Unifying Dynamical and Structural Stability of Equilibriums

We exhibit a fundamental relationship between measures of dynamical and structural stability of equilibriums, arising from real dynamical systems. We show that dynamical stability, quantified via systems local response to external perturbations, coincides with the minimal internal perturbation able to destabilize the equilibrium. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a system's local response to white-noise perturbations directly reflects the intensity of internal white noise that it can accommodate before asymptotic mean-square stability of the equilibrium is lost.Comment: 13 pages, 2 figure

Gyrotactic phytoplankton in laminar and turbulent flows: a dynamical systems approach

Gyrotactic algae are bottom heavy, motile cells whose swimming direction is determined by a balance between a buoyancy torque directing them upwards and fluid velocity gradients. Gyrotaxis has, in recent years, become a paradigmatic model for phytoplankton motility in flows. The essential attractiveness of this peculiar form of motility is the availability of a mechanistic description which, despite its simplicity, revealed predictive, rich in phenomenology, easily complemented to include the effects of shape, feed-back on the fluid and stochasticity (e.g. in cell orientation). In this review we consider recent theoretical, numerical and experimental results to discuss how, depending on flow properties, gyrotaxis can produce inhomogeneous phytoplankton distributions on a wide range of scales, from millimeters to kilometers, in both laminar and turbulent flows. In particular, we focus on the phenomenon of gyrotactic trapping in nonlinear shear flows and in fractal clustering in turbulent flows. We shall demonstrate the usefulness of ideas and tools borrowed from dynamical systems theory in explaining and interpreting these phenomena

Input-output analysis of stochastic base flow uncertainty

We adopt an input-output approach to analyze the effect of persistent white-in-time structured stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations. Such base flow variations enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitations. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analysis. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. For small-amplitude base flow perturbations, we bypass the need to solve large generalized Lyapunov equations by adopting a perturbation analysis. We use our framework to study the destabilizing effects of stochastic base flow variations in transitional parallel flows, and the reliability of numerically estimated mean velocity profiles in turbulent channel flows. We uncover the Reynolds number scaling of critically destabilizing perturbation variances and demonstrate how the wall-normal shape of base flow modulations can influence the amplification of various length scales. Furthermore, we explain the robust amplification of streamwise streaks in the presence of streamwise base flow variations by analyzing the dynamical structure of the governing equations as well as the Reynolds number dependence of the energy spectrum.Comment: 29 pages, 21 figure

On using extreme values to detect global stability thresholds in multi-stable systems: the case of transitional plane Couette flow

Extreme Value Theory (EVT) is exploited to determine the global stability threshold Rg of plane Couette flow ‚ the flow of a viscous fluid in the space between two parallel plates ‚ whose laminar or turbulent behavior depends on the Reynolds number R. Even if the existence of a global stability threshold has been detected in simulations and experiments, its numerical value has not been unequivocally defined. Rg is the value such that for R>Rg, turbulence is sustained, whereas for R>Rg, both the positive and negative extremes are bounded. As the critical Reynolds number is approached from above, the probability of observing a very low minimum increases causing asymmetries in the distributions of maxima and minima. On the other hand, the maxima distribution is unaffected as the fluctuations towards higher values of the perturbation energy remain bounded. This tipping point can be detected by fitting the data to the Generalized Extreme Value (GEV) distribution and by identifying Rg as the value of R such that the shape parameter of the GEV for the minima changes sign from negative to positive. The results are supported by the analysis of theoretical models which feature a bistable behavior

Life at the front of an expanding population

Recent microbial experiments suggest that enhanced genetic drift at the frontier of a two-dimensional range expansion can cause genetic sectoring patterns with fractal domain boundaries. Here, we propose and analyze a simple model of asexual biological evolution at expanding frontiers to explain these neutral patterns and predict the effect of natural selection. Our model attributes the observed gradual decrease in the number of sectors at the leading edge to an unbiased random walk of sector boundaries. Natural selection introduces a deterministic bias in the wandering of domain boundaries that renders beneficial mutations more likely to escape genetic drift and become established in a sector. We find that the opening angle of those sectors and the rate at which they become established depend sensitively on the selective advantage of the mutants. Deleterious mutations, on the other hand, are not able to establish a sector permanently. They can, however, temporarily "surf" on the population front, and thereby reach unusual high frequencies. As a consequence, expanding frontiers are susceptible to deleterious mutations as revealed by the high fraction of mutants at mutation-selection balance. Numerically, we also determine the condition at which the wild type is lost in favor of deleterious mutants (genetic meltdown) at a growing front. Our prediction for this error threshold differs qualitatively from existing well-mixed theories, and sets tight constraints on sustainable mutation rates for populations that undergo frequent range expansions.Comment: Updat

Resilience of dynamical systems

Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining how "globally stable" a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline "resilience". Unfortunately, there are many different variants and explanations of resilience, and often the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalized to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain non-autonomous perturbations to demonstrate, how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.Comment: 54 pages, 18 figure

Dynamics of marine zooplankton : social behavior ecological interactions, and physically-induced variability

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2008Marine ecosystems reflect the physical structure of their environment and the biological processes they carry out. This leads to spatial heterogeneity and temporal variability, some of which is imposed externally and some of which emerges from the ecological mechanisms themselves. The main focus of this thesis is on the formation of spatial patterns in the distribution of zooplankton arising from social interactions between individuals. In the Southern Ocean, krill often assemble in swarms and schools, the dynamics of which have important ecological consequences. Mathematical and numerical models are employed to study the interplay of biological and physical processes that contribute to the observed patchiness. The evolution of social behavior is simulated in a theoretical framework that includes zooplankton population dynamics, swimming behavior, and some aspects of the variability inherent to fluid environments. First, I formulate a model of resource utilization by a stage-structured predator population with density-dependent reproduction. Second, I incorporate the predator-prey dynamics into a spatially-explicit model, in which aggregations develop spontaneously as a result of linear instability of the uniform distribution. In this idealized ecosystem, benefits related to the local abundance of mates are offset by the cost of having to share resources with other group members. Third, I derive a weakly nonlinear approximation for the steady-state distributions of predator and prey biomass that captures the spatial patterns driven by social tendencies. Fourth, I simulate the schooling behavior of zooplankton in a variable environment; when turbulent flows generate patchiness in the resource field, schools can forage more efficiently than individuals. Taken together, these chapters demonstrate that aggregation/ schooling can indeed be the favored behavior when (i) reproduction (or other survival measures) increases with density in part of the range and (ii) mixing of prey into patches is rapid enough to offset the depletion. In the final two chapters, I consider sources of temporal variability in marine ecosystems. External perturbations amplified by nonlinear ecological interactions induce transient excursions away from equilibrium; in predator-prey dynamics the amplitude and duration of these transients are controlled by biological processes such as growth and mortality. In the Southern Ocean, large-scale winds associated with ENSO and the Southern Annular Mode cause convective mixing, which in turn drives air-sea fluxes of carbon dioxide and oxygen. Whether driven by stochastic fluctuations or by climatic phenomena, variability of the biogeochemical/physical environment has implications for ecosystem dynamics.Funding was provided by the Academic Programs Office of the MIT-WHOI Joint Program, an Ocean Ventures Fund Award, an Anonymous Ys Endowed Science Fellowship, and by NSF grants OCE-0221369 and OCE-336839