Abstract:
We study the evolution of quantum fluctuations in systems known as time crystals, hypothetical systems for which the lowest energy state performs a periodic motion. We first discuss some general properties shared by time crystals, and deduce the effective field theory parametrizing the evolution of their fluctuations. We show that these fluctuations fall into categories analogous to acoustic and optical phonons, encountered in conventional crystals. The acoustic phonons correspond to gapless Goldstone boson modes parametrizing the broken time translation invariance of the crystal, whereas the optical phonons are identified with modes perpendicular to the broken symmetry of the system, which generically remain gapped. We study how these two modes decay and interact together, and discuss some observable features that could be tested in experimental realizations of time crystals.

Abstract:
We consider here the dynamics of two polychaete populations based on a 20 yr temporal benthic survey of two muddy fine sand communities in the Bay of Morlaix, Western English Channel. These populations display high temporal variability, which is analyzed here using scaling approaches. We find that population densities have heavy tailed probability density functions. We analyze the dynamics of relative species abundance in two different communities of polychaetes by estimating in a novel way a "mean square drift" coefficient which characterizes their fluctuations in relative abundance over time. We show the usefulness of using new tools to approach and model such highly variable population dynamics in marine ecosystems.

Abstract:
Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.

Abstract:
Recent work in modeling the coupling between disease dynamics and dynamic social network geometry has led to the examination of how human interactions force a rewiring of connections in a population. Rewiring of the network may be considered an adaptive response to social forces due to disease spread, which in turn feeds back to the disease dynamics. Such epidemic models, called adaptive networks, have led to new dynamical instabilities along with the creation of multiple attracting states. The co-existence of several attractors is sensitive to internal and external fluctuations, and leads to enhanced stochastic oscillatory outbreaks and disease extinction. The aim of this paper is to explore the bifurcations of adaptive network models in the presence of fluctuations and to review some of the new fluctuation phenomena induced in adaptive networks.

Abstract:
The average economic agent is often used to model the dynamics of simple markets, based on the assumption that the dynamics of many agents can be averaged over in time and space. A popular idea that is based on this seemingly intuitive notion is to dampen electric power fluctuations from fluctuating sources (as e.g. wind or solar) via a market mechanism, namely by variable power prices that adapt demand to supply. The standard model of an average economic agent predicts that fluctuations are reduced by such an adaptive pricing mechanism. However, the underlying assumption that the actions of all agents average out on the time axis is not always true in a market of many agents. We numerically study an econophysics agent model of an adaptive power market that does not assume averaging a priori. We find that when agents are exposed to source noise via correlated price fluctuations (as adaptive pricing schemes suggest), the market may amplify those fluctuations. In particular, small price changes may translate to large load fluctuations through catastrophic consumer synchronization. As a result, an adaptive power market may cause the opposite effect than intended: Power fluctuations are not dampened but amplified instead.

Abstract:
For plane-wave and many-spiral states of the experimentally based Luo-Rudy 1 model of heart tissue in large (8 cm square) domains, we show that an explicit space-time-adaptive time-integration algorithm can achieve an order of magnitude reduction in computational effort and memory - but without a reduction in accuracy - when compared to an algorithm using a uniform space-time mesh at the finest resolution. Our results indicate that such an explicit algorithm can be extended straightforwardly to simulate quantitatively large-scale three-dimensional electrical dynamics over the whole human heart.

Abstract:
Two mathematical models of macroevolution are studied. These models have population dynamics at the species level, and mutations and extinction of species are also included. The population dynamics are updated by difference equations with stochastic noise terms that characterize population fluctuations. The effects of the stochastic population fluctuations on diversity and total population sizes on evolutionary time scales are studied. In one model, species can make either predator-prey, mutualistic, or competitive interactions, while the other model allows only predator-prey interactions. When the noise in the population dynamics is strong enough, both models show intermittent behavior and their power spectral densities show approximate $1/f$ fluctuations. In the noiseless limit, the two models have different power spectral densities. For the predator-prey model, $1/f^2$ fluctuations appears, indicating random-walk like behavior, while the other model still shows $1/f$ noise. These results indicate that stochastic population fluctuations may significantly affect long-time evolutionary dynamics.

Abstract:
We introduce an extension of the usual replicator dynamics to adaptive learning rates. We show that a population with a dynamic learning rate can gain an increased average payoff in transient phases and can also exploit external noise, leading the system away from the Nash equilibrium, in a reasonance-like fashion. The payoff versus noise curve resembles the signal to noise ratio curve in stochastic resonance. Seen in this broad context, we introduce another mechanism that exploits fluctuations in order to improve properties of the system. Such a mechanism could be of particular interest in economic systems.

Abstract:
The role of the selection pressure and mutation amplitude on the behavior of a single-species population evolving on a two-dimensional lattice, in a periodically changing environment, is studied both analytically and numerically. The mean-field level of description allows to highlight the delicate interplay between the different time-scale processes in the resulting complex dynamics of the system. We clarify the influence of the amplitude and period of the environmental changes on the critical value of the selection pressure corresponding to a phase-transition "extinct-alive" of the population. However, the intrinsic stochasticity and the dynamically-built in correlations among the individuals, as well as the role of the mutation-induced variety in population's evolution are not appropriately accounted for. A more refined level of description, which is an individual-based one, has to be considered. The inherent fluctuations do not destroy the phase transition "extinct-alive", and the mutation amplitude is strongly influencing the value of the critical selection pressure. The phase diagram in the plane of the population's parameters -- selection and mutation is discussed as a function of the environmental variation characteristics. The differences between a smooth variation of the environment and an abrupt, catastrophic change are also addressesd.

Abstract:
We show that the scale dependence of the fluctuations of the natural time itself under time reversal provides a useful tool for the discrimination of seismic electric signals (critical dynamics) from noises emitted from man made sources as well as for the determination of the scaling exponent. We present recent data of electric signals detected at the Earth's surface, which confirm that the value of the entropy in natural time as well as its value under time reversal are smaller than that of the entropy of a "uniform" distribution.