Food chains and webs in the environment can be modeled by systems of ordinary differential equations that approximate species or functional feeding group behavior with a variety of functional responses. We present here a new methodology for computing all equilibrium states and bifurcations of equilibria in food chain models. The methodology used is based on interval analysis, in particular an interval-Newton/generalized-bisection algorithm that provides a mathematical and computational guarantee that all roots of a nonlinear equation system are enclosed. The procedure is initialization-independent, and thus requires no a priori insights concerning the number of equilibrium states and bifurcations of equilibria or their approximate locations. The technique is tested using several example problems involving tritrophic food chains.
In many applications of interest in chemical engineering it is necessary to deal with nonlinear models of complex physical phenomena, on scales ranging from the macroscopic to the molecular. Frequently these are problems that require solving a nonlinear equation system and/or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. Interval analysis provides tools with which these reliability issues can be addressed, allowing such problems to be solved with complete certainty. This paper will focus on three types of applications: 1) parameter estimation in the modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) nonlinear dynamics, in particular the location of equilibrium states and bifurcations of equilibria in ecosystem models used to assess the risk associated with the introduction of new chemicals into the environment; 3) molecular modeling, with focus on transition state analysis of the diffusion of a sorbate molecule in a zeolite.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.