Computation of equilibrium states and bifurcations using interval analysis: Application to food chain models

Food chains and webs in the environment are highly nonlinear and interdependent systems. When these systems are modeled using simple sets of ordinary differential equations, these models can exhibit very rich and complex mathematical behaviors. We present here a new equation-solving technique for computing all equilibrium states and bifurcations of equilibria in food chain models. The method used is based on interval analysis, in particular an interval-Newton/generalized-bisection algorithm. Unlike the continuation methods often used in this context, the interval method provides a mathematical and computational guarantee that all roots of a nonlinear equation system are located. The technique is demonstrated using three different food chain models, and results of the computations are used to compare the models.