97 research outputs found
Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities
Many populations, e.g. of cells, bacteria, viruses, or replicating DNA
molecules, start small, from a few individuals, and grow large into a
noticeable fraction of the environmental carrying capacity . Typically, the
elements of the initiating, sparse set will not be hampering each other and
their number will grow from in a branching process or Malthusian
like, roughly exponential fashion, , where is the size at
discrete time , is the offspring mean per individual (at the
low starting density of elements, and large ), and a sum of i.i.d.
random variables. It will, thus, become detectable (i.e. of the same order as
) only after around generations, when its density will
tend to be strictly positive. Typically, this entity will be random, even if
the very beginning was not at all stochastic, as indicated by lower case ,
due to variations during the early development. However, from that time
onwards, law of large numbers effects will render the process deterministic,
though initiated by the random density at time log , expressed through the
variable . Thus, acts both as a random veil concealing the start and a
stochastic initial value for later, deterministic population density
development. We make such arguments precise, studying general density and also
system-size dependent, processes, as . As an intrinsic size
parameter, may also be chosen to be the time unit. The fundamental ideas
are to couple the initial system to a branching process and to show that late
densities develop very much like iterates of a conditional expectation
operator.Comment: presented at IV Workshop on Branching Processes and their
Applications at Badajoz, Spain, 10-13 April, 201
A new stochastic differential equation approach for waves in a random medium
We present a mathematical approach that simplifies the theoretical treatment
of electromagnetic localization in random media and leads to closed form
analytical solutions. Starting with the assumption that the dielectric
permittivity of the medium has delta-correlated spatial fluctuations, and using
the Ito lemma, we derive a linear stochastic differential equation for a one
dimensional random medium. The equation leads to localized wave solutions. The
localized wave solutions have a localization length that scales inversely with
the square of the frequency of the wave in the low frequency regime, whereas in
the high frequency regime, this length varies inversely with the frequency to
the power of two thirds
On the emergence of random initial conditions in fluid limits
The paper presents a phenomenon occurring in population processes that start
near zero and have large carrying capacity. By the classical result of
Kurtz~(1970), such processes, normalized by the carrying capacity, converge on
finite intervals to the solutions of ordinary differential equations, also
known as the fluid limit. When the initial population is small relative to
carrying capacity, this limit is trivial. Here we show that, viewed at suitably
chosen times increasing to infinity, the process converges to the fluid limit,
governed by the same dynamics, but with a random initial condition. This random
initial condition is related to the martingale limit of an associated linear
birth and death process
Large Deviations for processes on half-line
We consider a sequence of processes defined on half-line for all non negative
t. We give sufficient conditions for Large Deviation Principle (LDP) to hold in
the space of continuous functions with a new metric that is more sensitive to
behaviour at infinity than the uniform metric. LDP is established for Random
Walks, Diffusions, and CEV model of ruin, all defined on the half-line. LDP in
this space is "more precise" than that with the usual metric of uniform
convergence on compacts.Comment: 23 page
Persistence of Small Noise and Random initial conditions
The effect of small noise in a smooth dynamical system is negligible on any
finite time interval. Here we study situations when it persists on intervals
increasing to infinity. Such asymptotic regime occurs when the system starts
from initial condition, sufficiently close to an unstable fixed point. In this
case, under appropriate scaling, the trajectory converges to solution of the
unperturbed system, started from a certain {\em random} initial condition. In
this paper we consider the case of one dimensional diffusions on the positive
half line, which often arise as scaling limits in population dynamics
Extinction, Persistence, and Evolution
Extinction can occur for many reasons. We have a closer look at the most basic form, extinction of populations with stable but insufficient reproduction. Then we move on to competing populations and evolutionary suicide
Time-dependent probability density function in cubic stochastic processes
We report time-dependent Probability Density Functions (PDFs) for a nonlinear stochastic process with a cubic force by novel analytical and computational studies. Analytically, a transition probability is formulated by using a path integral and is computed by the saddle-point solution (instanton method) and a new nonlinear transformation of time. The predicted PDF p(x, t) is in general given as a time integral and useful PDFs with explicit dependence on x and t are presented in certain limits (e.g. in the short and long time limits). Numerical simulations of the FokkerPlanck equation provide exact time evolution of the PDFs and confirm analytical predictions in the limit of weak noise. In particular, we show that non-equilibrium PDFs behave drastically differently from the stationary PDFs in regards to the asymmetry (skewness) and kurtosis. Specifically, while stationary PDFs are symmetric, transient PDFs are skewed; transient PDFs are much broader than stationary PDFs, with the kurtosis larger and smaller than 3, respectively. We elucidate the effect of nonlinear interaction on the strong fluctuations and intermittency in relaxation process
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