4,708 research outputs found
Stochastic delocalization of finite populations
Heterogeneities in environmental conditions often induce corresponding
heterogeneities in the distribution of species. In the extreme case of a
localized patch of increased growth rates, reproducing populations can become
strongly concentrated at the patch despite the entropic tendency for population
to distribute evenly. Several deterministic mathematical models have been used
to characterize the conditions under which localized states can form, and how
they break down due to convective driving forces. Here, we study the
delocalization of a finite population in the presence of number fluctuations.
We find that any finite population delocalizes on sufficiently long time
scales. Depending on parameters, however, populations may remain localized for
a very long time. The typical waiting time to delocalization increases
exponentially with both population size and distance to the critical wind speed
of the deterministic approximation. We augment these simulation results by a
mathematical analysis that treats the reproduction and migration of individuals
as branching random walks subject to global constraints. For a particular
constraint, different from a fixed population size constraint, this model
yields a solvable first moment equation. We find that this solvable model
approximates very well the fixed population size model for large populations,
but starts to deviate as population sizes are small. The analytical approach
allows us to map out a phase diagram of the order parameter as a function of
the two driving parameters, inverse population size and wind speed. Our results
may be used to extend the analysis of delocalization transitions to different
settings, such as the viral quasi-species scenario
Ligand-receptor binding kinetics in surface plasmon resonance cells: A Monte Carlo analysis
Surface plasmon resonance (SPR) chips are widely used to measure association
and dissociation rates for the binding kinetics between two species of
chemicals, e.g., cell receptors and ligands. It is commonly assumed that
ligands are spatially well mixed in the SPR region, and hence a mean-field rate
equation description is appropriate. This approximation however ignores the
spatial fluctuations as well as temporal correlations induced by multiple local
rebinding events, which become prominent for slow diffusion rates and high
binding affinities. We report detailed Monte Carlo simulations of ligand
binding kinetics in an SPR cell subject to laminar flow. We extract the binding
and dissociation rates by means of the techniques frequently employed in
experimental analysis that are motivated by the mean-field approximation. We
find major discrepancies in a wide parameter regime between the thus extracted
rates and the known input simulation values. These results underscore the
crucial quantitative importance of spatio-temporal correlations in binary
reaction kinetics in SPR cell geometries, and demonstrate the failure of a
mean-field analysis of SPR cells in the regime of high Damk\"ohler number Da >
0.1, where the spatio-temporal correlations due to diffusive transport and
ligand-receptor rebinding events dominate the dynamics of SPR systems.Comment: 21 pages, 9 figure
Continuous and discrete models of cooperation in complex bacterial colonies
We study the effect of discreteness on various models for patterning in
bacterial colonies. In a bacterial colony with branching pattern, there are
discrete entities - bacteria - which are only two orders of magnitude smaller
than the elements of the macroscopic pattern. We present two types of models.
The first is the Communicating Walkers model, a hybrid model composed of both
continuous fields and discrete entities - walkers, which are coarse-graining of
the bacteria. Models of the second type are systems of reaction diffusion
equations, where the branching of the pattern is due to non-constant diffusion
coefficient of the bacterial field. The diffusion coefficient represents the
effect of self-generated lubrication fluid on the bacterial movement. We
implement the discreteness of the biological system by introducing a cutoff in
the growth term at low bacterial densities. We demonstrate that the cutoff does
not improve the models in any way. Its only effect is to decrease the effective
surface tension of the front, making it more sensitive to anisotropy. We
compare the models by introducing food chemotaxis and repulsive chemotactic
signaling into the models. We find that the growth dynamics of the
Communication Walkers model and the growth dynamics of the Non-Linear diffusion
model are affected in the same manner. From such similarities and from the
insensitivity of the Communication Walkers model to implicit anisotropy we
conclude that the increased discreteness, introduced be the coarse-graining of
the walkers, is small enough to be neglected.Comment: 16 pages, 10 figures in 13 gif files, to be published in proceeding
of CMDS
Mixed diffusive-convective relaxation of a broad beam of energetic particles in cold plasma
We revisit the applications of quasi-linear theory as a paradigmatic model
for weak plasma turbulence and the associated bump-on-tail problem. The work,
presented here, is built around the idea that large-amplitude or strongly
shaped beams do not relax through diffusion only and that there exists an
intermediate time scale where the relaxations are convective (ballistic-like).
