We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n = 0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n = 0. In scenario B there is an intermediate repelling point n = n1 between the attracting point n = 0 and another attracting point n = n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasi-stationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasi-stationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multi-step processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.
Understanding animal movement is essential to elucidate how animals interact, survive, and thrive in a changing world. Recent technological advances in data collection and management have transformed our understanding of animal “movement ecology” (the integrated study of organismal movement), creating a big-data discipline that benefits from rapid, cost-effective generation of large amounts of data on movements of animals in the wild. These high-throughput wildlife tracking systems now allow more thorough investigation of variation among individuals and species across space and time, the nature of biological interactions, and behavioral responses to the environment. Movement ecology is rapidly expanding scientific frontiers through large interdisciplinary and collaborative frameworks, providing improved opportunities for conservation and insights into the movements of wild animals, and their causes and consequences.
Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics -such as those determining population extinction, fixation or switching between different statesare presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.
Cells use genetic switches to shift between alternate gene expression states, e.g., to adapt to new environments or to follow a developmental pathway. Here, we study the dynamics of switching in a generic-feedback on/off switch. Unlike protein-only models, we explicitly account for stochastic fluctuations of mRNA, which have a dramatic impact on switch dynamics. Employing the WKB theory to treat the underlying chemical master equations, we obtain accurate results for the quasi-stationary distributions of mRNA and protein copy numbers and for the mean switching time, starting from either state. Our analytical results agree well with Monte Carlo simulations. Importantly, one can use the approach to study the effect of varying biological parameters on switch stability.
We suggest a general spectral method for calculating the statistics of multistep birth-death processes and chemical reactions of the type mA-->nA (m and n are positive integers) which possess an absorbing state. The method employs the generating function formalism in conjunction with the Sturm-Liouville theory of linear differential operators. It yields accurate results for the extinction statistics and for the quasistationary probability distribution, including large deviations, of the metastable state. The power of the method is demonstrated on the example of binary annihilation and triple branching 2A--> ø, A-->3A, representative of the rather general class of dissociation-recombination reactions.
Analysis of complex gene regulation networks gives rise to a landscape of metastable phenotypic states for cells. Heterogeneity within a population arises due to infrequent noise-driven transitions of individual cells between nearby metastable states. While most previous works have focused on the role of intrinsic fluctuations in driving such transitions, in this Letter we investigate the role of extrinsic fluctuations. First, we develop an analytical framework to study the combined effect of intrinsic and extrinsic noise on a toy model of a Boolean regulated genetic switch. We then extend these ideas to a more biologically relevant model with a Hill-like regulatory function. Employing our theory and Monte Carlo simulations, we show that extrinsic noise can significantly alter the lifetimes of the phenotypic states and may fundamentally change the escape mechanism. Finally, our theory can be readily generalized to more complex decision making networks in biology.
The extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment. We investigate this effect using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate. The population extinction is treated in eikonal approximation, where it is described as an instanton trajectory of a proper reaction Hamiltonian. The modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase. We calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations. The techniques include a first-order theory with respect to the modulation amplitude, a second-order theory in the spirit of the Kapitsa pendulum effect, and adiabatic theory valid for low modulation frequencies.
We apply the spectral method, recently developed by the authors, to calculate the statistics of a reaction-limited multistep birth-death process, or chemical reaction, that includes as elementary steps branching A-->2A and annihilation 2A-->0 . The spectral method employs the generating function technique in conjunction with the Sturm-Liouville theory of linear differential operators. We focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution (including large deviations) of the metastable long-lived state and for the extinction time statistics. The analytical results are in very good agreement with numerical calculations. Furthermore, we use this example to settle the issue of the "lacking" boundary condition in the spectral formulation.
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