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There are many scales at which to quantify stability in spatial and ecological networks. Local-scale analyses focus on specific nodes of the spatial network, while regional-scale analyses consider the whole network. Similarly, species- and community-level analyses either account for single species or for the whole community. Furthermore, stability itself can be defined in multiple ways, including resistance (the inverse of the relative displacement caused by a perturbation), initial resilience (the rate of return after a perturbation), and invariability (the inverse of the relative amplitude of the population fluctuations). Here, we analyze the scale-dependence of these stability properties. More specifically, we ask how spatial scale (local vs regional) and ecological scale (species vs community) influence these stability properties. We find that regional initial resilience is the weighted arithmetic mean of the local initial resiliences. The regional resistance is the harmonic mean of local resistances, which makes regional resistance particularly vulnerable to nodes with low stability, unlike regional initial resilience. Analogous results hold for the relationship between community- and species-level initial resilience and resistance. Both resistance and initial resilience are ``scale-free'' properties: regional and community values are simply the biomass-weighted means of the local and species values, respectively. Thus, one can easily estimate both stability metrics of whole networks from partial sampling. In contrast, invariability generally is greater at the regional and community-level than at the local and species-level, respectively. Hence, estimating the invariability of spatial or ecological networks from measurements at the local or species level is more complicated, requiring an unbiased estimate of the network (i.e. region or community) size.

Environmental fluctuations can create population-depleted areas and even extinct areas for the population. This effect is more severe in the presence of the Allee effect (decreasing growth rate at low population densities). Dispersal inside the habitat provides a rescue effect on population-depleted areas, enhancing the population resilience to environmental fluctuations. Habitat reduction decreases the effectiveness of the dispersal rescue mechanism. We report here how the population resilience to environmental fluctuations decreases when the dispersal length or the habitat size are reduced. The resilience reduction is characterized by a decrease of the extinction threshold for environmental fluctuations. The extinction threshold is shown to scale with the ratio between the dispersal length and the scale of environmental synchrony, i.e., it is the dispersal connection between non-environmentally-correlated regions that provides resilience to environmental fluctuations. Habitat reduction also decreases the resilience to environmental fluctuations, when the habitat size is similar to or smaller than the characteristic dispersal distances. The power laws of these scaling behaviors are characterized here. Alternative scaling functions with spatial scales of population synchrony are found to fit the simulations worse. These results support the dispersal length as the critical scale for extinction induced by habitat reduction., Comment: 24 pages

Many species are unsustainable at small population densities (Allee Effect), i.e., below a threshold named Allee threshold, the population decreases instead of growing. In a closed local population, environmental fluctuations always lead to extinction. Here, we show how, in spatially extended habitats, dispersal can lead to a sustainable population in a region, provided the amplitude of environmental fluctuations is below an extinction threshold. We have identified two types of sustainable populations: high-density and low-density populations (through a mean-field approximation, valid in the limit of large dispersal length). Our results show that patches where population is high, low or extinct, coexist when the population is close to global extinction (even for homogeneous habitats). The extinction threshold is maximum for characteristic dispersal distances much larger than the spatial scale of synchrony of environmental fluctuations. The extinction threshold increases proportionally to the square root of the dispersal rate and decreases with the Allee threshold. The low-density population solution can allow understanding difficulties in recovery after harvesting. This theoretical framework provides a novel approach to address other factors, such as habitat fragmentation or harvesting, impacting population resilience to environmental fluctuations., Comment: 32 pages, 9 figures, 1 table

In the present article we study the Inverse Electrodynamics Model. This model is a gauge and parity invariant non-linear Electrodynamics theory, which respects the conformal invariance of standard Electrodynamics. This modified Electrodynamics model, when minimally coupled to General Relativity, is compatible with static and spherically symmetric Reissner-Nordstrom-like black-hole solutions. However, these black-hole solutions present more complex thermodynamic properties than their Reissner-Nordstrom black-hole solutions counterparts in standard Electrodynamics. In particular, in the Inverse Model a new stability region, with both the heat capacity and the free energy negative, arises. Moreover, unlike the scenario in standard Electrodynamics, a sole transition phase is possible for a suitable choice in the set of parameters of these solutions., Comment: 9 pages, 2 figures, minor changes, published version at "Open Questions in Black Hole Physics" special issue, Universe 2015, 1, 412-421