Population extinction under bursty reproduction in a time-modulated environment

Abstract

In recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations. For example, nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size, or environmental variability in the form of time-varying reaction rates, have been separately found to dramatically impact the extinction risk of isolated populations. In this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation. Using the so-called momentum-space Wentzel-Kramers-Brillouin (WKB) approach and accounting for the explicit time dependence in the reaction rates, we arrive at a set of time-dependent Hamilton equations. To this end, we evaluate the population's extinction risk by finding the instanton of the time-perturbed Hamiltonian numerically, whereas analytical expressions are presented in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.

Heavy-tailed distributions in a stochastic gene autoregulation model

“…The near-deterministic regime of frequent and small bursts can be analysed using the Wentzel-Kramers-Brillouin (WKB) method; the WKB-approximate solutions closely agree with numerically obtained exact distributions even at moderate noise conditions [33]. Bursty production has been formulated and analysed with the WKB method also in the discrete state space [34][35][36][37][38]. Similar approaches have earlier been used in queueing systems [39,40].…”

Reconstructing an epigenetic landscape using a genetic ‘pulling’ approach

“…The matching conditions at times t = 0 and t = T are provided by the continuity of the functions q(t) and p(t) [73,74]. The pre-and post-perturbation segments must have a zero energy, E = 0, so they are parts of the original zero-energy trajectory, p 0 (q), see Eq.…”

“…To determine the energy E p , we demand that the duration of the perturbation be T [73,74]. Thus, we have:…”

Population Dynamics in a Changing Environment: Random versus Periodic Switching

“…In most studies, there is no interdependence between the fluctuations stemming from DN and EV, with growth rates often assumed to vary independently of the population size [6,7,[11][12][13][17][18][19][20][21][22]24,[27][28][29]31,[36][37][38]41,43]. Hence, there is as yet no systematic comparison of the dynamics under random and periodic switching: some works report that they lead to similar evolutionary processes while others find differences, see, e.g., Refs.…”

“…(S15) in [64]. P Kap ν is obtained from the master equation by using the WKB approximation [84] and the Kapitza method [12,24,85], i.e., separating the dynamics into fast and slow variables, see Section 2.2 of [64]. In Fig.…”