Applicability of the Fisher Equation to Bacterial Population Dynamics

The applicability of the Fisher equation, which combines diffusion with logistic nonlinearity, to population dynamics of bacterial colonies is studied with the help of explicit analytic solutions for the spatial distribution of a stationary bacterial population under a static mask. The mask protects the bacteria from ultraviolet light. The solution, which is in terms of Jacobian elliptic functions, is used to provide a practical prescription to extract Fisher equation parameters from observations and to decide on the validity of the Fisher equation.Comment: 5 pages, 3 figs. include

Terrace Standard, July, 09, 1997

Slowly strained solids deform via intermittent slips that exhibit a material-independent critical size distribution. Here, by comparing two disparate systems - granular materials and bulk metallic glasses - we show evidence that not only the statistics of slips but also their dynamics are remarkably similar, i.e. independent of the microscopic details of the material. By resolving and comparing the full time evolution of avalanches in bulk metallic glasses and granular materials, we uncover a regime of universal deformation dynamics. We experimentally verify the predicted universal scaling functions for the dynamics of individual avalanches in both systems, and show that both the slip statistics and dynamics are independent of the scale and details of the material structure and interactions, thus settling a long-standing debate as to whether or not the claim of universality includes only the slip statistics or also the slip dynamics. The results imply that the frictional weakening in granular materials and the interplay of damping, weakening and inertial effects in bulk metallic glasses have strikingly similar effects on the slip dynamics. These results are important for transferring experimental results across scales and material structures in a single theory of deformation dynamics

Disorder induced critical phenomena in magnetically glassy Cu-Al-Mn alloys

Measurements of magnetic hysteresis loops in Cu-Al-Mn alloys of different Mn content at low temperatures are presented. The loops are smooth and continuous above a certain temperature, but exhibit a magnetization discontinuity below that temperature. Scaling analysis suggest that this system displays a disorder induced phase transition line. Measurements allow to determine the critical exponents $\beta=0.03\pm 0.01$ and $\beta \delta = 0.4 \pm 0.1$ in agreement with those reported recently [Berger et al., Phys. Rev. Lett. {\bf 85}, 4176 (2000)]Comment: 4 pages, 5 figure

Depinning with dynamic stress overshoots: A hybrid of critical and pseudohysteretic behavior

A model of an elastic manifold driven through a random medium by an applied force F is studied focussing on the effects of inertia and elastic waves, in particular {\it stress overshoots} in which motion of one segment of the manifold causes a temporary stress on its neighboring segments in addition to the static stress. Such stress overshoots decrease the critical force for depinning and make the depinning transition hysteretic. We find that the steady state velocity of the moving phase is nevertheless history independent and the critical behavior as the force is decreased is in the same universality class as in the absence of stress overshoots: the dissipative limit which has been studied analytically. To reach this conclusion, finite-size scaling analyses of a variety of quantities have been supplemented by heuristic arguments. If the force is increased slowly from zero, the spectrum of avalanche sizes that occurs appears to be quite different from the dissipative limit. After stopping from the moving phase, the restarting involves both fractal and bubble-like nucleation. Hysteresis loops can be understood in terms of a depletion layer caused by the stress overshoots, but surprisingly, in the limit of very large samples the hysteresis loops vanish. We argue that, although there can be striking differences over a wide range of length scales, the universality class governing this pseudohysteresis is again that of the dissipative limit. Consequences of this picture for the statistics and dynamics of earthquakes on geological faults are briefly discussed.Comment: 43 pages, 57 figures (yes, that's a five followed by a seven), revte

Adaptation of Autocatalytic Fluctuations to Diffusive Noise

Evolution of a system of diffusing and proliferating mortal reactants is analyzed in the presence of randomly moving catalysts. While the continuum description of the problem predicts reactant extinction as the average growth rate becomes negative, growth rate fluctuations induced by the discrete nature of the agents are shown to allow for an active phase, where reactants proliferate as their spatial configuration adapts to the fluctuations of the catalysts density. The model is explored by employing field theoretical techniques, numerical simulations and strong coupling analysis. For d<=2, the system is shown to exhibits an active phase at any growth rate, while for d>2 a kinetic phase transition is predicted. The applicability of this model as a prototype for a host of phenomena which exhibit self organization is discussed.Comment: 6 pages 6 figur

Population dynamics in compressible flows

Organisms often grow, migrate and compete in liquid environments, as well as on solid surfaces. However, relatively little is known about what happens when competing species are mixed and compressed by fluid turbulence. In these lectures we review our recent work on population dynamics and population genetics in compressible velocity fields of one and two dimensions. We discuss why compressible turbulence is relevant for population dynamics in the ocean and we consider cases both where the velocity field is turbulent and when it is static. Furthermore, we investigate populations in terms of a continuos density field and when the populations are treated via discrete particles. In the last case we focus on the competition and fixation of one species compared to anotherComment: 16 pages, talk delivered at the Geilo Winter School 201

Can a Species Keep Pace with a Shifting Climate?

Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a one-dimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growth-diffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back

Non-local rheology in dense granular flows -- Revisiting the concept of fluidity

Granular materials belong to the class of amorphous athermal systems, like foams, emulsion or suspension they can resist shear like a solid, but flow like a liquid under a sufficiently large applied shear stress. They exhibit a dynamical phase transition between static and flowing states, as for phase transitions of thermodynamic systems, this rigidity transition exhibits a diverging length scales quantifying the degree of cooperatively. Several experiments have shown that the rheology of granular materials and emulsion is non-local, namely that the stress at a given location does not depend only on the shear rate at this location but also on the degree of mobility in the surrounding region. Several constitutive relations have recently been proposed and tested successfully against numerical and experimental results. Here we use discrete elements simulation of 2D shear flows to shed light on the dynamical mechanism underlying non-locality in dense granular flows