Mathematically and biologically consistent framework for presence-absence pairwise indices of diversity

Aim A large number of indices that compare two or more assemblages have been proposed or reinvented. The interpretation of the indices varies across the literature, despite efforts for clarification and unification. Most of the effort has focused on interdependence between the indices and the mathematics behind them. At the same time, following issues have been underestimated: (i) the difference between statistical independence of indices and the independence based on their informational value, and (ii) the inferences from the indices about diversity patterns and phenomena. Here we offer an alternative framework for diversity indices. Methods We distinguish different classes of dependence, and show that three indices which are mutually independent in terms of their information content are sufficient for appropriate inferences. This applies regardless of whether the indices are statistically correlated or not. We classify 20 existing indices into three main and four minor mutually independent families, and demonstrate how similarity between assemblages violates the stability of the families, confusing conceptually different patterns. We show what can be inferred about spatial diversity phenomena from different indices, demonstrate problems with most of the indices of nestedness, and show which combinations of indices may be used for meaningful ecological inference. Results and Main conclusions We demonstrate that no single index can properly filter out a single effect of a phenomena because the phenomena inevitably bound each other (e.g. species richness gradient bounds possible values of Jaccard index of community similarity). Consequently, inventing indices which seemingly purify these effect (e.g. pure turnover or pure nestedness) leads to misleading inference. In contrast, a proper inference is obtained by using a combination of classical indices from different, mutually independent families. Our framework provides a practical clue how to compare different indices across the literature.

Gradient bounds for solutions of nonlinear strictly elliptic equations, applications and extensions

In this short manuscript, we briefly recall some well-known methods for obtaining gradient bounds of viscosity solutions for elliptic and parabolic equations. The two methods we focus here are the idea coming from Ishii- Lions’ method applies for strictly elliptic equations and the other one is socalled weak Bernstein’s method developed by Barles applies for degenerate equations. We present some results based on these methods and their applications. We also discuss some promising extensions in the future.