1,481 research outputs found
Perturbation expansions at large order: Results for scalar field theories revisited
The question of the asymptotic form of the perturbation expansion in scalar
field theories is reconsidered. Renewed interest in the computation of terms in
the epsilon-expansion, used to calculate critical exponents, has been
frustrated by the differing and incompatible results for the high-order
behaviour of the perturbation expansion reported in the literature. We identify
the sources of the errors made in earlier papers, correct them, and obtain a
consistent set of results. We focus on phi^4 theory, since this has been the
most studied and is the most widely used, but we also briefly discuss analogous
results for phi^N theory, with N>4. This reexamination of the structure of
perturbation expansions raises issues concerning the renormalisation of
non-perturbative effects and the nature of the Feynman diagrams at large order,
which we discuss.Comment: 14 page
Modelling Food Webs
We review theoretical approaches to the understanding of food webs. After an
overview of the available food web data, we discuss three different classes of
models. The first class comprise static models, which assign links between
species according to some simple rule. The second class are dynamical models,
which include the population dynamics of several interacting species. We focus
on the question of the stability of such webs. The third class are species
assembly models and evolutionary models, which build webs starting from a few
species by adding new species through a process of "invasion" (assembly models)
or "speciation" (evolutionary models). Evolutionary models are found to be
capable of building large stable webs.Comment: 34 pages, 2 figures. To be published in "Handbook of graphs and
networks" S. Bornholdt and H. G. Schuster (eds) (Wiley-VCH, Berlin
Modes of competition and the fitness of evolved populations
Competition between individuals drives the evolution of whole species.
Although the fittest individuals survive the longest and produce the most
offspring, in some circumstances the resulting species may not be optimally
fit. Here, using theoretical analysis and stochastic simulations of a simple
model ecology, we show how the mode of competition can profoundly affect the
fitness of evolved species. When individuals compete directly with one another,
the adaptive dynamics framework provides accurate predictions for the number
and distribution of species, which occupy positions of maximal fitness. By
contrast, if competition is mediated by the consumption of a common resource
then demographic noise leads to the stabilization of species with near minimal
fitness.Comment: 11 pages, 6 figure
Predator-prey cycles from resonant amplification of demographic stochasticity
In this paper we present the simplest individual level model of predator-prey
dynamics and show, via direct calculation, that it exhibits cycling behavior.
The deterministic analogue of our model, recovered when the number of
individuals is infinitely large, is the Volterra system (with density-dependent
prey reproduction) which is well-known to fail to predict cycles. This
difference in behavior can be traced to a resonant amplification of demographic
fluctuations which disappears only when the number of individuals is strictly
infinite. Our results indicate that additional biological mechanisms, such as
predator satiation, may not be necessary to explain observed predator-prey
cycles in real (finite) populations.Comment: 4 pages, 2 figure
Block Spins for Partial Differential Equations
We investigate the use of renormalisation group methods to solve partial
differential equations (PDEs) numerically. Our approach focuses on
coarse-graining the underlying continuum process as opposed to the conventional
numerical analysis method of sampling it. We calculate exactly the
coarse-grained or `perfect' Laplacian operator and investigate the numerical
effectiveness of the technique on a series of 1+1-dimensional PDEs with varying
levels of smoothness in the dynamics: the diffusion equation, the
time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the
damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is
superior to conventional sampling-based discretisations in representing
faithfully the dynamics with a large grid spacing, introducing no detectable
lattice artifacts as long as there is a natural ultra-violet cut off in the
problem. We discuss limitations and open problems of this approach.Comment: 8 pages, RevTeX, 8 figures, contribution to L.P. Kadanoff festschrift
(J. Stat. Phys
Synchronisation of stochastic oscillators in biochemical systems
A formalism is developed which describes the extent to which stochastic
oscillations in biochemical models are synchronised. It is based on the
calculation of the complex coherence function within the linear noise
approximation. The method is illustrated on a simple example and then applied
to study the synchronisation of chemical concentrations in social amoeba. The
degree to which variation of rate constants in different cells and the volume
of the cells affects synchronisation of the oscillations is explored, and the
phase lag calculated. In all cases the analytical results are shown to be in
good agreement with those obtained through numerical simulations
Analytic solution of Hubbell's model of local community dynamics
Recent theoretical approaches to community structure and dynamics reveal that
many large-scale features of community structure (such as species-rank
distributions and species-area relations) can be explained by a so-called
neutral model. Using this approach, species are taken to be equivalent and
trophic relations are not taken into account explicitly. Here we provide a
general analytic solution to the local community model of Hubbell's neutral
theory of biodiversity by recasting it as an urn model i.e.a Markovian
description of states and their transitions. Both stationary and time-dependent
distributions are analysed. The stationary distribution -- also called the
zero-sum multinomial -- is given in closed form. An approximate form for the
time-dependence is obtained by using an expansion of the master equation. The
temporal evolution of the approximate distribution is shown to be a good
representation for the true temporal evolution for a large range of parameter
values.Comment: 10 pages, 2 figure
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