Uses and abuses of fractal methodology in ecology

Fractals have found widespread application in a range of scientific fields, including ecology. This rapid growth has produced substantial new insights, but has also spawned confusion and a host of methodological problems. In this paper, we review the value of fractal methods, in particular for applications to spatial ecology, and outline potential pitfalls. Methods for measuring fractals in nature and generating fractal patterns for use in modelling are surveyed. We stress the limitations and the strengths of fractal models. Strictly speaking, no ecological pattern can be truly fractal, but fractal methods may nonetheless provide the most efficient tool available for describing and predicting ecological patterns at multiple scales.

Fractal species distributions do not produce power-law species-area relationships

We derive the species-area relationship (SAR) expected from an assemblage of fractally distributed species. If species have truly fractal spatial distributions with different fractal dimensions, we show that the expected SAR is not the classical power-law function, as suggested recently in the literature. This analytically derived SAR has a distinctive shape that is not commonly observed in nature: upward-accelerating richness with increasing area (when plotted on log-log axes). This suggests that, in reality, most species depart from true fractal spatial structure. We demonstrate the fitting of a fractal SAR using two plant assemblages (Alaskan trees and British grasses). We show that in both cases, when modelled as fractal patterns, the modelled SAR departs from the observed SAR in the same way, in accord with the theory developed here. The challenge is to identify how species depart from fractality, either individually or within assemblages, and more importantly to suggest reasons why species distributions are not self-similar and what, if anything, this can tell us about the spatial processes involved in their generation.

The area-independent effects of habitat complexity on biodiversity vary between regions.

Potential explanatory variables often co-vary in studies of species richness. Where topography varies within a survey it is difficult to separate area and habitat-diversity effects. Topographically complex surfaces may contain more species due to increased habitat diversity or as a result of increased area per se. Fractal geometry can be used to adjust species richness estimates to control for increases in area on complex surfaces. Application of fractal techniques to a survey of rocky shores demonstrated an unambiguous area-independent effect of topography on species richness in the Isle of Man. In contrast, variation in species richness in south-west England reflected surface availability alone. Multivariate tests and variation in limpet abundances also demonstrated regional variation in the area-independent effects of topography. Community composition did not vary with increasing surface complexity in south-west England. These results suggest large-scale gradients in the effects of heterogeneity on community processes or demography.

Polymer conditioning of alum sludge and discrepancies between estimates of the optimum dosage

The paper outlines the effects of polymer conditioning on alum sludge properties, such as floc size, density, fractal dimension (DF) and rheological properties. Experimental results demonstrate that polymer conditioning of alum sludge leads to: larger floc size with a plateau reached in higher doses; higher densities associated with higher doses; increased degree of compactness; and an initial decrease followed by an increase of supernatant viscosity with continued increase in polymer dose. The secondary focus of this paper dwells on a comparison of the estimates of optimum dose using different criteria that emanate from established dewatering tests such as CST, SRF, liquid phase viscosity and modified SRF as well as a simple settlement test in terms of CML30. Alum sludge was derived from a water works treating coloured, low-turbidity raw waters.

The role of Mittag-Leffler functions in anomalous relaxation

The inertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al. [Phys. Rev. E, 65, 32 102 (2002)] to describe fractional dynamics and anomalous rotational diffusion. The linear- response theory of the normalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular-velocity correlation function was parametrized via fractal Mittag-Leffler functions. Here we apply the latter method and complex-contour integral- representation methods to determine the original time-dependent amplitude as an inverse Laplace transform using both analytical and numerical approaches, as appropriate. (C) 2004 Elsevier B.V. All rights reserved.

Self-affine crossover length in a layered silicate deposit

Self-affine dehydrated colloidal deposits on fresh mica surfaces of the synthetic layered silicate 2:1 smectite clay laponite have been studied by means of atomic force microscopy (AFM). AFM images of these prepared assemblies of sol and gel aggregates have been analyzed both by means of standard AFM Fourier software and a wavelet method. The deposited surfaces show a persistence to antipersistent crossover with a clay concentration dependent crossover length. It is concluded that the crossover length is associated with aggregate size, and further that the persistent roughness at small length scales signals near compact clusters of fractal dimension three, whereas the antipersistent roughness at large length scales signals a sedimentation process.

Periodically loaded 1-D metallodielectric electromagnetic bandgap structures for miniaturisation and bandwidth enhancement

Periodic loading of 1-D metallodielectric electromagnetic bandgap (MEBG) structures has been rigorously investigated. Miniaturised and broadband MEBG structures have been produced by means of periodically loading a dipole array. A study has been carried out with regard to the loading mechanism, the number of stubs, the topology of the structure and the order of loading. Simulations have been carried out using a method of moments based software. First order uniform loading stubs have yielded a significant size reduction of the MEBG array and the bandwidth has doubled. Good agreement between simulations and measurements has been achieved. The current distribution on the proposed structure has been studied, yielding valuable insight. An interdigital topology has resulted in further miniaturisation and bandwidth enhancement. Fractal-type arrays have been produced after applying second order loading. A maximum miniaturisation of 2.5:1 has been achieved.

