Reliable Computation of Equilibrium States and Bifurcations in Nonlinear Dynamics

A problem of frequent interest in the analysis of nonlinear ODE systems is the location of equilibrium states and bifurcations. Through the use of bifurcation diagrams, a large amount of information concerning the number and stability of equilibria in an ODE model can be concisely represented. Bifurcations of equilibria are typically found by solving a nonlinear algebraic system consisting of the equilibrium (steady-state) conditions along with one or more augmenting functions [1]. Typically this equation system is solved using some continuationbased tool (e.g., AUTO). However, in general, these methods do not provide any guarantee that all bifurcations will be found, and are often initialization sensitive. Thus, without some a priori knowledge of system behavior, one may not know with complete certainty if all bifurcation curves have been identified and explored. In this presentation, we explore the use of intervalNewton techniques for identifying, with certainty, all equilibrium states and all codimension-1 and codimension-2 bifurcations of interest within specified model parameter intervals. A problem of particular interest is the identification of equilibrium states and bifurcations in food chain and food web models. These models are descriptive of a wide range of behavior in the environment, and are potentially useful in applications such as environmental risk assessment (e.g., [2]). While these models often appear to be mathematically simple, they frequently exhibit a rich mathematical behavior in how the number and stability of equilibria vary with changes in the model parameters (e.g., [3]). Thus, bifurcation analysis is quite useful when characterizing ecological system behavior as it allows for the concise study of a wide range of model parameter values. In this presentation, we will focus on the solution of tritrophic food chains in a chemostat (e.g., Canale’s model), applying an interval methodology to ensure that all equilibrium states and bifurcations of interest are identified.