  As proposed by Alan Turing in 1952 as a ubiquitous mechanism for
nonequilibrium pattern formation, diffusional effects may destabilize uniform
distributions of reacting chemical species and lead to both spatially and
temporally heterogeneous patterns. While stationary Turing patterns are broadly
known, the oscillatory instability, leading to traveling waves in continuous
media 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 ecological
metapopulations of biological species with dispersal connections between
habitats. Remarkably, the oscillatory Turing instability does not lead to wave
patterns in networks, but to spontaneous development of heterogeneous
oscillations and possible extinction of some species, even though they are
absent for isolated populations. Furthermore, our theoretical analysis reveals
that this instability is more common in ecological metapopulations than in
chemical reactions. Indeed, we find the instabilities for all possible food
webs with three predator or prey species, under various assumptions about the
mobility of individual species and nonlinear interactions between them.
Therefore, we suggest that the oscillatory Turing instability is generic and
must play a fundamental role in metapopulation dynamics, providing a common
mechanism for dispersal-induced destabilization of ecosystems.
