enszer et al. [2], which exploits the Taylor models used in VSPODE to obtain probability bounds for the state variables. As opposed to traditional sampling methods, in which it is impossible to consider the complete space of uncertainties in a finite number of trials, this method provides completely rigorous results. Finally, to aid in dealing with a potential bifurcation (or separatrix) within the intervals of uncertainty, we use the method of Gwaltney et al. [3], who use an interval-Newton approach to rigorously identify and enclose bifurcations in nonlinear dynamic systems.

To test the approach that we have developed, we investigate the well-known tritrophic Rosenzweig-MacArthur system, a nonlinear food chain model that simulates an ecosystem of three species (trophic levels). This model is known to exhibit a number of bifurcations and to admit qualitatively different long-term behavior depending on the initial conditions and/or parameter values [4]. We compare the results obtained using our approach to those obtained using standard numerical integration schemes and sampling methods.

[1] Y. Lin and M. A. Stadtherr. Validated solutions of initial value problems for parametric ODEs. Appl. Num. Math., 57:1145--1162, 2007.

[2] J. A. Enszer, Y. Lin, S, Ferson, G. F. Corliss, and M. A. Stadtherr. Probability bounds analysis for nonlinear dynamic process models. AIChE J., in press, 2010.

[3] C. R. Gwaltney, M. P. Styczynski, and M. A. Stadtherr. Reliable computation of equilibrium states and bifurcations in food chain models. Comput. Chem. Eng., 28:1981--1996, 2004.

[4] A. Gragnani, O. De Feo, and S. Rinaldi. Food chains in the chemostat: Relationships between mean yield and complex dynamics. Bull. Math. Biol., 60:703--719, 1998.