The species-area relationship (SAR) has been used to predict the numbers of species going extinct due to habitat loss, but other researchers have maintained that SARs overestimate extinctions and instead one should use the endemics-area relationship (EAR) to predict extinctions. Here, we employ spatially explicit simulations of large numbers of species in spatially heterogeneous landscapes to investigate SARs and extinctions in a dynamic context. The EAR gives the number of species going extinct immediately after habitat loss, but typically many other species have unviable populations in the remaining habitat and go extinct soon afterwards. We conclude that the EAR underestimates extinctions due to habitat loss, the continental SAR (with slope~0.1 or somewhat less) gives a good approximation of short-term extinctions, while the island SAR calculated for discrete fragments of habitat (with slope~0.25) predicts the long-term extinctions. However, when the remaining area of land-covering habitat such as forest is roughly less than 20% of the total landscape and the habitat is highly fragmented, all current SARs underestimate extinction rate. We show how the 'fragmentation effect' can be incorporated into a predictive SAR model. When the remaining habitat is highly fragmented, an effective way to combat the fragmentation effect is to aggregate habitat fragments into clusters rather than to place them randomly across the landscape.

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Ω(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Ω(log * n) rounds [Chung et al. 2014].

The species-area relationship (SAR) gives a quantitative description of the increasing number of species in a community with increasing area of habitat. In conservation, SARs have been used to predict the number of extinctions when the area of habitat is reduced. Such predictions are most needed for landscapes rather than for individual habitat fragments, but SAR-based predictions of extinctions for landscapes with highly fragmented habitat are likely to be biased because SAR assumes contiguous habitat. In reality, habitat loss is typically accompanied by habitat fragmentation. To quantify the effect of fragmentation in addition to the effect of habitat loss on the number of species, we extend the power-law SAR to the species-fragmented area relationship. This model unites the single-species metapopulation theory with the multispecies SAR for communities. We demonstrate with a realistic simulation model and with empirical data for forest-inhabiting subtropical birds that the species-fragmented area relationship gives a far superior prediction than SAR of the number of species in fragmented landscapes. The results demonstrate that for communities of species that are not well adapted to live in fragmented landscapes, the conventional SAR underestimates the number of extinctions for landscapes in which little habitat remains and it is highly fragmented.extinction threshold | habitat conversion | metapopulation capacity | Atlantic forest | Nagoya biodiversity agreement T he species-area relationship (SAR) describes a very general pattern in the occurrence of species, which is fundamental to community ecology (1), biogeography (2), and macroecology (3). Since the 1920s (4, 5), SARs have been applied to describe the occurrence of a wide range of organisms on true islands (6-8), in fragments of distinct habitat (9, 10), and in parts of more arbitrarily delimited contiguous landscapes (1, 3). In the past decades, SAR has become an important concept and a tool also in conservation biology, where it has been used to make broad assessments of species extinctions from habitat loss (11-18). These calculations have been criticized for various reasons (17,(19)(20)(21), but minimally SAR provides a valuable point of reference for the threat that habitat loss poses to biodiversity.SARs are typically applied to a set of habitat fragments within a single landscape, but in conservation, in contrast, the essential question is how many species will persist in different landscapes (regions) with dissimilar amounts of habitat rather than in different fragments within a single landscape. This creates a problem: Habitat loss is virtually always accompanied by fragmentation (22-24), and hence the remaining habitat is not contiguous, unlike assumed by SAR, at the landscape level. In other words, SAR does not account for any adverse effects of fragmentation on the occurrence of species (25, 26). Fragmentation matters whenever individual habitat fragments are small enough to reduce the viability of the respective local populations (27, 28). ...

LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log * n), or Θ(n), and the design of optimal algorithms can be fully automated.This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log * n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log * n) or Θ(n) in 2-dimensional grids.Nevertheless, if we correctly guess that the complexity of a problem is Θ(log * n), we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form A • S k , where A is a finite function, S k is an algorithm for finding a maximal independent set in kth power of the grid, and k is a constant.Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations. arXiv:1702.05456v2 [cs.DC] 24 May 2017 1.1 Problem setting: LCL problems on grids 92 33 77 57 49 26 71 79 8 62 48 24 31 21 15 30 60 67 0 5 17 95 23 47 87 80 25 38 20 64 45 61 91 51 69 1 74 55 3 98 88 99 58 53 63 40 16 2 39Grids. In this work, we study distributed algorithms in a setting where the underlying input graph is a grid. Specifically, we consider the complexity of locally checkable labelling problems, or LCL problems, in the standard LOCAL model of distributed complexity, and consider graphs that are toroidal two-dimensional n × n grids with a consistent orientation; we focus on the two-dimensional case for concreteness, but most of our results generalise to d-dimensional grids of arbitrary dimensions. This setting occupies a middle ground between the wellunderstood directed n-cycles [10,32], where all solvable LCL problems are known to have deterministic time complexity either O(1), Θ(log * n) or Θ(n), and the more complicated setting of general n-vertex graphs, where intermediate problems with time complexities such as Θ(log n) are known to exist, even for bounded-degree graphs. Grid-like systems with local dynamics also occur frequently in the study of real-world phenomena. However, grids have so far not been systematically studied from a distributed computing perspective.LOCAL model and LCL problems. In the LOCAL model of distributed computing, nodes are labelled with unique numerical identifiers with O(log n) bits. A time-t algorithm in this model is simply a mapping from radius-t neighbourhoods to local outputs; equivalently, it can be interpreted as a message-passing algorithm in which the nodes exchange messages for t synchronous rounds and then announce their local outputs.LCL problems are graph problems for which the feasibility of a solution can be verified by checking the solution for each O(1)-radius neighbourhood; if all local neighbourhoods look valid, the s...