Neuropeptide-like protein diversity in phylum Nematoda

This study reports the identification of nematode neuropeptide-like protein (nlp) sequelogs from the GenBank expressed sequence tag (EST) database, using BLAST (Basic Local Alignment Search Tool) search methodology. Search strings derived from peptides encoded by the 45 known Caenorhabatitis elegans nlp genes were used to identify more than 1000 ESTs encoding a total of 26 multi-species nlp sequelogs. The remaining 18 nlps (nlp-4, -16, -24 through -36, -39, -41 and -45) were identified only in C elegans, while the sole EST representative of nlp-23 was from Caenorhabditis remanei. Several ESTs encoding putative antibacterial peptides similar to those encoded by the C elegans genes nlp-24-33 were observed in several parasite species. A novel gene (nlp-46) was identified, encoding a single, amidated dodecapeptide (NIA[I/T]GR[G/A]DG[F/L]RPG) in eight species. Secretory signal peptides were identified in at least one species representing each nlp sequelog, confirming that all 46 nematode nlp genes encode secretory peptides. A random sub-set of C elegans NLPs was tested physiologically in Ascaris suum ovijector and body wall muscle bioassays. None of the peptides tested were able to modulate ovijector activity, while only three displayed measurable myoactivity on somatic body wall muscle. AFAAGWNRamide (from nlp-23) and AVNPFLDSIamide (nlp-3) both produced a relaxation of body wall muscle, while AIPFNGGMYamide (nlp-10) induced a transient contraction. Numerical analyses of nip-encoding ESTs demonstrate that nlp-3, -13, -14, -15 and -18 are amongst the most highly represented transcripts in the dataset. Using available bioinformatics resources, this study delineates the nlp complement of phylum Nematoda, providing a rich source of neuropeptide ligands for deorphanisation of nematode neuropeptide receptors. (C) 2008 Australian Society for Parasitology Inc. Published by Elsevier Ltd. All rights reserved.

Exploratory analysis of spatiotemporal patterns of cellular automata by clustering compressibility

In this paper we study the classification of spatiotemporal pattern of one-dimensional cellular automata (CA) whereas the classification comprises CA rules including their initial conditions. We propose an exploratory analysis method based on the normalized compression distance (NCD) of spatiotemporal patterns which is used as dissimilarity measure for a hierarchical clustering. Our approach is different with respect to the following points. First, the classification of spatiotemporal pattern is comparative because the NCD evaluates explicitly the difference of compressibility among two objects, e.g., strings corresponding to spatiotemporal patterns. This is in contrast to all other measures applied so far in a similar context because they are essentially univariate. Second, Kolmogorov complexity, which underlies the NCD, was used in the classification of CA with respect to their spatiotemporal pattern. Third, our method is semiautomatic allowing us to investigate hundreds or thousands of CA rules or initial conditions simultaneously to gain insights into their organizational structure. Our numerical results are not only plausible confirming previous classification attempts but also shed light on the intricate influence of random initial conditions on the classification results.

Burrow architecture and digging activity in the Cape dune mole rat

While females are traditionally thought to invest more time and energy into parental care than males, males often invest more resources into searching and displaying for mates, obtaining mates and in male-male conflict. Solitary subterranean mammals perform these activities in a particularly challenging niche, necessitating energetically expensive burrowing to both search for mates and forage for food. This restriction presumably affects males more than females as the former are thought to dig longer tunnels that cover greater distances to search for females. We excavated burrow systems of male and female Cape dune mole rats Bathyergus suillus the, largest truly subterranean mammal, to investigate whether male burrows differ from those of females in ways that reflect mate searching by males. We consider burrow architecture (length, internal dimensions, fractal dimension of tunnel systems, number of nesting chambers and mole mounds on the surface) in relation to mating strategy. Males excavated significantly longer burrow systems with higher fractal dimensions and larger burrow areas than females. Male burrow systems were also significantly farther from one another than females were from other females' burrow systems. However, no sex differences were evident in tunnel cross-sectional area, mass of soil excavated per mound, number of mounds produced per unit burrow length or mass of soil excavated per burrow system. Hence, while males may use their habitat differently from females, they do not appear to differ in the dimensions of the tunnels they create. Thus, exploration and use of the habitat differs between the sexes, which may be a consequence of sex differences in mating behaviour and greater demands for food.

Genetic Algorithm Search for Predictive Patterns in Multidimensional Time Series

Based on an algorithm for pattern matching in character strings, we implement a pattern matching machine that searches for occurrences of patterns in multidimensional time series. Before the search process takes place, time series are encoded in user-designed alphabets. The patterns, on the other hand, are formulated as regular expressions that are composed of letters from these alphabets and operators. Furthermore, we develop a genetic algorithm to breed patterns that maximize a user-defined fitness function. In an application to financial data, we show that patterns bred to predict high exchange rates volatility in training samples retain statistically significant predictive power in validation samples.

Recent advances in the analysis of behavioural organization and interpretation as indicators of animal welfare

While the incorporation of mathematical and engineering methods has greatly advanced in other areas of the life sciences, they have been under-utilized in the field of animal welfare. Exceptions are beginning to emerge and share a common motivation to quantify 'hidden' aspects in the structure of the behaviour of an individual, or group of animals. Such analyses have the potential to quantify behavioural markers of pain and stress and quantify abnormal behaviour objectively. This review seeks to explore the scope of such analytical methods as behavioural indicators of welfare. We outline four classes of analyses that can be used to quantify aspects of behavioural organization. The underlying principles, possible applications and limitations are described for: fractal analysis, temporal methods, social network analysis, and agent-based modelling and simulation. We hope to encourage further application of analyses of behavioural organization by highlighting potential applications in the assessment of animal welfare, and increasing awareness of the scope for the development of new mathematical methods in this area.

A similarity measure for graphs with low computational complexity

We present and analyze an algorithm to measure the structural similarity of generalized trees, a new graph class which includes rooted trees. For this, we represent structural properties of graphs as strings and define the similarity of two Graphs as optimal alignments of the corresponding property stings. We prove that the obtained graph similarity measures are so called Backward similarity measures. From this we find that the time complexity of our algorithm is polynomial and, hence, significantly better than the time complexity of classical graph similarity methods based on isomorphic relations. (c) 2006 Elsevier Inc. All rights reserved.