Qualitative behavior of predator-prey communities

We examine a set of n-species predator-prey models which incorporate functional responses of the predators to their prey and non-linear intraspecific interactions. We review the limitations of a linear analysis around equilibrium states and provide an extension of the Routh-Hurwitz criteria to the non-linear regime by using Birkhoff's normal forms of differential equations. Qualitative properties like the orbit structure around isolated singularities become clear in this method. It is possible to obtain the radius of the torus and the renormalized frequencies when the eigenvalues of the community matrix have small positive real parts, and to classify different topological structures near bifurcation values of a convenient set of control parameters. Examples of two, three, and four species are analyzed in the context of normal forms. We conclude with some suggestions concerning the coupling of a small subsystem to a large community, and the relations between the graphical method of isoclines and normal forms. This correspondence indicates a road to generalize the study of many-dimensional systems when the intuition provided by the graphical method fails.