SUPPORTING INFORMATION from Critical transitions in spatial systems induced by Ornstein–Uhlenbeck noise: spatial mutual information as a precursor

Complex dynamical systems are subject to perturbations across space and time, which can induce a critical transition or tipping in the state of the system. External perturbations are often correlated in time and can interplay with the underlying nonlinearity of the spatial system, affecting the occurrence of critical transitions. Theoretical analysis of the spatial system perturbed by the Ornstein–Uhlenbeck (OU) correlated noise poses challenges beyond the white noise assumptions and is yet to be done. Here, we resort to the mean-field approximation of a spatially extended system perturbed with OU noise and obtain the stationary probability density function deriving the Fokker–Planck equation for the same. This allows us to determine the role of diffusion and noise on the resilience of the spatial system. While the theoretical analysis guides us on the landscape of tipping thresholds of the system, critical transitions customary to a variety of systems, require a priori prediction. Here, we propose a probabilistic information-based indicator—spatial mutual information—that can successfully forecast tippings, complementing the previously developed spatial indicators. Further, validating its reliability on empirical data, we show that spatial mutual information serves as a robust indicator capturing information characteristic to an imminent tipping reaching peaks in its vicinity

Rhythmogenesis, birhythmicity and chaos in a metapopulation model

We study a mean-field coupled ecological model of metapopulation in which species from spatially fragmented habitats are connected through dispersal. The interplay between dispersal and heterogeneous characteristics of connected habitats determines the mechanisms in promoting dispersal-induced synchrony and stability of metapopulation. Depending on dispersal rate and environmental heterogeneity, the mean-field coupling acts as a feedback to the system which enables various rhythmic processes such as rhythmogenesis and birhythmicity. In addition to that, heterogeneity determines period-doubling of limit cycle which further leads to chaotic oscillation