It is argued that general mathematical-statistical theorems imply that standard statistical analysis techniques of inter-individual variation are invalid to investigate developmental processes. Developmental processes have to be analyzed at the level of individual subjects, using time series data characterizing the patterns of intra-individual variation. It is shown that standard statistical techniques based on the analysis of inter-individual variation appear to be insensitive to the presence of arbitrary large degrees of inter-individual heterogeneity in the population. An important class of nonlinear epigenetic models of neural growth is described which can explain the occurrence of such heterogeneity in brain structures and behavior. Links with models of developmental instability are discussed. A simulation study based on a chaotic growth model illustrates the invalidity of standard analysis of inter-individual variation, whereas time series analysis of intra-individual variation is able to recover the true state of affairs. © 2007 Wiley Periodicals, Inc. Dev Psychobiol 50: 60–69, 2008.

On the implications of the classical ergodic theorems: analysis of developmental processes has to focus on intra-individual variation.

The second part of this synthetic work presents and discusses the most spectacular and successful applications of the irreversible thermodynamics with internal variables. These include viscosity in both fluids and solids (in the former case, in complex fluids and structurally complex flows), viscoplasticity and rate-independent plasticity in small and finite strains, damage and cyclic plasticity, electric and magnetic relaxation, magnetic and electric hysteresis, normal and semi-conduction, superconductivity of deformable solids, and ferrofluids. In all cases the internal variables of interest are given some physical significance in terms of quantities defined at a sublevel of description. The relevant internal variables may be as varied as second-order tensors, real or complex valued scalars, and polar or axial vectors. Furthermore, the role played by internal variables in wave-propagation problems is emphasized through appropriate examples. The presentation ends with reaction-diffusion problems. This is illustrated by damage, plastic-strain localization and models for nerve-pulse dynamics. Globally, we have all ingredients of a truly post-Duhemian irreversible thermodynamics of complex behaviors.

Thermodynamics with Internal Variables. Part II. Applications

Effect of solutes on dislocation motion - A phase-field simulation

We investigate systematically the effect of small-scale biological and physical processes on the generation of plankton patchiness, and on the biological production averaged over scales much larger than those of the patches. We use an excitable reaction-diffusion model with one physical dimension and two biological compartments: zooplanktonic predators Z and phytoplanktonic prey P. It is assumed that the assimilation efficiency is of order 0.1. With sinusoidal initial conditions, the maximum in domain-averaged P is about six times that which would be estimated assuming spatial homogeneity. With random initial conditions similar enhancement factors are obtained and patches emerge of characteristic size, proportional to the square root of the diffusivity as in classical work on simpler systems. Both the patch size and the production show large variances across ensembles of integrations initialized randomly but with the same mean. For both sinusoidal and random initial conditions, the most extreme effect is obtained when the phytoplankton growth rate is between one and four times the rate of diffusion, and when the mean initial condition lies close to the excitation threshold; the effect of varying the initial conditions is also considered. A bi-stable version of the plankton dynamics produces a quasi-stable patchy state of the model with mean properties far from those of either equilibrium. These results illustrate both the desirability of high-resolution observations of plankton, and that caution is required in using 'mean field' plankton dynamics in large-scale biophysical models of the ocean.

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Plankton patchiness and its effect on larger-scale productivity

The impact of colloids on the migration and dispersal of radionuclides within fractured rock

Basic considerations Pattern formation Plane waves A geometrical theory for waves Nonlinear dispersal mechanisms.

Peter Grindrod

Papers

Patterns and waves : the theory and applications of reaction-diffusion equations