Are the spatial patterns of weeds scale-invariant?

In previous empirical and modelling studies of rare species and weeds, evidence of fractal behaviour has been found. We propose that weeds in modern agricultural systems may be managed close to critical population dynamic thresholds, below which their rates of increase will be negative and where scale-invariance may be expected as a consequence. We collected detailed spatial data on five contrasting species over a period of three years in a primarily arable field. Counts in 20×20 cm contiguous quadrats, 225,000 in 1998 and 84,375 thereafter, could be re-structured into a wide range of larger quadrat sizes. These were analysed using three methods based on correlation sum, incidence and conditional incidence. We found non-trivial scale invariance for species occurring at low mean densities and where they were strongly aggregated. The fact that the scale-invariance was not found for widespread species occurring at higher densities suggests that the scaling in agricultural weed populations may, indeed, be related to critical phenomena.

Nonlinear data assimilation in geosciences: an extremely efficient particle filter

Almost all research fields in geosciences use numerical models and observations and combine these using data-assimilation techniques. With ever-increasing resolution and complexity, the numerical models tend to be highly nonlinear and also observations become more complicated and their relation to the models more nonlinear. Standard data-assimilation techniques like (ensemble) Kalman filters and variational methods like 4D-Var rely on linearizations and are likely to fail in one way or another. Nonlinear data-assimilation techniques are available, but are only efficient for small-dimensional problems, hampered by the so-called ‘curse of dimensionality’. Here we present a fully nonlinear particle filter that can be applied to higher dimensional problems by exploiting the freedom of the proposal density inherent in particle filtering. The method is illustrated for the three-dimensional Lorenz model using three particles and the much more complex 40-dimensional Lorenz model using 20 particles. By also applying the method to the 1000-dimensional Lorenz model, again using only 20 particles, we demonstrate the strong scale-invariance of the method, leading to the optimistic conjecture that the method is applicable to realistic geophysical problems. Copyright c 2010 Royal Meteorological Society

Classifying general nonlinear force laws in cell-based models via the continuum limit

though discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

From a discrete to a continuum model of cell dynamics in one dimension

Multiscale modeling is emerging as one of the key challenges in mathematical biology. However, the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions. For example, at the cellular scale, how can the appropriate discrete cell-level model be identified in a given context? Additionally, how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior? In order to begin to address such questions, we consider a discrete one-dimensional cell-based model in which cells are assumed to interact via linear springs. From the discrete equations of motion, the continuous Rouse [P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)] model is obtained. This formalism readily allows the definition of a cell number density for which a nonlinear "fast" diffusion equation is derived. Excellent agreement is demonstrated between the continuum and discrete models. Subsequently, via the incorporation of cell division, we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail. In the limit of stiff springs, where cells can be considered to be incompressible, we show that cell velocity can be directly related to cell production. This assumption is frequently made in the literature but our derivation places limits on its validity. Finally, the model is compared with a model of a similar form recently derived for a different discrete cell-based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models.

Effect of shear rupture on aggregate scale formation in sea ice

A discrete element model is used to study shear rupture of sea ice under convergent wind stresses. The model includes compressive, tensile, and shear rupture of viscous elastic joints connecting floes that move under the action of the wind stresses. The adopted shear rupture is governed by Coulomb’s criterion. The ice pack is a 400 km long square domain consisting of 4 km size floes. In the standard case with tensile strength 10 times smaller than the compressive strength, under uniaxial compression the failure regime is mainly shear rupture with the most probable scenario corresponding to that with the minimum failure work. The orientation of cracks delineating formed aggregates is bimodal with the peaks around the angles given by the wing crack theory determining diamond-shaped blocks. The ice block (floe aggregate) size decreases as the wind stress gradient increases since the elastic strain energy grows faster leading to a higher speed of crack propagation. As the tensile strength grows, shear rupture becomes harder to attain and compressive failure becomes equally important leading to elongation of blocks perpendicular to the compression direction and the blocks grow larger. In the standard case, as the wind stress confinement ratio increases the failure mode changes at a confinement ratio within 0.2–0.4, which corresponds to the analytical critical confinement ratio of 0.32. Below this value, the cracks are bimodal delineating diamond shape aggregates, while above this value failure becomes isotropic and is determined by small-scale stress anomalies due to irregularities in floe shape.

Psychometric properties of the parent-infant caregiving touch scale

Recent work in animals suggests that the extent of early tactile stimulation by parents of offspring is an important element in early caregiving. We evaluate the psychometric properties of a new parent-report measure designed to assess frequency of tactile stimulation across multiple caregiving domains in infancy. We describe the full item set of the Parent-Infant Caregiving Touch Scale (PICTS) and, using data from a UK longitudinal Child Health and Development Study, the response frequencies and factor structure and whether it was invariant over two time points in early development (5 and 9 weeks). When their infant was 9 weeks old, 838 mothers responded on the PICTS while a stratified subsample of 268 mothers completed PICTS at an earlier 5 week old assessment (229 responded on both occasions). Three PICTS factors were identified reflecting stroking, holding and affective communication. These were moderately to strongly correlated at each of the two time points of interest and were unrelated to, and therefore distinct from, a traditional measure of maternal sensitivity at 7-months. A wholly stable psychometry over 5 and 9-week assessments was not identified which suggests that behavior profiles differ slightly for younger and older infants. Tests of measurement invariance demonstrated that all three factors are characterized by full configural and metric invariance, as well as a moderate degree of evidence of scalar invariance for the stroking factor. We propose the PICTS as a valuable new measure of important aspects of caregiving in infancy.