512 research outputs found
Does a Single Zealot Affect an Infinite Group of Voters ?
A method for studying exact properties of a class of {\it inhomogeneous}
stochastic many-body systems is developed and presented in the framework of a
voter model perturbed by the presence of a ``zealot'', an individual allowed to
favour an opinion. We compute exactly the magnetization of this model and find
that in one (1d) and two dimensions (2d) it evolves, algebraically () in 1d and much slower () in 2d, towards the unanimity
state chosen by the zealot. In higher dimensions the stationary magnetization
is no longer uniform: the zealot cannot influence all the individuals.
Implications to other physical problems are also pointed out.Comment: 4 pages, 2-column revtex4 forma
Fixation and Polarization in a Three-Species Opinion Dynamics Model
Motivated by the dynamics of cultural change and diversity, we generalize the
three-species constrained voter model on a complete graph introduced in [J.
Phys. A 37, 8479 (2004)]. In this opinion dynamics model, a population of size
N is composed of "leftists" and "rightists" that interact with "centrists": a
leftist and centrist can both become leftists with rate (1+q)/2 or centrists
with rate (1-q)/2 (and similarly for rightists and centrists), where q denotes
the bias towards extremism (q>0) or centrism (q<0). This system admits three
absorbing fixed points and a "polarization" line along which a frozen mixture
of leftists and rightists coexist. In the realm of Fokker-Planck equation, and
using a mapping onto a population genetics model, we compute the fixation
probability of ending in every absorbing state and the mean times for these
events. We therefore show, especially in the limit of weak bias and large
population size when |q|~1/N and N>>1, how fluctuations alter the mean field
predictions: polarization is likely when q>0, but there is always a finite
probability to reach a consensus; the opposite happens when q<0. Our findings
are corroborated by stochastic simulations.Comment: 6 pages in EPL format, 3 color figures (6 panels). Minor
modifications. To appear in EPL (Europhysics Letters
Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations
We study the oscillatory dynamics in the generic three-species
rock-paper-scissors games with mutations. In the mean-field limit, different
behaviors are found: (a) for high mutation rate, there is a stable interior
fixed point with coexistence of all species; (b) for low mutation rates, there
is a region of the parameter space characterized by a limit cycle resulting
from a Hopf bifurcation; (c) in the absence of mutations, there is a region
where heteroclinic cycles yield oscillations of large amplitude (not robust
against noise). After a discussion on the main properties of the mean-field
dynamics, we investigate the stochastic version of the model within an
individual-based formulation. Demographic fluctuations are therefore naturally
accounted and their effects are studied using a diffusion theory complemented
by numerical simulations. It is thus shown that persistent erratic oscillations
(quasi-cycles) of large amplitude emerge from a noise-induced resonance
phenomenon. We also analytically and numerically compute the average escape
time necessary to reach a (quasi-)cycle on which the system oscillates at a
given amplitude.Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biolog
Fixation in Evolutionary Games under Non-Vanishing Selection
One of the most striking effect of fluctuations in evolutionary game theory
is the possibility for mutants to fixate (take over) an entire population.
Here, we generalize a recent WKB-based theory to study fixation in evolutionary
games under non-vanishing selection, and investigate the relation between
selection intensity w and demographic (random) fluctuations. This allows the
accurate treatment of large fluctuations and yields the probability and mean
times of fixation beyond the weak selection limit. The power of the theory is
demonstrated on prototypical models of cooperation dilemmas with multiple
absorbing states. Our predictions compare excellently with numerical
simulations and, for finite w, significantly improve over those of the
Fokker-Planck approximation.Comment: 4 figures, to appear in EPL (Europhysics Letters
Particle Statistics and Population Dynamics
We study a master equation system modelling a population dynamics problem in
a lattice. The problem is the calculation of the minimum size of a refuge that
can protect a population from hostile external conditions, the so called
critical patch size problem. We analize both cases in which the particles are
considered fermions and bosons and show using exact analitical methods that,
while the Fermi-Dirac statistics leads to certain extinction for any refuge
size, the Bose-Eistein statistics allows survival even for the minimal refuge
Spatial stochastic predator-prey models
We consider a broad class of stochastic lattice predator-prey models, whose
main features are overviewed. In particular, this article aims at drawing a
picture of the influence of spatial fluctuations, which are not accounted for
by the deterministic rate equations, on the properties of the stochastic
models. Here, we outline the robust scenario obeyed by most of the lattice
predator-prey models with an interaction "a' la Lotka-Volterra". We also show
how a drastically different behavior can emerge as the result of a subtle
interplay between long-range interactions and a nearest-neighbor exchange
process.Comment: 5 pages, 2 figures. Proceedings paper of the workshop "Stochastic
models in biological sciences" (May 29 - June 2, 2006 in Warsaw) for the
Banach Center Publication
- …