Destabilization of synchronous periodic solutions for patch models.

We study the destabilization of synchronous periodic solutions for general patch-models with cross-diffusion-like couplings, where the underlying kinetic systems have stable periodic solutions bifurcating from double homoclinic loops. We first reduce the stability problem of synchronous periodic solutions for patch models into that for lower-dimensional systems, then analyze the destabilization by using the Floquet theory and solving a class of boundary value problems. After establishing the Fredholm alternative properties for an auxiliary linear operator, we give the characteristic function to determine the Floquet spectra for the reduced systems, and then the conditions for the destabilization. Finally, we apply the main results to a patch model with a two-dimensional kinetic system. [ABSTRACT FROM AUTHOR]