Finite-Time Lyapunov Exponents of Deep Neural Networks.

We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualize the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.

Inertia Induces Strong Orientation Fluctuations of Nonspherical Atmospheric Particles.

The orientation of nonspherical particles in the atmosphere, such as volcanic ash and ice crystals, influences their residence times and the radiative properties of the atmosphere. Here, we demonstrate experimentally that the orientation of heavy submillimeter spheroids settling in still air exhibits decaying oscillations, whereas it relaxes monotonically in liquids. Theoretical analysis shows that these oscillations are due to particle inertia, caused by the large particle-fluid mass-density ratio. This effect must be accounted for to model solid particles in the atmosphere.

Lagrangian Supersaturation Fluctuations at the Cloud Edge.

Evaporation of cloud droplets accelerates when turbulence mixes dry air into the cloud, affecting droplet-size distributions in atmospheric clouds, combustion sprays, and jets of exhaled droplets. The challenge is to model local correlations between droplet numbers, sizes, and supersaturation, which determine supersaturation fluctuations along droplet paths (Lagrangian fluctuations). We derived a statistical model that accounts for these correlations. Its predictions are in quantitative agreement with results of direct numerical simulations, and explain the key mechanisms at play.

Alignment of Nonspherical Active Particles in Chaotic Flows.

We study the orientation statistics of spheroidal, axisymmetric microswimmers, with shapes ranging from disks to rods, swimming in chaotic, moderately turbulent flows. Numerical simulations show that rodlike active particles preferentially align with the flow velocity. To explain the underlying mechanism, we solve a statistical model via the perturbation theory. We show that such an alignment is caused by correlations of fluid velocity and its gradients along particle paths combined with fore-aft symmetry breaking due to both swimming and particle nonsphericity. Remarkably, the discovered alignment is found to be a robust kinematical effect, independent of the underlying flow evolution. We discuss its possible relevance for aquatic ecology.

Statistical Model for the Orientation of Nonspherical Particles Settling in Turbulence.

The orientation of small anisotropic particles settling in a turbulent fluid determines some essential properties of the suspension. We show that the orientation distribution of small heavy spheroids settling through turbulence can be accurately predicted by a simple Gaussian statistical model that takes into account particle inertia and provides a quantitative understanding of the orientation distribution on the problem parameters when fluid inertia is negligible. Our results open the way to a parametrization of the distribution of ice crystals in clouds, and potentially lead to an improved understanding of radiation reflection or particle aggregation through collisions in clouds.

One-Parameter Scaling Theory for DNA Extension in a Nanochannel.

Experiments measuring DNA extension in nanochannels are at odds with even the most basic predictions of current scaling arguments for the conformations of confined semiflexible polymers such as DNA. We show that a theory based on a weakly self-avoiding, one-dimensional "telegraph" process collapses experimental data and simulation results onto a single master curve throughout the experimentally relevant region of parameter space and explains the mechanisms at play.

Interpreting the genomic landscape of speciation: a road map for finding barriers to gene flow.

Speciation, the evolution of reproductive isolation among populations, is continuous, complex, and involves multiple, interacting barriers. Until it is complete, the effects of this process vary along the genome and can lead to a heterogeneous genomic landscape with peaks and troughs of differentiation and divergence. When gene flow occurs during speciation, barriers restricting gene flow locally in the genome lead to patterns of heterogeneity. However, genomic heterogeneity can also be produced or modified by variation in factors such as background selection and selective sweeps, recombination and mutation rate variation, and heterogeneous gene density. Extracting the effects of gene flow, divergent selection and reproductive isolation from such modifying factors presents a major challenge to speciation genomics. We argue one of the principal aims of the field is to identify the barrier loci involved in limiting gene flow. We first summarize the expected signatures of selection at barrier loci, at the genomic regions linked to them and across the entire genome. We then discuss the modifying factors that complicate the interpretation of the observed genomic landscape. Finally, we end with a road map for future speciation research: a proposal for how to account for these modifying factors and to progress towards understanding the nature of barrier loci. Despite the difficulties of interpreting empirical data, we argue that the availability of promising technical and analytical methods will shed further light on the important roles that gene flow and divergent selection have in shaping the genomic landscape of speciation.

Neutral processes forming large clones during colonization of new areas.

