8,842 research outputs found
Extinction in neutrally stable stochastic Lotka-Volterra models
Populations of competing biological species exhibit a fascinating interplay
between the nonlinear dynamics of evolutionary selection forces and random
fluctuations arising from the stochastic nature of the interactions. The
processes leading to extinction of species, whose understanding is a key
component in the study of evolution and biodiversity, are influenced by both of
these factors.
In this paper, we investigate a class of stochastic population dynamics
models based on generalized Lotka-Volterra systems. In the case of neutral
stability of the underlying deterministic model, the impact of intrinsic noise
on the survival of species is dramatic: it destroys coexistence of interacting
species on a time scale proportional to the population size. We introduce a new
method based on stochastic averaging which allows one to understand this
extinction process quantitatively by reduction to a lower-dimensional effective
dynamics. This is performed analytically for two highly symmetrical models and
can be generalized numerically to more complex situations. The extinction
probability distributions and other quantities of interest we obtain show
excellent agreement with simulations.Comment: 14 pages, 7 figure
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Large scale analytic calculations in quantum field theories
We present a survey on the mathematical structure of zero- and single scale
quantities and the associated calculation methods and function spaces in higher
order perturbative calculations in relativistic renormalizable quantum field
theories.Comment: 25 pages Latex, 1 style fil
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
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