A topological theorem for nonlinear dynamics in chemical and ecological networks.

A topological theorem is given which places restrictions on the numbers and types of critical points in nonlinear chemical and ecological systems, provided certain stringent conditions on the flows on the boundary of the concentration space are satisfied. The sets of critical points that are found for stable and unstable oscillations, and for competitive exclusion, represent the simplest sets of critical points that satisfy the topological theorem. The restrictions that are imposed on bifurcations of the dynamics are discussed.