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Biodiversity offsetting is a conservation tool that compensates for the residual loss of biodiversity at one geographic locality with an equivalent gain elsewhere. The goal is generally to ensure no net loss of biodiversity after development. One path to no net loss is preventing the imminent loss of biodiversity at the offset location through increased conservation; a strategy known as averted-loss. No net loss is most feasible in these instances when (a) the initial loss of biodiversity is small, (b) the background decline in biodiversity is rapid and (c) conservation interventions are effective. This study, however, demonstrates that the functional form of the biodiversity trajectory also plays a considerable role in determining the feasibility of achieving no net loss. Using a simple framework of equations with fraction exponents, I demonstrate that the time required to reach no net loss is generally longer when background biodiversity trajectories are concave, compared to when they are convex. Moreover, the net effect of conservation must be greater for concave biodiversity trajectories if no net loss is to be reached within time-frames similar to convex trajectories. Incorporating different forms of biodiversity trajectories into existing frameworks may have major implications for the feasibility of averted-loss biodiversity offsets. [ABSTRACT FROM AUTHOR]

It is well known that the extinction properties along lines of sight to Type Ia supernovae (SNe Ia) are described by steep extinction curves with unusually low total-to-selective extinction ratios of R V = 1.0 − 2.0 . In order to reveal the properties of interstellar dust that causes such peculiar extinction laws, we perform the fitting calculations to the measured extinction curves by applying a two-component dust model composed of graphite and silicate. As for the size distribution of grains, we consider two function forms of the power-law and lognormal distributions. We find that the steep extinction curves derived from the one-parameter formula by Cardelli et al. (1989) with R V =2.0, 1.5, and 1.0 can be reasonably explained even by the simple power-law dust model that has a fixed power index of − 3.5 with the maximum cut-off radii of a max ≃ 0.13 μ m , 0.094 µm, and 0.057 μ m , respectively. These maximum cut-off radii are smaller than a max ≃ 0.24 μ m considered to be valid in the Milky Way, clearly demonstrating that the interstellar dust responsible for steep extinction curves is highly biased to smaller sizes. We show that the lognomal size distribution can also lead to good fits to the extinction curves with R V = 1.0 – 3.1 by taking the appropriate combinations of the relevant parameters. We discuss that the extinction data at ultraviolet wavelengths are essential for constraining the composition and size distribution of interstellar dust. [ABSTRACT FROM AUTHOR]