1. Aquatic plants are a key component of spatial heterogeneity in a waterscape, contributing to habitat complexity and helping determine diversity at various spatial scales. Theoretically, the more complex a habitat, the higher the number of species present. 2. Few empirical data are available to test the hypothesis that complexity increases diversity in aquatic communities (e.g. Jeffries, 1993). Fractal dimension has become widely applied in ecology as a tool to quantify the degree of complexity at different scales. 3. We investigated the hypothesis that complexity in vegetated habitat in two tropical lagoons mediates littoral invertebrate number of taxa (S) and density (N). Aquatic macrophyte habitat complexity was defined using a fractal dimension and a gradient of natural plant complexities. We also considered plant area, plant identity and, only for S, invertebrate density as additional explanatory variables. 4. Our results indicate that habitat complexity provided by the different architectures of aquatic plants, significantly affects both S and total N. However, number of individuals (as a result of passive sampling) also helps to account for S and, together with plant identity and area, contributes to the determination of N. We suggest that measurements of structural complexity, measured through fractal geometry, should be included in studies aimed at explaining attributes of attached invertebrates at small (e.g. plant or leaf) scales.

No abstract

No abstract

Two empirical, but plausible, previously published independent generalizations of the standard Poisson−Nernst−Planck (PNP) continuum diffusion model for mobile-charge conduction in liquids and solids are discussed, their responses are compared, and their physical appropriateness and usefulness for data fitting are investigated. They both involve anomalous diffusion of PNPA type with power-law frequency-response elements involving fractional exponents. Both models apply to finite-length regions of material between completely blocking electrodes and, for simplicity, deal primarily with positive and negative charge carriers of equal valence numbers and mobilities. The charge carriers may be either ionic or electronic. The first PNPA model, model A, involves the common separation of the expression for ordinary PNP impedance into an interface diffusion part and a high-frequency limiting conductance and capacitance, followed by the replacement of all normal diffusion elements in the former by anomalous ones. It predicts the presence of the usual PNP plateau in the real part of the total impedance below the Debye relaxation frequency, followed at sufficiently low frequencies by an anomalous-diffusion power-law increase above the plateau. The second model, C, alternatively generalizes the normal time derivatives in the continuity equation by replacing them with fractional ones and leads to no plateau, except in the PNP limit, but instead predicts an immediate power-law increase as the frequency decreases below the Debye relaxation one. Fitting of experimental frequency response data sets for three disparate materials leads to much poorer fits for model C than for model A.

No abstract

The influence of the ions on the electrochemical impedance of a cell is calculated in the framework of a complete model in which the fractional drift-diffusion problem is analytically solved. The resulting distribution of the electric field inside the sample is determined by solving Poisson's equation. The theoretical model to determine the electrical impedance we are proposing here is based on the fractional derivative of distributed order on the diffusion equation. We argue that this is the more convenient and physically significant approach to account for the enormous variety of the diffusive regimes in a real cell. The frequency dependence of the real and imaginary parts of the impedance are shown to be very similar to the ones experimentally obtained in a large variety of electrolytic samples.

Several indices have been created to measure diversity, and the most frequently used are the Shannon-Wiener (H) and Simpson (D) indices along with the number of species (S) and evenness (E). Controversies about which index should be used are common in literature. However, a generalized entropy (Tsallis entropy) has the potential to solve part of these problems. Here we explore a family of diversity indices (S q ; where q is the Tsallis index) and evenness (E q ), based on Tsallis entropy that incorporates the most used indices. It approaches S when q 00, H when q 01 and gives D when q 0 2. In general, varying the value of the Tsallis index (q), S q varies from emphasis on species richness (qB1) to emphasis on dominance (q 1). Similarly, E q also works as a tool to investigate diversity. In particular, for a given community, its minimum value represents the maximum deviation from homogeneity (E q

The electrical response of an electrolytic cell in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order is analyzed. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poisson's equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.