Spectral stability correspondence between networks and continuous media: Theory and applications to population dynamics.

We investigate the stability of synchronized oscillations in coupled nonlinear systems by establishing a spectral correspondence between continuous linear shift-invariant (LSI) media and discrete networks. In this framework, Fourier modes of a continuous spatial operator and eigenmodes of a network coupling matrix are treated as spectral parameters of the same master stability function. This correspondence allows finite-wavenumber instabilities of continuous media to be translated into predictable instability windows in the network coupling space. Applying the framework to zero-row-sum Metzler coupling matrices and using a competitive Lotka-Volterra model as a paradigm, we show that synchronization may exhibit reentrant behavior: it is stable for weak coupling, lost within intermediate coupling intervals, and restored at stronger coupling. The framework also reveals a distinction between undirected and directed networks. For undirected networks, the relevant spectra are real and the resulting instability mechanism is analogous to that of standard reaction-diffusion systems with real wavenumbers. Directed networks, however, can possess complex spectra. We show that such complex spectral modes can induce quasiperiodic bifurcations of the synchronized state, leading to dynamical regimes that are inaccessible to standard real-wavenumber reflection-invariant reaction-diffusion models.