3,206 research outputs found
Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models
We study the general properties of stochastic two-species models for
predator-prey competition and coexistence with Lotka-Volterra type interactions
defined on a -dimensional lattice. Introducing spatial degrees of freedom
and allowing for stochastic fluctuations generically invalidates the classical,
deterministic mean-field picture. Already within mean-field theory, however,
spatial constraints, modeling locally limited resources, lead to the emergence
of a continuous active-to-absorbing state phase transition. Field-theoretic
arguments, supported by Monte Carlo simulation results, indicate that this
transition, which represents an extinction threshold for the predator
population, is governed by the directed percolation universality class. In the
active state, where predators and prey coexist, the classical center
singularities with associated population cycles are replaced by either nodes or
foci. In the vicinity of the stable nodes, the system is characterized by
essentially stationary localized clusters of predators in a sea of prey. Near
the stable foci, however, the stochastic lattice Lotka-Volterra system displays
complex, correlated spatio-temporal patterns of competing activity fronts.
Correspondingly, the population densities in our numerical simulations turn out
to oscillate irregularly in time, with amplitudes that tend to zero in the
thermodynamic limit. Yet in finite systems these oscillatory fluctuations are
quite persistent, and their features are determined by the intrinsic
interaction rates rather than the initial conditions. We emphasize the
robustness of this scenario with respect to various model perturbations.Comment: 19 pages, 11 figures, 2-column revtex4 format. Minor modifications.
Accepted in the Journal of Statistical Physics. Movies corresponding to
Figures 2 and 3 are available at
http://www.phys.vt.edu/~tauber/PredatorPrey/movies
Global attractors and extinction dynamics of cyclically competing species
Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases
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
Inference in Nonlinear Systems with Unscented Kalman Filters
An increasing number of scientific disciplines, most notably the life sciences and
health care, have become more quantitative, describing complex systems with coupled nonlinear
di↵erential equations. While powerful algorithms for numerical simulations from these systems
have been developed, statistical inference of the system parameters is still a challenging problem.
A promising approach is based on the unscented Kalman filter (UKF), which has seen
a variety of recent applications, from soft tissue mechanics to chemical kinetics. The present
study investigates the dependence of the accuracy of parameter estimation on the initialisation.
Based on three toy systems that capture typical features of real-world complex systems: limit
cycles, chaotic attractors and intrinsic stochasticity, we carry out repeated simulations on a large
range of independent data instantiations. Our study allows a quantification of the accuracy of
inference, measured in terms of two alternative distance measures in function and parameter
space, in dependence on the initial deviation from the ground truth
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
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