In species reproducing both sexually and asexually clones are often more common in recently established populations. Earlier studies have suggested that this pattern arises due to natural selection favouring generally or locally successful genotypes in new environments. Alternatively, as we show here, this pattern may result from neutral processes during species' range expansions. We model a dioecious species expanding into a new area in which all individuals are capable of both sexual and asexual reproduction, and all individuals have equal survival rates and dispersal distances. Even under conditions that favour sexual recruitment in the long run, colonization starts with an asexual wave. After colonization is completed, a sexual wave erodes clonal dominance. If individuals reproduce more than one season, and with only local dispersal, a few large clones typically dominate for thousands of reproductive seasons. Adding occasional long-distance dispersal, more dominant clones emerge, but they persist for a shorter period of time. The general mechanism involved is simple: edge effects at the expansion front favour asexual (uniparental) recruitment where potential mates are rare. Specifically, our model shows that neutral processes (with respect to genotype fitness) during the population expansion, such as random dispersal and demographic stochasticity, produce genotype patterns that differ from the patterns arising in a selection model. The comparison with empirical data from a post-glacially established seaweed species (Fucus radicans) shows that in this case, a neutral mechanism is strongly supported.

Angular Dynamics of a Small Particle in Turbulence.

We compute the angular dynamics of a neutrally buoyant nearly spherical particle immersed in an unsteady fluid. We assume that the particle is small, that its translational slip velocity is negligible, and that unsteady and convective inertia are small perturbations. We derive an approximation for the torque on the particle that determines the first inertial corrections to Jeffery's equation. These corrections arise as a consequence of local vortex stretching and can be substantial in turbulence, where local vortex stretching is strong and closely linked to the irreversibility of turbulence.

Preferential Sampling and Small-Scale Clustering of Gyrotactic Microswimmers in Turbulence.

Recent studies show that spherical motile microorganisms in turbulence subject to gravitational torques gather in down-welling regions of the turbulent flow. By analyzing a statistical model we analytically compute how shape affects the dynamics, preferential sampling, and small-scale spatial clustering. We find that oblong organisms may spend more time in up-welling regions of the flow, and that all organisms are biased to regions of positive fluid-velocity gradients in the upward direction. We analyze small-scale spatial clustering and find that oblong particles may either cluster more or less than spherical ones, depending on the strength of the gravitational torques.

Statistical model for collisions and recollisions of inertial particles in mixing flows.

Finding a quantitative description of the rate of collisions between small particles suspended in mixing flows is a long-standing problem. Here we investigate the validity of a parameterisation of the collision rate for identical particles subject to Stokes force, based on results for relative velocities of heavy particles that were recently obtained within a statistical model for the dynamics of turbulent aerosols. This model represents the turbulent velocity fluctuations by Gaussian random functions. We find that the parameterisation gives quantitatively good results in the limit where the "ghost-particle approximation" applies. The collision rate is a sum of two contributions due to "caustics" and to "clustering". Within the statistical model we compare the relative importance of these two collision mechanisms. The caustic formation rate is high when the particle inertia becomes large, and we find that caustics dominate the collision rate as soon as they form frequently. We compare the magnitude of the caustic contribution to the collision rate to the formation rate of caustics.

Tumbling of small axisymmetric particles in random and turbulent flows.

We analyze the tumbling of small nonspherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients, which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, Phys. Rev. Lett. 109, 134501 (2012)]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.

Demography-adjusted tests of neutrality based on genome-wide SNP data.

Tests of the neutral evolution hypothesis are usually built on the standard null model which assumes that mutations are neutral and the population size remains constant over time. However, it is unclear how such tests are affected if the last assumption is dropped. Here, we extend the unifying framework for tests based on the site frequency spectrum, introduced by Achaz and Ferretti, to populations of varying size. Key ingredients are the first two moments of the site frequency spectrum. We show how these moments can be computed analytically if a population has experienced two instantaneous size changes in the past. We apply our method to data from ten human populations gathered in the 1000 genomes project, estimate their demographies and define demography-adjusted versions of Tajima's D, Fay & Wu's H, and Zeng's E. Our results show that demography-adjusted test statistics facilitate the direct comparison between populations and that most of the differences among populations seen in the original unadjusted tests can be explained by their underlying demographies. Upon carrying out whole-genome screens for deviations from neutrality, we identify candidate regions of recent positive selection. We provide track files with values of the adjusted and unadjusted tests for upload to the UCSC genome browser.

Evolutionary branching in a stochastic population model with discrete mutational steps.

Evolutionary branching is analysed in a stochastic, individual-based population model under mutation and selection. In such models, the common assumption is that individual reproduction and life career are characterised by values of a trait, and also by population sizes, and that mutations lead to small changes ϵ in trait value. Then, traditionally, the evolutionary dynamics is studied in the limit ϵ→0. In the present approach, small but non-negligible mutational steps are considered. By means of theoretical analysis in the limit of infinitely large populations, as well as computer simulations, we demonstrate how discrete mutational steps affect the patterns of evolutionary branching. We also argue that the average time to the first branching depends in a sensitive way on both mutational step size and population size.

