We present an experiment that systematically probes the basins of attraction of two fixed points of a nonlinear nanomechanical resonator and maps them out with high resolution. We observe a separatrix which progressively alters shape for varying drive strength and changes the relative areas of the two basins of attraction. The observed separatrix is blurred due to ambient fluctuations, including residual noise in the drive system, which cause uncertainty in the preparation of an initial state close to the separatrix. We find a good agreement between the experimentally mapped and theoretically calculated basins of attraction. DOI: 10.1103/PhysRevLett.99.207201 PACS numbers: 85.85.+j, 05.45.ÿa In the last few years the dimensions of mechanical devices have been scaled deep into the submicrometer regime, which resulted in the increased detection sensitivity of extremely small physical quantities [1,2] and in an enhancement of the significance of nonlinear dynamics in such devices [3]. Deliberately operating the system in the nonlinear regime can improve precision of some experimental measurements on nanoscale. For example, a nonlinear resonator can be employed to suppress amplifier noise in an oscillator circuit [4], noise-induced switching between two stable states in a nonlinear beam resonator enables precision measurement of the resonant frequency [5], and the sensitivity of a resonator for mass detection can be improved when the resonator is driven into a region of nonlinear oscillations [6]. Finally, in a Josephson junction, which is dynamically similar to a mechanical resonator in the nonlinear regime, the bistable state of the nonlinear system can be used as a bifurcation amplifier to perform a nondissipative, low-backaction measurement of the phase across the junction [7]. When nanomechanical devices reach the quantum-limited regime [8], a nanomechanical version of such an amplifier could be used for a similar sensitive low-back-action measurement of the state of a quantum mechanical resonator. The nonlinear response of nanomechanical resonators could also be used to detect transition from the classical to quantum regime [9,10]. It is therefore important to understand nonlinear dynamics of these systems well, so that we can fully realize their potential in expanding our experimental capabilities.Although some work has been done with parametric systems [11,12], the majority of nonlinear nanoscale systems that have been studied are directly driven [5,13,14] and the dominant nonlinearity in the restoring force is cubic, also known as Duffing nonlinearity. When a system is driven strongly, the Duffing nonlinearity causes the resonance response curve to become asymmetric. When the resonance is pulled far enough to one side, hysteretic behavior is observed as two stable states appear in the system [15]. The stable states, known as ''attractors'' or ''fixed points'', correspond to the points in state space to which the system converges with time. For each attractor, a set of initial states that dynamically evo...

As the single opportunity for plants to move, seed dispersal has an important impact on plant fitness, species distributions and patterns of biodiversity. However, models that predict dynamics such as risk of extinction, range shifts and biodiversity loss tend to rely on the mean value of parameters and rarely incorporate realistic dispersal mechanisms. By focusing on the mean population value, variation among individuals or variability caused by complex spatial and temporal dynamics is ignored. This calls for increased efforts to understand individual variation in dispersal and integrate it more explicitly into population and community models involving dispersal. However, the sources, magnitude and outcomes of intraspecific variation in dispersal are poorly characterized, limiting our understanding of the role of dispersal in mediating the dynamics of communities and their response to global change. In this manuscript, we synthesize recent research that examines the sources of individual variation in dispersal and emphasize its implications for plant fitness, populations and communities. We argue that this intraspecific variation in seed dispersal does not simply add noise to systems, but, in fact, alters dispersal processes and patterns with consequences for demography, communities, evolution and response to anthropogenic changes. We conclude with recommendations for moving this field of research forward.

We develop a perturbation method for studying quasineutral competition in a broad class of stochastic competition models and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic susceptible-infected-susceptible (SIS) model. Here we extend previous results due to Parsons and Quince [Theor. Popul. Biol. 72, 468 (2007)], Parsons et al. [Theor. Popul. Biol. 74, 302 (2008)], and Lin, Kim, and Doering [J. Stat. Phys. 148, 646 (2012)]. The second model, a two-strain generalization of the stochastic susceptible-infected-recovered (SIR) model with population turnover, has not been studied previously. In each of the two models, when the basic reproduction numbers of the two strains are identical, a system with an infinite population size approaches a point on the deterministic coexistence line (CL): a straight line of fixed points in the phase space of subpopulation sizes. Shot noise drives one of the strain populations to fixation, and the other to extinction, on a time scale proportional to the total population size. Our perturbation method explicitly tracks the dynamics of the probability distribution of the subpopulations in the vicinity of the CL. We argue that, whereas the slow strain has a competitive advantage for mathematically "typical" initial conditions, it is the fast strain that is more likely to win in the important situation when a few infectives of both strains are introduced into a susceptible population.

As a function of packing fraction at zero temperature and applied stress, an amorphous packing of spheres exhibits a jamming transition where the system is sensitive to boundary conditions even in the thermodynamic limit. Upon further compression, the system should become insensitive to boundary conditions provided it is sufficiently large. Here we explore the linear response to a large class of boundary perturbations in 2 and 3 dimensions. We consider each finite packing with periodic-boundary conditions as the basis of an infinite square or cubic lattice and study properties of vibrational modes at arbitrary wave vector. We find that the stability of such modes be understood in terms of a competition between plane waves and the anomalous vibrational modes associated with the jamming transition; infinitesimal boundary perturbations become irrelevant for systems that are larger than a length scale that characterizes the transverse excitations. This previously identified length diverges at the jamming transition.Comment: 8 pages, 5 figure

Although dispersal is generally viewed as a crucial determinant for the fitness of any organism, our understanding of its role in the persistence and spread of plant populations remains incomplete. Generalizing and predicting dispersal processes are challenging due to context dependence of seed dispersal, environmental heterogeneity and interdependent processes occurring over multiple spatial and temporal scales. Current population models often use simple phenomenological descriptions of dispersal processes, limiting their ability to examine the role of population persistence and spread, especially under global change. To move seed dispersal ecology forward, we need to evaluate the impact of any single seed dispersal event within the full spatial and temporal context of a plant’s life history and environmental variability that ultimately influences a population’s ability to persist and spread. In this perspective, we provide guidance on integrating empirical and theoretical approaches that account for the context dependency of seed dispersal to improve our ability to generalize and predict the consequences of dispersal, and its anthropogenic alteration, across systems. We synthesize suitable theoretical frameworks for this work and discuss concepts, approaches and available data from diverse subdisciplines to help operationalize concepts, highlight recent breakthroughs across research areas and discuss ongoing challenges and open questions. We address knowledge gaps in the movement ecology of seeds and the integration of dispersal and demography that could benefit from such a synthesis. With an interdisciplinary perspective, we will be able to better understand how global change will impact seed dispersal processes, and potential cascading effects on plant population persistence, spread and biodiversity.