Reliable Location of Equilibrium States and Bifurcations in Nonlinear Dynamical Systems With Applications in Food Web Modeling and Chemical Engineering

The characterization of complex physical phenomena, often modeled as nonlinear dynamical systems, is a problem of frequent interest in science and engineering. The topology of a nonlinear system may qualitatively change (bifurcate with changes in model parameters. Location of equilibrium states and bifurcations requires the solution of a nonlinear algebraic system, but standard methods used to solve such systems are inherently fallible and may fail to find all solutions. Complete knowledge of the location and nature of all solutions to a set of nonlinear equations is necessary to reliably analyze model systems. In this work, a new method for bifurcation analysis was developed based upon interval mathematics, specifically an interval-Newton approach combined with generalized bisection. This new bifurcation analysis technique is completely reliable, providing a mathematical and computational guarantee that all equilibrium states and bifurcations within parameter intervals of interest will be located.

Modeling food chains and food webs is the application of particular interest here because ecosystem models aid the proactive assessment and management of environmental risks. A new class of compounds known as room temperature ionic liquids (RTILs) motivates this interest since RTILs show great potential in several industrial applications. RTILs are nonvolatile; therefore, their use may be a ÌøåÀå_greenÌøåÀå_ alternative to volatile organic compounds, a major source of air pollution. The environmental risks of RTILs to aqueous ecosystems should beassessed proactively, prior to industrial scale use; ecosystem modeling is a key tool needed to accomplish this. Bifurcation analysis of food chain and food web models aids in determining ecologically relevant parameters from a risk management perspective. Using mathematical models to study species interdependence and contaminant effects in ecosystems may elucidate what steps can be taken to mitigate the environmental consequences of pollution.

The new bifurcation analysis method was tested by application to a tritrophic Rosenzweig-MacArthur model and two variations thereof, a tritrophic system in a chemostat (CanaleÌøåÀå_s model), and an experimentally verified algae-rotifer food chain model. The technique was also applied to a seven species and a twelve species food web model, neither having been previously solved. Bifurcation analysis of a modification of CanaleÌøåÀå_s model was used to study contamination effects under different scenarios of lethality to species in the food chain.

The new analysis technique was also used to study a continuously stirred tank reactor (CSTR) model. Such models are known to produce isolated solution branches (isola), which standard methods may fail to locate. Locating isola is critical to the full characterization of reactor behavior. The interval-based technique succeeded in locating all solutions, including those on isola.