Metapopulation dynamics on the brink of extinction.

We analyse metapopulation dynamics in terms of an individual-based, stochastic model of a finite metapopulation. We suggest a new approach, using the number of patches in the population as a large parameter. This approach does not require that the number of individuals per patch is large, neither is it necessary to assume a time-scale separation between local population dynamics and migration. Our approach makes it possible to accurately describe the dynamics of metapopulations consisting of many small patches. We focus on metapopulations on the brink of extinction. We estimate the time to extinction and describe the most likely path to extinction. We find that the logarithm of the time to extinction is proportional to the product of two vectors, a vector characterising the distribution of patch population sizes in the quasi-steady state, and a vector-related to Fisher's reproduction vector-that quantifies the sensitivity of the quasi-steady state distribution to demographic fluctuations. We compare our analytical results to stochastic simulations of the model, and discuss the range of validity of the analytical expressions. By identifying fast and slow degrees of freedom in the metapopulation dynamics, we show that the dynamics of large metapopulations close to extinction is approximately described by a deterministic equation originally proposed by Levins (1969). We were able to compute the rates in Levins' equation in terms of the parameters of our stochastic, individual-based model. It turns out, however, that the interpretation of the dynamical variable depends strongly on the intrinsic growth rate and carrying capacity of the patches. Only when the local growth rate and the carrying capacity are large does the slow variable correspond to the number of patches, as envisaged by Levins. Last but not least, we discuss how our findings relate to other, widely used metapopulation models.

Climate change tendencies observable in the rainfall measurements since 1950 in the federal land of North Rhine-Westphalia and their consequences for urban hydrology.

Climate change is present in climatological models - but did we already observe changes in the past measurement data? For the state of North Rhine Westphalia, the rainfall measurements since 1950 have been systematically analysed in order to find out whether there have already been trends and whether the behaviour of rainfall has changed in time. More than 600 station series have been screened for use in the project and quality controlled. Implausible data were discarded. For the analysis, standard values such as yearly sums, half-yearly sums, monthly sums, number of dry days, number of days with precipitation above a threshold, partial time series and extreme values statistics have been calculated and evaluated. Results show that also in the past 50 years, changes in precipitation regime could be observed. These changes have been regionally different. Consequences for urban hydrology include a development of more flexible design approaches.

Sequential Markov coalescent algorithms for population models with demographic structure.

We analyse sequential Markov coalescent algorithms for populations with demographic structure: for a bottleneck model, a population-divergence model, and for a two-island model with migration. The sequential Markov coalescent method is an approximation to the coalescent suggested by McVean and Cardin, and by Marjoram and Wall. Within this algorithm we compute, for two individuals randomly sampled from the population, the correlation between times to the most recent common ancestor and the linkage probability corresponding to two different loci with recombination rate R between them. These quantities characterise the linkage between the two loci in question. We find that the sequential Markov coalescent method approximates the coalescent well in general in models with demographic structure. An exception is the case where individuals are sampled from populations separated by reduced gene flow. In this situation, the correlations may be significantly underestimated. We explain why this is the case.

Multiple regimes of diffusion.

We consider the diffusion of independent particles experiencing random accelerations by a space- and time-dependent force as well as viscous damping. This model can exhibit several asymptotic behaviors, depending upon the limiting cases which are considered, some of which have been discussed in earlier work. Here, we explore the full space of dimensionless parameters and present an "asymptotic phase diagram" which delineates the limiting regimes.

Heavy particles in a persistent random flow with traps.

We study a one-dimensional model for heavy particles in a compressible fluid. The fluid-velocity field is modeled by a persistent Gaussian random function, and the particles are assumed to be weakly inertial. Since one-dimensional fluid-velocity fields are always compressible, the model exhibits spatial trapping regions where particles tend to accumulate. We determine the statistics of fluid-velocity gradients in the vicinity of these traps and show how this allows one to determine the spatial Lyapunov exponent and the rate of caustic formation. We compare our analytical results with numerical simulations of the model and explore the limits of validity of the theory. Finally, we discuss implications for higher-dimensional systems.

Distribution of label spacings for genome mapping in nanochannels.

In genome mapping experiments, long DNA molecules are stretched by confining them to very narrow channels, so that the locations of sequence-specific fluorescent labels along the channel axis provide large-scale genomic information. It is difficult, however, to make the channels narrow enough so that the DNA molecule is fully stretched. In practice, its conformations may form hairpins that change the spacings between internal segments of the DNA molecule, and thus the label locations along the channel axis. Here, we describe a theory for the distribution of label spacings that explains the heavy tails observed in distributions of label spacings in genome mapping experiments.