  The method of generalized modeling has been applied successfully in many
different contexts, particularly in ecology and systems biology. It can be used
to analyze the stability and bifurcations of steady-state solutions. Although
many dynamical systems in mathematical biology exhibit steady-state behaviour
one also wants to understand nonlocal dynamics beyond equilibrium points. In
this paper we analyze predator-prey dynamical systems and extend the method of
generalized models to periodic solutions. First, we adapt the equilibrium
generalized modeling approach and compute the unique Floquet multiplier of the
periodic solution which depends upon so-called generalized elasticity and scale
functions. We prove that these functions also have to satisfy a flow on
parameter (or moduli) space. Then we use Fourier analysis to provide computable
conditions for stability and the moduli space flow. The final stability
analysis reduces to two discrete convolutions which can be interpreted to
understand when the predator-prey system is stable and what factors enhance or
prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm
for parameter space based on nonlinear optimization and the Fast Fourier
Transform which enables us to gain a statistical understanding of the stability
properties of periodic predator-prey dynamics.
