Diffusion-limited aggregation as branched growth

I present a first-principles theory of diffusion-limited aggregation in two dimensions. A renormalized mean-field approximation gives the form of the unstable manifold for branch competition, following the method of Halsey and Leibig [Phys. Rev. A {\bf 46}, 7793 (1992)]. This leads to a result for the cluster dimensionality, D \approx 1.66, which is close to numerically obtained values. In addition, the multifractal exponent \tau(3) = D in this theory, in agreement with a proposed `electrostatic' scaling law.Comment: 13 pages, one figure not included (available by request, by ordinary mail), Plain Te

Terrestrial movement energetics: current knowledge and its application to the optimising animal

ABSTRACTThe energetic cost of locomotion can be a substantial proportion of an animal's daily energy budget and thus key to its ecology. Studies on myriad species have added to our knowledge about the general cost of animal movement, including the effects of variations in the environment such as terrain angle. However, further such studies might provide diminishing returns on the development of a deeper understanding of how animals trade-off the cost of movement with other energy costs, and other ecological currencies such as time. Here, I propose the ‘individual energy landscape’ as an approach to conceptualising the choices facing the optimising animal. In this Commentary, first I outline previous broad findings about animal walking and running locomotion, focusing in particular on the use of net cost of transport as a metric of comparison between species, and then considering the effects of environmental perturbations and other extrinsic factors on movement costs. I then introduce and explore the idea that these factors combine with the behaviour of the animal in seeking short-term optimality to create that animal's individual energy landscape – the result of the geographical landscape and environmental factors combined with the animal's selected trade-offs. Considering an animal's locomotion energy expenditure within this context enables hard-won empirical data on transport costs to be applied to questions about how an animal can and does move through its environment to maximise its fitness, and the relative importance, or otherwise, of locomotion energy economy.</jats:p

Exact Multifractal Spectra for Arbitrary Laplacian Random Walks

Iterated conformal mappings are used to obtain exact multifractal spectra of the harmonic measure for arbitrary Laplacian random walks in two dimensions. Separate spectra are found to describe scaling of the growth measure in time, of the measure near the growth tip, and of the measure away from the growth tip. The spectra away from the tip coincide with those of conformally invariant equilibrium systems with arbitrary central charge $c\leq 1$, with $c$ related to the particular walk chosen, while the scaling in time and near the tip cannot be obtained from the equilibrium properties.Comment: 4 pages, 3 figures; references added, minor correction

A different angle: comparative analyses of whole-animal transport costs running uphill

Comparative work on animals' costs of terrestrial locomotion has focussed on the underpinning physiology and biomechanics. Often, much of an animal's energy budget is spent on moving around thus there is also value in interpreting such data from an ecological perspective. When animals move through their environment they encounter topographical variation, and this is a key factor that can dramatically affect their energy expenditure. We collated published data on the costs for birds and mammals to locomote terrestrially on inclines, and investigated the scaling relationships using a phylogenetically informed approach. We show that smaller animals have a greater mass-specific cost of transport on inclines across the body mass range analysed. We also demonstrate that the increase in cost for smaller animals to run up a slope relative to along a flat surface is comparatively low. Heavier animals show larger absolute and relative increases in energy cost to travel uphill. Consideration of all aspects of the cost of incline locomotion – absolute, relative, and mass-specific – provides a fuller understanding of the interactions between transport costs, body mass, incline gradient and phylogeny, and enables us to consider their ecological implications, which we couch within the context of the ‘energy landscape‘.</jats:p

Measuring Energetics and Behaviour Using Accelerometry in Cane Toads Bufo marinus

Cane toads Bufo marinus were introduced to Australia as a control agent but now have a rapidly progressing invasion front and damage new habitats they enter. Predictive models that can give expansion rates as functions of energy supply and feeding ground distribution could help to maximise control efficiency but to date no study has measured rates of field energy expenditure in an amphibian. In the present study we used the accelerometry technique to generate behavioural time budgets and, through the derivation of ODBA (overall dynamic body acceleration), to obtain estimates of energetics in free ranging cane toads. This represents the first time that accelerometers have been used to not only quantify the behaviour of animals but also assign to those behaviours rates of energy expenditure. Firstly, laboratory calibrations between ODBA and metabolic rate were obtained and used to generate a common prediction equation for the subject toads (R2 = 0.74). Furthermore, acceleration data recorded during different behaviours was studied to ascertain threshold values for objectively defining behaviour categories. Importantly, while subsequent accelerometer field deployments were relatively short they agreed with previous studies on the proportion of time that cane toads locomote yet suggest that the metabolic rate of cane toads in the wild may sometimes be considerably higher than might be assumed based on data for other species

The branching structure of diffusion-limited aggregates

I analyze the topological structures generated by diffusion-limited aggregation (DLA), using the recently developed "branched growth model". The computed bifurcation number B for DLA in two dimensions is B ~ 4.9, in good agreement with the numerically obtained result of B ~ 5.2. In high dimensions, B -> 3.12; the bifurcation ratio is thus a decreasing function of dimensionality. This analysis also determines the scaling properties of the ramification matrix, which describes the hierarchy of branches.Comment: 6 pages, 1 figure, Euro-LaTeX styl

Stability of Monomer-Dimer Piles

We measure how strong, localized contact adhesion between grains affects the maximum static critical angle, theta_c, of a dry sand pile. By mixing dimer grains, each consisting of two spheres that have been rigidly bonded together, with simple spherical monomer grains, we create sandpiles that contain strong localized adhesion between a given particle and at most one of its neighbors. We find that tan(theta_c) increases from 0.45 to 1.1 and the grain packing fraction, Phi, decreases from 0.58 to 0.52 as we increase the relative number fraction of dimer particles in the pile, nu_d, from 0 to 1. We attribute the increase in tan(theta_c(nu_d)) to the enhanced stability of dimers on the surface, which reduces the density of monomers that need to be accomodated in the most stable surface traps. A full characterization and geometrical stability analysis of surface traps provides a good quantitative agreement between experiment and theory over a wide range of nu_d, without any fitting parameters.Comment: 11 pages, 12 figures consisting of 21 eps files, submitted to PR