912 research outputs found
Metastability and anomalous fixation in evolutionary games on scale-free networks
We study the influence of complex graphs on the metastability and fixation
properties of a set of evolutionary processes. In the framework of evolutionary
game theory, where the fitness and selection are frequency-dependent and vary
with the population composition, we analyze the dynamics of snowdrift games
(characterized by a metastable coexistence state) on scale-free networks. Using
an effective diffusion theory in the weak selection limit, we demonstrate how
the scale-free structure affects the system's metastable state and leads to
anomalous fixation. In particular, we analytically and numerically show that
the probability and mean time of fixation are characterized by stretched
exponential behaviors with exponents depending on the network's degree
distribution.Comment: 5 pages, 4 figures, to appear in Physical Review Letter
Temporal and dimensional effects in evolutionary graph theory
The spread in time of a mutation through a population is studied analytically
and computationally in fully-connected networks and on spatial lattices. The
time, t_*, for a favourable mutation to dominate scales with population size N
as N^{(D+1)/D} in D-dimensional hypercubic lattices and as N ln N in
fully-connected graphs. It is shown that the surface of the interface between
mutants and non-mutants is crucial in predicting the dynamics of the system.
Network topology has a significant effect on the equilibrium fitness of a
simple population model incorporating multiple mutations and sexual
reproduction. Includes supplementary information.Comment: 6 pages, 4 figures Replaced after final round of peer revie
Preservation of information in a prebiotic package model
The coexistence between different informational molecules has been the
preferred mode to circumvent the limitation posed by imperfect replication on
the amount of information stored by each of these molecules. Here we reexamine
a classic package model in which distinct information carriers or templates are
forced to coexist within vesicles, which in turn can proliferate freely through
binary division. The combined dynamics of vesicles and templates is described
by a multitype branching process which allows us to write equations for the
average number of the different types of vesicles as well as for their
extinction probabilities. The threshold phenomenon associated to the extinction
of the vesicle population is studied quantitatively using finite-size scaling
techniques. We conclude that the resultant coexistence is too frail in the
presence of parasites and so confinement of templates in vesicles without an
explicit mechanism of cooperation does not resolve the information crisis of
prebiotic evolution.Comment: 9 pages, 8 figures, accepted version, to be published in PR
Evolutionary dynamics on degree-heterogeneous graphs
The evolution of two species with different fitness is investigated on
degree-heterogeneous graphs. The population evolves either by one individual
dying and being replaced by the offspring of a random neighbor (voter model
(VM) dynamics) or by an individual giving birth to an offspring that takes over
a random neighbor node (invasion process (IP) dynamics). The fixation
probability for one species to take over a population of N individuals depends
crucially on the dynamics and on the local environment. Starting with a single
fitter mutant at a node of degree k, the fixation probability is proportional
to k for VM dynamics and to 1/k for IP dynamics.Comment: 4 pages, 4 figures, 2 column revtex4 format. Revisions in response to
referee comments for publication in PRL. The version on arxiv.org has one
more figure than the published PR
Nongaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process
The appropriate description of fluctuations within the framework of
evolutionary game theory is a fundamental unsolved problem in the case of
finite populations. The Moran process recently introduced into this context
[Nowak et al., Nature (London) 428, 646 (2004)] defines a promising standard
model of evolutionary game theory in finite populations for which analytical
results are accessible. In this paper, we derive the stationary distribution of
the Moran process population dynamics for arbitrary games for the
finite size case. We show that a nonvanishing background fitness can be
transformed to the vanishing case by rescaling the payoff matrix. In contrast
to the common approach to mimic finite-size fluctuations by Gaussian
distributed noise, the finite size fluctuations can deviate significantly from
a Gaussian distribution.Comment: 4 pages (2 figs). Published in Physical Review E (Rapid
Communications
Stochasticity and evolutionary stability
In stochastic dynamical systems, different concepts of stability can be
obtained in different limits. A particularly interesting example is
evolutionary game theory, which is traditionally based on infinite populations,
where strict Nash equilibria correspond to stable fixed points that are always
evolutionarily stable. However, in finite populations stochastic effects can
drive the system away from strict Nash equilibria, which gives rise to a new
concept for evolutionary stability. The conventional and the new stability
concepts may apparently contradict each other leading to conflicting
