  In this paper, we study the destabilization of synchronous periodic solutions
for patch models. By applying perturbation theory for matrices, we derive
asymptotic expressions of the Floquet spectra and provide a destabilization
criterion for synchronous periodic solutions arising from closed orbits or
degenerate Hopf bifurcations in terms of period functions. Finally, we apply
the main results to the well-known two-patch Holling-Tanner model.