We cast this novel idea in the rigorous form of a self-consistent nonlinear
dynamical model, which generalizes the classic equations of the quasi-linear
theory to "broad" beams with internal structure. We also present numerical
simulation results of the relaxation of a broad beam of energetic particles in
cold plasma. These generally demonstrate the mixed diffusive-convective
features of supra-thermal particle transport; and essentially depend on
nonlinear wave-particle interactions and phase-space structures. Taking into
account modes of the stable linear spectrum is crucial for the self-consistent
evolution of the distribution function and the fluctuation intensity spectrum.Comment: 25 pages, 15 figure
Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection-diffusion-reaction problems
In this paper a scheme for approximating solutions of convection-diffusion-reaction equations by Markov jump processes is studied. The general principle of the method of lines reduces evolution partial differential equations to semidiscrete approximations consisting of systems of ordinary differential equations. Our approach is to use for this resulting system a stochastic scheme which is essentially a direct simulation of the corresponding infinitesimal dynamics. This implies automatically the time adaptivity and, in one space dimension, stable approximations of diffusion operators on non-uniform grids and the possibility of using moving cells for the transport part, all within the framework of an explicit method. We present several results in one space dimension including free boundary problems, but the general algorithm is simple, flexible and on uniform grids it can be formulated for general evolution partial differential equations in arbitrary space dimensions
The exit-time problem for a Markov jump process
The purpose of this paper is to consider the exit-time problem for a
finite-range Markov jump process, i.e, the distance the particle can jump is
bounded independent of its location. Such jump diffusions are expedient models
for anomalous transport exhibiting super-diffusion or nonstandard normal
diffusion. We refer to the associated deterministic equation as a
volume-constrained nonlocal diffusion equation. The volume constraint is the
nonlocal analogue of a boundary condition necessary to demonstrate that the
nonlocal diffusion equation is well-posed and is consistent with the jump
process. A critical aspect of the analysis is a variational formulation and a
recently developed nonlocal vector calculus. This calculus allows us to pose
nonlocal backward and forward Kolmogorov equations, the former equation
granting the various moments of the exit-time distribution.Comment: 15 pages, 7 figure
Particle Density Estimation with Grid-Projected Adaptive Kernels
The reconstruction of smooth density fields from scattered data points is a
procedure that has multiple applications in a variety of disciplines, including
Lagrangian (particle-based) models of solute transport in fluids. In random
walk particle tracking (RWPT) simulations, particle density is directly linked
to solute concentrations, which is normally the main variable of interest, not
just for visualization and post-processing of the results, but also for the
computation of non-linear processes, such as chemical reactions. Previous works
have shown the superiority of kernel density estimation (KDE) over other
methods such as binning, in terms of its ability to accurately estimate the
"true" particle density relying on a limited amount of information. Here, we
develop a grid-projected KDE methodology to determine particle densities by
applying kernel smoothing on a pilot binning; this may be seen as a "hybrid"
approach between binning and KDE. The kernel bandwidth is optimized locally.
Through simple implementation examples, we elucidate several appealing aspects
of the proposed approach, including its computational efficiency and the
possibility to account for typical boundary conditions, which would otherwise
be cumbersome in conventional KDE
Why are probabilistic laws governing quantum mechanics and neurobiology?
We address the question: Why are dynamical laws governing in quantum
mechanics and in neuroscience of probabilistic nature instead of being
deterministic? We discuss some ideas showing that the probabilistic option
offers advantages over the deterministic one.Comment: 40 pages, 8 fig
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