579 research outputs found
Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions
We study a variant of the cyclic Lotka-Volterra model with three-agent
interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors
game, the model describes an ideal ecosystem in which cyclic competition among
three species develops through cooperative predation. Its rate equations in a
well-mixed environment display a degenerate Hopf bifurcation, occurring as
reactions involving two predators plus one prey have the same rate as reactions
involving two preys plus one predator. We estimate the magnitude of the
stochastic noise at the bifurcation point, where finite size effects turn
neutrally stable orbits into erratically diverging trajectories. In particular,
we compare analytic predictions for the extinction probability, derived in the
Fokker-Planck approximation, with numerical simulations based on the Gillespie
stochastic algorithm. We then extend the analysis of the phase portrait to
heterogeneous rates. In a well-mixed environment, we observe a continuum of
degenerate Hopf bifurcations, generalizing the above one. Neutral stability
ensues from a complex equilibrium between different reactions. Remarkably, on a
two-dimensional lattice, all bifurcations disappear as a consequence of the
spatial locality of the interactions. In the second part of the paper, we
investigate the effects of mobility in a lattice metapopulation model with
patches hosting several agents. We find that strategies propagate along the
arms of rotating spirals, as they usually do in models of cyclic dominance. We
observe propagation instabilities in the regime of large wavelengths. We also
examine three-agent interactions inducing nonlinear diffusion.Comment: 22 pages, 13 figures. v2: version accepted for publication in EPJ
Mesoscopic Interactions and Species Coexistence in Evolutionary Game Dynamics of Cyclic Competitions
Date of Acceptance: 27/11/2014Peer reviewedPublisher PD
Spatiotemporal dynamics in a spatial plankton system
In this paper, we investigate the complex dynamics of a spatial plankton-fish
system with Holling type III functional responses. We have carried out the
analytical study for both one and two dimensional system in details and found
out a condition for diffusive instability of a locally stable equilibrium.
Furthermore, we present a theoretical analysis of processes of pattern
formation that involves organism distribution and their interaction of
spatially distributed population with local diffusion. The results of numerical
simulations reveal that, on increasing the value of the fish predation rates,
the sequences spots spot-stripe mixtures
stripes hole-stripe mixtures holes wave pattern is
observed. Our study shows that the spatially extended model system has not only
more complex dynamic patterns in the space, but also has spiral waves.Comment: Published Pape
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
Noise and Correlations in a Spatial Population Model with Cyclic Competition
Noise and spatial degrees of freedom characterize most ecosystems. Some
aspects of their influence on the coevolution of populations with cyclic
interspecies competition have been demonstrated in recent experiments [e.g. B.
Kerr et al., Nature {\bf 418}, 171 (2002)]. To reach a better theoretical
understanding of these phenomena, we consider a paradigmatic spatial model
where three species exhibit cyclic dominance. Using an individual-based
description, as well as stochastic partial differential and deterministic
reaction-diffusion equations, we account for stochastic fluctuations and
spatial diffusion at different levels, and show how fascinating patterns of
entangled spirals emerge. We rationalize our analysis by computing the
spatio-temporal correlation functions and provide analytical expressions for
the front velocity and the wavelength of the propagating spiral waves.Comment: 4 pages of main text, 3 color figures + 2 pages of supplementary
material (EPAPS Document). Final version for Physical Review Letter
Co-existence in the two-dimensional May-Leonard model with random rates
We employ Monte Carlo simulations to numerically study the temporal evolution
and transient oscillations of the population densities, the associated
frequency power spectra, and the spatial correlation functions in the
(quasi-)steady state in two-dimensional stochastic May--Leonard models of
mobile individuals, allowing for particle exchanges with nearest-neighbors and
hopping onto empty sites. We therefore consider a class of four-state
three-species cyclic predator-prey models whose total particle number is not
conserved. We demonstrate that quenched disorder in either the reaction or in
the mobility rates hardly impacts the dynamical evolution, the emergence and
structure of spiral patterns, or the mean extinction time in this system. We
also show that direct particle pair exchange processes promote the formation of
regular spiral structures. Moreover, upon increasing the rates of mobility, we
observe a remarkable change in the extinction properties in the May--Leonard
system (for small system sizes): (1) As the mobility rate exceeds a threshold
that separates a species coexistence (quasi-)steady state from an absorbing
state, the mean extinction time as function of system size N crosses over from
a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence;
(2) the measured histogram of extinction times displays a corresponding
crossover from an (approximately) exponential to a Gaussian distribution. The
latter results are found to hold true also when the mobility rates are randomly
distributed.Comment: 9 pages, 4 figures; to appear in Eur. Phys. J. B (2011
Stochastic population dynamics in spatially extended predator-prey systems
Spatially extended population dynamics models that incorporate intrinsic
noise serve as case studies for the role of fluctuations and correlations in
biological systems. Including spatial structure and stochastic noise in
predator-prey competition invalidates the deterministic Lotka-Volterra picture
of neutral population cycles. Stochastic models yield long-lived erratic
population oscillations stemming from a resonant amplification mechanism. In
spatially extended predator-prey systems, one observes noise-stabilized
activity and persistent correlations. Fluctuation-induced renormalizations of
the oscillation parameters can be analyzed perturbatively. The critical
dynamics and the non-equilibrium relaxation kinetics at the predator extinction
threshold are characterized by the directed percolation universality class.
Spatial or environmental variability results in more localized patches which
enhances both species densities. Affixing variable rates to individual
particles and allowing for trait inheritance subject to mutations induces fast
evolutionary dynamics for the rate distributions. Stochastic spatial variants
of cyclic competition with rock-paper-scissors interactions illustrate
connections between population dynamics and evolutionary game theory, and
demonstrate how space can help maintain diversity. In two dimensions,
three-species cyclic competition models of the May-Leonard type are
characterized by the emergence of spiral patterns whose properties are
elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to
general food networks can be classified on the mean-field level, which provides
both a fundamental understanding of ensuing cooperativity and emergence of
alliances. Novel space-time patterns emerge as a result of the formation of
competing alliances, such as coarsening domains that each incorporate
rock-paper-scissors competition games
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