11,499 research outputs found
Stochastic Porous Media Equation and Self-Organized Criticality
The existence and uniqueness of nonnegative strong solutions for stochastic
porous media equations with noncoercive monotone diffusivity function and
Wiener forcing term is proven. The finite time extinction of solutions with
high probability is also proven in 1-D. The results are relevant for
self-organized critical behaviour of stochastic nonlinear diffusion equations
with critical states.Comment: 29 pages, BiBoS-Preprint No. 07-11-26
Circular Stochastic Fluctuations in SIS Epidemics with Heterogeneous Contacts Among Sub-populations
The conceptual difference between equilibrium and non-equilibrium steady
state (NESS) is well established in physics and chemistry. This distinction,
however, is not widely appreciated in dynamical descriptions of biological
populations in terms of differential equations in which fixed point, steady
state, and equilibrium are all synonymous. We study NESS in a stochastic SIS
(susceptible-infectious-susceptible) system with heterogeneous individuals in
their contact behavior represented in terms of subgroups. In the infinite
population limit, the stochastic dynamics yields a system of deterministic
evolution equations for population densities; and for very large but finite
system a diffusion process is obtained. We report the emergence of a circular
dynamics in the diffusion process, with an intrinsic frequency, near the
endemic steady state. The endemic steady state is represented by a stable node
in the deterministic dynamics; As a NESS phenomenon, the circular motion is
caused by the intrinsic heterogeneity within the subgroups, leading to a broken
symmetry and time irreversibility.Comment: 29 pages, 5 figure
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
Cyclic dominance of species has been identified as a potential mechanism to
maintain biodiversity, see e.g. B. Kerr, M. A. Riley, M. W. Feldman and B. J.
M. Bohannan [Nature {\bf 418}, 171 (2002)] and B. Kirkup and M. A. Riley
[Nature {\bf 428}, 412 (2004)]. Through analytical methods supported by
numerical simulations, we address this issue by studying the properties of a
paradigmatic non-spatial three-species stochastic system, namely the
`rock-paper-scissors' or cyclic Lotka-Volterra model. While the deterministic
approach (rate equations) predicts the coexistence of the species resulting in
regular (yet neutrally stable) oscillations of the population densities, we
demonstrate that fluctuations arising in the system with a \emph{finite number
of agents} drastically alter this picture and are responsible for extinction:
After long enough time, two of the three species die out. As main findings we
provide analytic estimates and numerical computation of the extinction
probability at a given time. We also discuss the implications of our results
for a broad class of competing population systems.Comment: 12 pages, 9 figures, minor correction
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