predictions in large yet finite populations. We show that the two concepts can
be derived from the frequency dependent Moran process in different limits. Our
results help to determine the appropriate stability concept in large finite
populations. The general validity of our findings is demonstrated showing that
the same results are valid employing vastly different co-evolutionary
processes
Universality of weak selection
Weak selection, which means a phenotype is slightly advantageous over
another, is an important limiting case in evolutionary biology. Recently it has
been introduced into evolutionary game theory. In evolutionary game dynamics,
the probability to be imitated or to reproduce depends on the performance in a
game. The influence of the game on the stochastic dynamics in finite
populations is governed by the intensity of selection. In many models of both
unstructured and structured populations, a key assumption allowing analytical
calculations is weak selection, which means that all individuals perform
approximately equally well. In the weak selection limit many different
microscopic evolutionary models have the same or similar properties. How
universal is weak selection for those microscopic evolutionary processes? We
answer this question by investigating the fixation probability and the average
fixation time not only up to linear, but also up to higher orders in selection
intensity. We find universal higher order expansions, which allow a rescaling
of the selection intensity. With this, we can identify specific models which
violate (linear) weak selection results, such as the one--third rule of
coordination games in finite but large populations.Comment: 12 pages, 3 figures, accepted for publication in Physical Review
The statistical mechanics of a polygenic characterunder stabilizing selection, mutation and drift
By exploiting an analogy between population genetics and statistical
mechanics, we study the evolution of a polygenic trait under stabilizing
selection, mutation, and genetic drift. This requires us to track only four
macroscopic variables, instead of the distribution of all the allele
frequencies that influence the trait. These macroscopic variables are the
expectations of: the trait mean and its square, the genetic variance, and of a
measure of heterozygosity, and are derived from a generating function that is
in turn derived by maximizing an entropy measure. These four macroscopics are
enough to accurately describe the dynamics of the trait mean and of its genetic
variance (and in principle of any other quantity). Unlike previous approaches
that were based on an infinite series of moments or cumulants, which had to be
truncated arbitrarily, our calculations provide a well-defined approximation
procedure. We apply the framework to abrupt and gradual changes in the optimum,
as well as to changes in the strength of stabilizing selection. Our
approximations are surprisingly accurate, even for systems with as few as 5
loci. We find that when the effects of drift are included, the expected genetic
variance is hardly altered by directional selection, even though it fluctuates
in any particular instance. We also find hysteresis, showing that even after
averaging over the microscopic variables, the macroscopic trajectories retain a
memory of the underlying genetic states.Comment: 35 pages, 8 figure
Shift in critical temperature for random spatial permutations with cycle weights
We examine a phase transition in a model of random spatial permutations which
originates in a study of the interacting Bose gas. Permutations are weighted
according to point positions; the low-temperature onset of the appearance of
arbitrarily long cycles is connected to the phase transition of Bose-Einstein
condensates. In our simplified model, point positions are held fixed on the
fully occupied cubic lattice and interactions are expressed as Ewens-type
weights on cycle lengths of permutations. The critical temperature of the
transition to long cycles depends on an interaction-strength parameter
. For weak interactions, the shift in critical temperature is expected
to be linear in with constant of linearity . Using Markov chain
Monte Carlo methods and finite-size scaling, we find .
This finding matches a similar analytical result of Ueltschi and Betz. We also
examine the mean longest cycle length as a fraction of the number of sites in
long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial
permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures
which appeared at the end of the PDF
Evolutionary game theory in growing populations
Existing theoretical models of evolution focus on the relative fitness
advantages of different mutants in a population while the dynamic behavior of
the population size is mostly left unconsidered. We here present a generic
stochastic model which combines the growth dynamics of the population and its
internal evolution. Our model thereby accounts for the fact that both
evolutionary and growth dynamics are based on individual reproduction events
and hence are highly coupled and stochastic in nature. We exemplify our
approach by studying the dilemma of cooperation in growing populations and show
that genuinely stochastic events can ease the dilemma by leading to a transient
but robust increase in cooperationComment: 4 pages, 2 figures and 2 pages supplementary informatio
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