114 research outputs found
Forcing anomalous scaling on demographic fluctuations
We discuss the conditions under which a population of anomalously diffusing
individuals can be characterized by demographic fluctuations that are
anomalously scaling themselves. Two examples are provided in the case of
individuals migrating by Gaussian diffusion, and by a sequence of L\'evy
flights.Comment: 5 pages 2 figure
Particle Dispersion on Rapidly Folding Random Hetero-Polymers
We investigate the dynamics of a particle moving randomly along a disordered
hetero-polymer subjected to rapid conformational changes which induce
superdiffusive motion in chemical coordinates. We study the antagonistic
interplay between the enhanced diffusion and the quenched disorder. The
dispersion speed exhibits universal behavior independent of the folding
statistics. On the other hand it is strongly affected by the structure of the
disordered potential. The results may serve as a reference point for a number
of translocation phenomena observed in biological cells, such as protein
dynamics on DNA strands.Comment: 4 pages, 4 figure
Levy Flights in Inhomogeneous Media
We investigate the impact of external periodic potentials on superdiffusive
random walks known as Levy flights and show that even strongly superdiffusive
transport is substantially affected by the external field. Unlike ordinary
random walks, Levy flights are surprisingly sensitive to the shape of the
potential while their asymptotic behavior ceases to depend on the Levy index
. Our analysis is based on a novel generalization of the Fokker-Planck
equation suitable for systems in thermal equilibrium. Thus, the results
presented are applicable to the large class of situations in which
superdiffusion is caused by topological complexity, such as diffusion on folded
polymers and scale-free networks.Comment: 4 pages, 4 figure
Spatial clustering of interacting bugs: Levy flights versus Gaussian jumps
A biological competition model where the individuals of the same species
perform a two-dimensional Markovian continuous-time random walk and undergo
reproduction and death is studied. The competition is introduced through the
assumption that the reproduction rate depends on the crowding in the
neighborhood. The spatial dynamics corresponds either to normal diffusion
characterized by Gaussian jumps or to superdiffusion characterized by L\'evy
flights. It is observed that in both cases periodic patterns occur for
appropriate parameters of the model, indicating that the general macroscopic
collective behavior of the system is more strongly influenced by the
competition for the resources than by the type of spatial dynamics. However,
some differences arise that are discussed.Comment: This version incorporates in the text the correction published as an
Erratum in Europhysics Letters (EPL) 95, 69902 (2011) [doi:
10.1209/0295-5075/95/69902
The fractional Keller-Segel model
The Keller-Segel model is a system of partial differential equations
modelling chemotactic aggregation in cellular systems. This model has blowing
up solutions for large enough initial conditions in dimensions d >= 2, but all
the solutions are regular in one dimension; a mathematical fact that crucially
affects the patterns that can form in the biological system. One of the
strongest assumptions of the Keller-Segel model is the diffusive character of
the cellular motion, known to be false in many situations. We extend this model
to such situations in which the cellular dispersal is better modelled by a
fractional operator. We analyze this fractional Keller-Segel model and find
that all solutions are again globally bounded in time in one dimension. This
fact shows the robustness of the main biological conclusions obtained from the
Keller-Segel model
Sheared bioconvection in a horizontal tube
The recent interest in using microorganisms for biofuels is motivation enough
to study bioconvection and cell dispersion in tubes subject to imposed flow. To
optimize light and nutrient uptake, many microorganisms swim in directions
biased by environmental cues (e.g. phototaxis in algae and chemotaxis in
bacteria). Such taxes inevitably lead to accumulations of cells, which, as many
microorganisms have a density different to the fluid, can induce hydrodynamic
instabilites. The large-scale fluid flow and spectacular patterns that arise
are termed bioconvection. However, the extent to which bioconvection is
affected or suppressed by an imposed fluid flow, and how bioconvection
influences the mean flow profile and cell transport are open questions. This
experimental study is the first to address these issues by quantifying the
patterns due to suspensions of the gravitactic and gyrotactic green
biflagellate alga Chlamydomonas in horizontal tubes subject to an imposed flow.
With no flow, the dependence of the dominant pattern wavelength at pattern
onset on cell concentration is established for three different tube diameters.
For small imposed flows, the vertical plumes of cells are observed merely to
bow in the direction of flow. For sufficiently high flow rates, the plumes
progressively fragment into piecewise linear diagonal plumes, unexpectedly
inclined at constant angles and translating at fixed speeds. The pattern
wavelength generally grows with flow rate, with transitions at critical rates
that depend on concentration. Even at high imposed flow rates, bioconvection is
not wholly suppressed and perturbs the flow field.Comment: 19 pages, 9 figures, published version available at
http://iopscience.iop.org/1478-3975/7/4/04600
Fractional Klein-Kramers equation for superdiffusive transport: normal versus anomalous time evolution in a differential L{\'e}vy walk model
We introduce a fractional Klein-Kramers equation which describes
sub-ballistic superdiffusion in phase space in the presence of a
space-dependent external force field. This equation defines the differential
L{\'e}vy walk model whose solution is shown to be non-negative. In the velocity
coordinate, the probability density relaxes in Mittag-Leffler fashion towards
the Maxwell distribution whereas in the space coordinate, no stationary
solution exists and the temporal evolution of moments exhibits a competition
between Brownian and anomalous contributions.Comment: 4 pages, REVTe
Optimal search strategies for hidden targets
What is the fastest way of finding a randomly hidden target? This question of
general relevance is of vital importance for foraging animals. Experimental
observations reveal that the search behaviour of foragers is generally
intermittent: active search phases randomly alternate with phases of fast
ballistic motion. In this letter, we study the efficiency of this type of two
states search strategies, by calculating analytically the mean first passage
time at the target. We model the perception mecanism involved in the active
search phase by a diffusive process. In this framework, we show that the search
strategy is optimal when the average duration of "motion phases" varies like
the power either 3/5 or 2/3 of the average duration of "search phases",
depending on the regime. This scaling accounts for experimental data over a
wide range of species, which suggests that the kinetics of search trajectories
is a determining factor optimized by foragers and that the perception activity
is adequately described by a diffusion process.Comment: 4 pages, 5 figures. to appear in Phys. Rev. Let
The one-dimensional Keller-Segel model with fractional diffusion of cells
We investigate the one-dimensional Keller-Segel model where the diffusion is
replaced by a non-local operator, namely the fractional diffusion with exponent
. We prove some features related to the classical
two-dimensional Keller-Segel system: blow-up may or may not occur depending on
the initial data. More precisely a singularity appears in finite time when
and the initial configuration of cells is sufficiently concentrated.
On the opposite, global existence holds true for if the initial
density is small enough in the sense of the norm.Comment: 12 page
Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation
equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate
diffusion. The focus of this paper is the unification and generalization of the
well-posedness theory of these models. We prove local well-posedness on bounded
domains for dimensions and in all of space for , the
uniqueness being a result previously not known for PKS with degenerate
diffusion. We generalize the notion of criticality for PKS and show that
subcritical problems are globally well-posed. For a fairly general class of
problems, we prove the existence of a critical mass which sharply divides the
possibility of finite time blow up and global existence. Moreover, we compute
the critical mass for fully general problems and show that solutions with
smaller mass exists globally. For a class of supercritical problems we prove
finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page
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