Primes and prime ideals in short intervals

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.Comment: minor change to Proposition

Forecasting potential invaders to prevent future biological invasions worldwide

International audienceThe ever‐increasing and expanding globalisation of trade and transport underpins the escalating global problem of biological invasions. Developing biosecurity infrastructures is crucial to anticipate and prevent the transport and introduction of invasive alien species. Still, robust and defensible forecasts of potential invaders are rare, especially for species without known invasion history. Here, we aim to support decision‐making by developing a quantitative invasion risk assessment tool based on invasion syndromes (i.e., generalising typical attributes of invasive alien species). We implemented a workflow based on ‘Multiple Imputation with Chain Equation’ to estimate invasion syndromes from imputed datasets of species' life‐history and ecological traits and macroecological patterns. Importantly, our models disentangle the factors explaining (i) transport and introduction and (ii) establishment. We showcase our tool by modelling the invasion syndromes of 466 amphibians and reptile species with invasion history. Then, we project these models to amphibians and reptiles worldwide (16,236 species [c.76% global coverage]) to identify species with a risk of being unintentionally transported and introduced, and risk of establishing alien populations. Our invasion syndrome models showed high predictive accuracy with a good balance between specificity and generality. Unintentionally transported and introduced species tend to be common and thrive well in human‐disturbed habitats. In contrast, those with established alien populations tend to be large‐sized, are habitat generalists, thrive well in human‐disturbed habitats, and have large native geographic ranges. We forecast that 160 amphibians and reptiles without known invasion history could be unintentionally transported and introduced in the future. Among them, 57 species have a high risk of establishing alien populations. Our reliable, reproducible, transferable, statistically robust and scientifically defensible quantitative invasion risk assessment tool is a significant new addition to the suite of decision‐support tools needed for developing a future‐proof preventative biosecurity globally

Global correlates of terrestrial and marine coverage by protected areas on islands

Many islands are biodiversity hotspots but also extinction epicenters. In addition to strong cultural connections to nature, islanders derive a significant part of their economy and broader wellbeing from this biodiversity. Islands are thus considered as the socio-ecosystems most vulnerable to species and habitat loss. Yet, the extent and key correlates of protected area coverage on islands is still unknown. Here we assess the relative influence of climate, geography, habitat diversity, culture, resource capacity, and human footprint on terrestrial and marine protected area coverage across 2323 inhabited islands globally. We show that, on average, 22% of terrestrial and 13% of marine island areas are under protection status, but that half of all islands have no protected areas. Climate, diversity of languages, human population density and development are strongly associated with differences observed in protected area coverage among islands. Our study suggests that economic development and population growth may critically limit the amount of protection on islands. © 2020, The Author(s)