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Register a new userExtinctions of coupled populations, and rare-event dynamics under non-Gaussian noise
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The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time. It provides a detailed description of the statistics of histories leading to the rare event. The framework applies to rare events in a broad class of systems driven by non-Gaussian noise. Secondly, applying this framework to the population-dynamics model, we find a novel phase transition in its extinction behavior. Strikingly, a patch which is a sink (where individuals die more than are born), can nonetheless reduce the probability of extinction, even if it normally lowers the populations size and growth rate.
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Recently, a first step was made by the authors towards a systematic investigation of the effect of reaction-step-size noise - uncertainty in the step size of the reaction - on the dynamics of stochastic populations. This was done by investigating the effect of bursty influx on the switching dynamics of stochastic populations. Here we extend this formalism to account for bursty reproduction processes, and improve the accuracy of the formalism to include subleading-order corrections. Bursty reproduction appears in various contexts, where notable examples include bursty viral production from infected cells, and reproduction of mammals involving varying number of offspring. The main question we quantitatively address is how bursty reproduction affects the overall fate of the population. We consider two complementary scenarios: population extinction and population survival; in the former a population gets extinct after maintaining a long-lived metastable state, whereas in the latter a population proliferates despite undergoing a deterministic drift towards extinction. In both models reproduction occurs in bursts, sampled from an arbitrary distribution. In the extinction problem, we show that bursty reproduction broadens the quasi-stationary distribution of population sizes in the metastable state, which results in an exponential decrease of the mean time to extinction. In the survival problem, bursty reproduction yields an exponential increase in survival probability of the population. Close to the bifurcation limit our analytical results simplify considerably and are shown to depend solely on the mean and variance of the burst-size distribution. Our formalism is demonstrated on several realistic distributions which all compare well with numerical Monte-Carlo simulations.
We introduce a software generator for a class of emph{colored} (self-correlated) and emph{non-Gaussian} noise, whose statistics and spectrum depend upon only two parameters, $q$ and $tau$. Inspired by Tsallis nonextensive formulation of statistical physics, this so-called $q$-distribution is a handy source of self-correlated noise for a large variety of applications. The $q$-noise---which tends smoothly for $q=1$ to Ornstein--Uhlenbeck noise with autocorrelation $tau$---is generated via a stochastic differential equation, using the Heun method (a second order Runge--Kutta type integration scheme). The algorithm is implemented as a stand-alone library in texttt{c++}, available as open source in the texttt{Github} repository. The noises statistics can be chosen at will, by varying only parameter $q$: it has compact support for $q<1$ (sub-Gaussian regime) and finite variance up to $q=5/3$ (supra-Gaussian regime). Once $q$ has been fixed, the noises autocorrelation can be tuned up independently by means of parameter $tau$. This software provides a tool for modeling a large variety of real-world noise types, and is suitable to study the effects of correlation and deviations from the normal distribution in systems of stochastic differential equations which may be relevant for a wide variety of technological applications, as well as for the understanding of situations of biological interest. Applications illustrating how the noise statistics affects the response of a variety of nonlinear systems are briefly discussed. In many of these examples, the systems response turns out to be optimal for some $q eq1$.
We study numerically the behavior of RNA secondary structures under influence of a varying external force. This allows to measure the work $W$ during the resulting fast unfolding and refolding processes. Here, we investigate a medium-size hairpin structure. Using a sophisticated large-deviation algorithm, we are able to measure work distributions with high precision down to probabilities as small as $10^{-46}$. Due to this precision and by comparison with exact free-energy calculations we are able to verify the theorems of Crooks and Jarzynski. Furthermore, we analyze force-extension curves and the configurations of the secondary structures during unfolding and refolding for typical equilibrium processes and non-equilibrium processes, conditioned to selected values of the measured work $W$, typical and rare ones. We find that the non-equilibrium processes where the work values are close to those which are most relevant for applying Crooks and Jarzynski theorems, respectively, are most and quite similar to the equilibrium processes. Thus, a similarity of equilibrium and non-equilibrium behavior with respect to a mere scalar variable, which occurs with a very small probability but can be generated in a controlled but non-targeted way, is related to a high similarity for the set of configurations sampled along the full dynamical trajectory.
We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can be observed in the model of a particles motion in a random landscape of diffusion coefficients slowly varying in space. Our conclusion is that such behavior is extremely unlikely in the situations when the particles, introduced into the system at random at $t=0$, are observed from the preparation of the system on. However, it indeed may arise in the case when the diffusion (as described in Ito interpretation) is observed under equilibrated conditions. This paradigmatic situation can be translated into the model of the diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind of the diffusing diffusivity model.
For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction, and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.
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