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Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are a prominent example. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. While mean-field approaches may guide recovery strategies by indicating the conditions needed to destabilize undesired states, these approaches are not accurately capturing the transition process toward the desired state of spatially-extended systems in stochastic environments. The difficulty is rooted in the lack of mathematical tools to analyze systems with high dimensionality, nonlinearity, and stochastic effects. We bridge this gap by developing new mathematical tools that employ nucleation theory in spatially-embedded systems to advance resilience restoration. We examine our approach on systems following mutualistic dynamics and diffusion models, finding that systems may exhibit single-cluster or multi-cluster phases depending on their sizes and noise strengths, and also construct a new scaling law governing the restoration time for arbitrary system size and noise strength in two-dimensional systems. This approach is not limited to ecosystems and has applications in various dynamical systems, from biology to infrastructural systems.
A new type of noise-induced synchronous behavior is described. This phenomenon, called incomplete noise-induced synchronization, arises for one-dimensional Ginzburg-Landau equations driven by common noise. The mechanisms resulting in the incomplete n
Noise through its interaction with the nonlinearity of the living systems can give rise to counter-intuitive phenomena. In this paper we shortly review noise induced effects in different ecosystems, in which two populations compete for the same resou
Ecologists have long suspected that species are more likely to interact if their traits match in a particular way. For example, a pollination interaction may be more likely if the proportions of a bees tongue fit a plants flower shape. Empirical esti
We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multi
A Belief Propagation approach has been recently proposed for the zero-patient problem in a SIR epidemics. The zero-patient problem consists in finding the initial source of an epidemic outbreak given observations at a later time. In this work, we stu