Angular velocity variations and stability of spatially explicit prey-predator systems

The linear instability of Lotka-Volterra orbits in the homogenous manifold of a two-patch system is analyzed. The origin of these orbits instability in the absence of prey migration is revealed to be the dependence of the angular velocity on the azimuthal angle; in particular, the system desynchronizes at the exit from the slow part of the trajectory. Using this insight, an analogous model of a two coupled oscillator is presented and shown to yield the same type of linear instability. This enables one to incorporate the linear instability within a recently presented general framework that allows for comparison of all known stabilization mechanisms and for simple classification of observed oscillations.