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As proposed by Alan Turing in 1952 as a ubiquitous mechanism fornonequilibrium pattern formation, diffusional effects may destabilize uniformdistributions of reacting chemical species and lead to both spatially andtemporally heterogeneous patterns. While stationary Turing patterns are broadlyknown, the oscillatory instability, leading to traveling waves in continuousmedia and also called the wave bifurcation, is rare for chemical systems. Here,we extend the analysis by Turing to general networks and apply it to ecologicalmetapopulations of biological species with dispersal connections betweenhabitats. Remarkably, the oscillatory Turing instability does not lead to wavepatterns in networks, but to spontaneous development of heterogeneousoscillations and possible extinction of some species, even though they areabsent for isolated populations. Furthermore, our theoretical analysis revealsthat this instability is more common in ecological metapopulations than inchemical reactions. Indeed, we find the instabilities for all possible foodwebs with three predator or prey species, under various assumptions about themobility of individual species and nonlinear interactions between them.Therefore, we suggest that the oscillatory Turing instability is generic andmust play a fundamental role in metapopulation dynamics, providing a commonmechanism for dispersal-induced destabilization of ecosystems.

Dispersal-induced destabilization of metapopulations and oscillatory Turing patterns in ecological networks