854 research outputs found
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
Range expansion with mutation and selection: dynamical phase transition in a two-species Eden model
The colonization of unoccupied territory by invading species, known as range expansion, is a spatially heterogeneous non-equilibrium growth process. We introduce a two-species Eden growth model to analyze the interplay between uni-directional (irreversible) mutations and selection at the expanding front. While the evolutionary dynamics leads to coalescence of both wild-type and mutant clusters, the non-homogeneous advance of the colony results in a rough front. We show that roughening and domain dynamics are strongly coupled, resulting in qualitatively altered bulk and front properties. For beneficial mutations the front is quickly taken over by mutants and growth proceeds Eden-like. In contrast, if mutants grow slower than wild-types, there is an antagonism between selection pressure against mutants and growth by the merging of mutant domains with an ensuing absorbing state phase transition to an all-mutant front. We find that surface roughening has a marked effect on the critical properties of the absorbing state phase transition. While reference models, which keep the expanding front flat, exhibit directed percolation critical behavior, the exponents of the two-species Eden model strongly deviate from it. In turn, the mutation-selection process induces an increased surface roughness with exponents distinct from that of the classical Eden model
Master equations
The dynamics of a complex physical, biological, or chemical systems can often be modelled in terms of a continuous-time Markov process. The governing equations of these processes are the Fokker-Planck and the master equation. Both equations assume that the future of a system depends only on its current state, memories of its past having been wiped out by randomizing forces. Whereas the Fokker-Planck equation describes a system that evolves continuously from one state to another, the master equation models a system that performs jumps between its states.
In this thesis, we focus on master equations. We first present a comprehensive mathematical framework for the analytical and numerical analysis of master equations in chapter I. Special attention is given to their representation by path integrals. In the subsequent chapters, master equations are applied to the study of physical and biological systems. In chapter II, we study the stochastic and deterministic evolution of zero-sum games and thereby explain a condensation phenomenon expected in driven-dissipative bosonic quantum systems. Afterwards, in chapter III, we develop a coarse-grained model of microbial range expansions and use it to predict which of three strains of Escherichia coli survive such an expansion
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
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
Stochastic population oscillations in spatial predator-prey models
It is well-established that including spatial structure and stochastic noise
in models for predator-prey interactions invalidates the classical
deterministic Lotka-Volterra picture of neutral population cycles. In contrast,
stochastic models yield long-lived, but ultimately decaying erratic population
oscillations, which can be understood through a resonant amplification
mechanism for density fluctuations. In Monte Carlo simulations of spatial
stochastic predator-prey systems, one observes striking complex spatio-temporal
structures. These spreading activity fronts induce persistent correlations
between predators and prey. In the presence of local particle density
restrictions (finite prey carrying capacity), there exists an extinction
threshold for the predator population. The accompanying continuous
non-equilibrium phase transition is governed by the directed-percolation
universality class. We employ field-theoretic methods based on the Doi-Peliti
representation of the master equation for stochastic particle interaction
models to (i) map the ensuing action in the vicinity of the absorbing state
phase transition to Reggeon field theory, and (ii) to quantitatively address
fluctuation-induced renormalizations of the population oscillation frequency,
damping, and diffusion coefficients in the species coexistence phase.Comment: 14 pages, 6 figures, submitted to J. Phys C: Conf. Ser. (2011
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